title
stringlengths 1
131
| url
stringlengths 33
167
| content
stringlengths 0
637k
|
|---|---|---|
A first idea of quantum field theory -- Scattering | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Scattering |
see _[[A first idea of quantum field theory -- Interacting quantum fields]]_ |
A first idea of quantum field theory -- Spacetime | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Spacetime |
## Spacetime
{#Spacetime}
[[relativistic field theory|Relativistic field theory]] takes place on _[[spacetime]]_.
The concept of [[spacetime]] makes sense for every [[dimension]] $p+1$ with $p \in \mathbb{N}$.
The [[observable universe]] has macroscopic dimension $3+1$, but [[quantum field theory]] generally makes sense
also in lower and in higher dimensions. For instance quantum field theory in dimension 0+1 is the "[[worldline]]"
theory of [[particles]], also known as _[[quantum mechanics]]_; while quantum field theory in dimension $\gt p+1$
may be "[[Kaluza-Klein compactifications|KK-compactified]]" to an "[[effective field theory|effective]]"
field theory in dimension $p+1$ which generally looks more complicated than its higher dimensional incarnation.
However, every realistic field theory, and also most of the non-realistic field theories of interest, contain
[[spinor fields]] such as the [[Dirac field]] (example \ref{LagrangianDensityForDiracField} below) and
the precise nature and behaviour of [[spinors]] does depend sensitively on spacetime dimension.
In fact the theory of relativistic spinors is mathematically most natural in just the following four
spacetime dimensions:
$$
p +1 = \phantom{AAAAA} \array{ 2+1,\; & 3+1,\; & \, & 5+1,\; &\, & \, & \, & \, 9+1 }
$$
In the literature one finds these four dimensions advertized for two superficially unrelated reasons:
1. in precisely these dimensions "[[twistors]]" exist (see [there](twistor+space#TwistorSpace));
1. in precisely these dimensions "[[Green-Schwarz superstrings|GS-superstrings]]" exist (see [there](geometry+of+physics+--+fundamental+super+p-branes#TheSuperStringAndTheSuperMembrane)).
However, both these explanations have a common origin in something simpler and deeper:
Spacetime in these dimensions appears from the "[[Pauli matrices]]" with entries in the real [[normed division algebras]].
(In fact it goes [deeper still](geometry+of+physics+--+supersymmetry#SupersymmetryFromTheSuperpoint), but this will not concern us here.)
This we explain now, and then we use this to obtain a slick handle on [[spinors]] in these dimensions,
using simple [[linear algebra]] over the four [[real number|real]] [[normed division algebras]].
At the end (in remark \ref{TwoComponentSpinorNotation}) we give a dictionary that expresses these constructions in terms
of the "two-component spinor notation" that is traditionally used in physics texts (remark \ref{TwoComponentSpinorNotation} below).
The relation between _[[real spin representations and division algebras]]_,
is originally due to [Kugo-Townsend 82](geometry+of+physics+--+supersymmetry#KugoTownsend82), [Sudbery 84](geometry+of+physics+--+supersymmetry#Sudbery84) and others. We follow the streamlined discussion in [Baez-Huerta 09](geometry+of+physics+--+supersymmetry#BaezHuerta09) and [Baez-Huerta 10](geometry+of+physics+--+supersymmetry#BaezHuerta10).
A key extra structure that the [[spinors]] impose on the underlying [[Cartesian space]] of [[spacetime]]
is its _[[causal structure]]_, which determines which points in [[spacetime]] ("[[events]]") are in
the [[future]] or the [[past]] of other points (def. \ref{SpacelikeTimelikeLightlike} below). This [[causal structure]] will turn out to tightly control the [[quantum field theory]] on [[spacetime]] in terms of the "[[causal additivity]] of the [[S-matrix]]" (prop. \ref{TimeOrderedProductInducesPerturbativeSMatrix} below) and the induced "[[causal locality]]" of the [[algebra of quantum observables]] (prop. \ref{PerturbativeQuantumObservablesIsLocalnet} below).
To prepare the discussion of these constructions, we end this chapter with some basics on the [[causal structure]] of [[Minkowski spacetime]].
$\,$
1. [Real division algebras](#RealDivisionAlgebras)
1. [Spacetime in dimensions 3, 4, 6 and 10](#MinkowskiSpacetimeInCriticalDimensions)
1. [Lorentz group and Spin group](#LorentzGroupAndSpinGroup)
1. [Spinors in dimensions 3, 4, 6 and 10](#InTermsOfNormedDivisionAlgebraInDimension3To10)
1. [Causal structure](#CausalStructure)
$\,$
**Real division algebras**
{#RealDivisionAlgebras}
To amplify the following pattern and to fix our notation for algebra generators, recall these definitions:
+-- {: .num_defn #TheComplexNumbers}
###### Definition
**([[complex numbers]])**
The _[[complex numbers]]_ $\mathbb{C}$ is the [[commutative algebra]] over the [[real numbers]] $\mathbb{R}$ which is [[generators and relations|generated]] from one generators $\{e_1\}$ subject to the [[generators and relations|relation]]
* $(e_1)^2 = -1$.
=--
+-- {: .num_defn #TheQuaternions}
###### Definition
**([[quaternions]])**
The _[[quaternions]]_ $\mathbb{H}$ is the [[associative algebra]] over the [[real numbers]] which is [[generators and relations|generated]] from three generators $\{e_1, e_2, e_3\}$ subject to the [[generators and relations|relations]]
<div style="float:right;margin:0 20px 10px 20px;">
<img src="https://ncatlab.org/nlab/files/QuaternionMultiplicationTable.jpg" width="300" alt="quaternion multiplication table">
</div>
1. for all $i$
$(e_i)^2 = -1$
1. for $(i,j,k)$ a cyclic [[permutation]] of $(1,2,3)$ then
1. $e_i e_j = e_k$
1. $e_j e_i = -e_k$
> (graphics grabbed from [Baez 02](geometry+of+physics+--+supersymmetry#Baez02))
=--
+-- {: .num_defn #TheOctonions}
###### Definition
**([[octonions]])**
The _[[octonions]]_ $\mathbb{O}$ is the [[nonassociative algebra]] over the [[real numbers]] which is [[generators and relations|generated]] from seven generators $\{e_1, \cdots, e_7\}$ subject to the [[generators and relations|relations]]
<div style="float:right;margin:0 20px 10px 20px;">
<img src="https://ncatlab.org/nlab/files/OctonionMultiplicationTable.jpg" width="400" alt="octonion multiplication table">
</div>
1. for all $i$
$(e_i)^2 = -1$
1. for $e_i \to e_j \to e_k$ an edge or circle in the diagram shown (a labeled version of the [[Fano plane]]) then
1. $e_i e_j = e_k$
1. $e_j e_i = -e_k$
and all relations obtained by cyclic [[permutation]] of the indices in these equations.
> (graphics grabbed from [Baez 02](geometry+of+physics+--+supersymmetry#Baez02))
=--
One defines the following operations on these real algebras:
+-- {: .num_defn #Conjugation}
###### Definition
**(conjugation, [[real part]], [[imaginary part]] and [[absolute value]])**
For $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}\}$, let
$$
(-)^\ast
\;\colon\;
\mathbb{K}
\longrightarrow
\mathbb{K}
$$
be the [[antihomomorphism]] of real algebras
$$
\begin{aligned}
(r a)^\ast = r a^\ast &, \text{for}\;\; r \in \mathbb{R}, a \in \mathbb{K}
\\
(a b)^\ast = b^\ast a^\ast &,\text{for}\;\; a,b \in \mathbb{K}
\end{aligned}
$$
given on the generators of def. \ref{TheComplexNumbers}, def. \ref{TheQuaternions}
and def. \ref{TheOctonions} by
$$
(e_i)^\ast = - e_i
\,.
$$
This operation makes $\mathbb{K}$ into a [[star algebra]].
For the [[complex numbers]] $\mathbb{C}$ this is called _[[complex conjugation]]_, and in general we call it _conjugation_.
Let then
$$
Re \;\colon\; \mathbb{K} \longrightarrow \mathbb{R}
$$
be the [[function]]
$$
Re(a) \;\coloneqq\; \tfrac{1}{2}(a + a^\ast)
$$
("[[real part]]") and
$$
Im \;\colon\; \mathbb{K} \longrightarrow \mathbb{R}
$$
be the [[function]]
$$
Im(a) \;\coloneqq \; \tfrac{1}{2}(a - a^\ast)
$$
("[[imaginary part]]").
It follows that for all $a \in \mathbb{K}$ then the product of a with its conjugate is in the real [[center]]
of $\mathbb{K}$
$$
a a^\ast = a^\ast a \;\in \mathbb{R} \hookrightarrow \mathbb{K}
$$
and we write the [[square root]] of this expression as
$$
{\vert a\vert}
\;\coloneqq\;
\sqrt{a a^\ast}
$$
called the _[[norm]]_ or _[[absolute value]]_ [[function]]
$$
{\vert -\vert}
\;\colon\;
\mathbb{K}
\longrightarrow
\mathbb{R}
\,.
$$
This norm operation clearly satisfies the following properties (for all $a,b \in \mathbb{K}$)
1. $\vert a \vert \geq 0$;
1. ${\vert a \vert } = 0 \;\;\;\;\; \Leftrightarrow\;\;\;\;\;\; a = 0$;
1. ${\vert a b \vert } = {\vert a \vert} {\vert b \vert}$
and hence makes $\mathbb{K}$ a [[normed algebra]].
Since $\mathbb{R}$ is a [[division algebra]], these relations immediately imply that each $\mathbb{K}$ is a [[division algebra]], in that
$$
a b = 0 \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; a = 0 \;\; \text{or} \;\; b = 0
\,.
$$
Hence the conjugation operation makes $\mathbb{K}$ a [[real numbers|real]] [[normed division algebra]].
=--
+-- {: .num_remark #SequenceOfInclusionsOfRealNormedDivisionAlgebras}
###### Remark
**(sequence of inclusions of real [[normed division algebras]])**
Suitably embedding the sets of generators in def. \ref{TheComplexNumbers}, def. \ref{TheQuaternions} and def. \ref{TheOctonions}
into each other yields sequences of real [[star-algebra]] [[monomorphisms|inclusions]]
$$
\mathbb{R}
\hookrightarrow
\mathbb{C}
\hookrightarrow
\mathbb{H}
\hookrightarrow
\mathbb{O}
\,.
$$
For example for the first two inclusions we may send each generator to the generator of the same name, and for the
last inclusion me may choose
$$
\array{
1 &\mapsto& 1
\\
e_1 &\mapsto & e_3
\\
e_2 &\mapsto& e_4
\\
e_3 &\mapsto& e_6
}
$$
=--
+-- {: .num_prop #HurwitzTheorem}
###### Proposition
**([[Hurwitz theorem]]: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ are the normed real division algebras)**
The four algebras of [[real numbers]] $\mathbb{R}$, [[complex numbers]] $\mathbb{C}$,
[[quaternions]] $\mathbb{H}$ and [[octonions]] $\mathbb{O}$
from def. \ref{TheComplexNumbers}, def. \ref{TheQuaternions} and def. \ref{TheOctonions} respectively,
which are real [[normed division algebras]] via def. \ref{Conjugation},
are, up to [[isomorphism]], the _only_ real normed division algebras that exist.
=--
+-- {: .num_remark}
###### Remark
**([[Cayley-Dickson construction]] and [[sedenions]])**
While prop. \ref{HurwitzTheorem} says that the sequence from remark \ref{SequenceOfInclusionsOfRealNormedDivisionAlgebras}
$$
\mathbb{R}
\hookrightarrow
\mathbb{C}
\hookrightarrow
\mathbb{H}
\hookrightarrow
\mathbb{O}
$$
is maximal in the [[category]] of real normed non-associative [[division algebras]], there is a pattern that does
continue if one disregards the division algebra property. Namely each step in this sequence is given by
a construction called _forming the [[Cayley-Dickson double algebra]]_. This continues to an unbounded sequence of
real nonassociative star-algebras
$$
\mathbb{R}
\hookrightarrow
\mathbb{C}
\hookrightarrow
\mathbb{H}
\hookrightarrow
\mathbb{O}
\hookrightarrow
\mathbb{S}
\hookrightarrow
\cdots
$$
where the next algebra $\mathbb{S}$ is called the _[[sedenions]]_.
=--
What actually matters for the following relation of the real normed division algebras to [[real spin representations]]
is that they are also [[alternative algebras]]:
+-- {: .num_defn #AlternativeAlgebra}
###### Definition
**([[alternative algebras]])**
Given any [[non-associative algebra]] $A$, then the trilinear map
$$
[-,-,-] \;-\; A \otimes A \otimes A \longrightarrow A
$$
given on any elements $a,b,c \in A$ by
$$
[a,b,c] \coloneqq (a b) c - a (b c)
$$
is called the _[[associator]]_ (in analogy with the _[[commutator]]_ $[a,b] \coloneqq a b - b a$ ).
If the associator is completely antisymmetric (in that for any [[permutation]] $\sigma$ of three elements then $[a_{\sigma_1}, a_{\sigma_2}, a_{\sigma_3}] = (-1)^{\vert \sigma\vert} [a_1, a_2, a_3]$ for $\vert \sigma \vert$ the [[signature of a permutation|signature of the permutation]]) then $A$ is called an _[[alternative algebra]]_.
If the [[characteristic]] of the [[ground field]] is different from 2, then alternativity is
readily seen to be equivalent to the conditions that for all $a,b \in A$ then
$$
(a a)b = a (a b)
\;\;\;\;\;
\text{and}
\;\;\;\;\;
(a b) b = a (b b)
\,.
$$
=--
We record some basic properties of associators in alternative star-algebras that we need below:
+-- {: .num_prop #PropertiesOfAssociatorInAlternativeAlgebra}
###### Proposition
**(properties of [[alternative algebra|alternative]] [[star algebras]])**
Let $A$ be an [[alternative algebra]] (def. \ref{AlternativeAlgebra}) which is also a [[star algebra]]. Then (using def. \ref{Conjugation}):
1. the [[associator]] vanishes when at least one argument is [[real part|real]]
$$
[Re(a),b,c]
$$
1. the [[associator]] changes sign when one of its arguments is conjugated
$$
[a,b,c] = -[a^\ast,b,c]
\,;
$$
1. the [[associator]] vanishes when one of its arguments is the conjugate of another
$$
[a,a^\ast, b] = 0
\,;
$$
1. the [[associator]] is purely [[imaginary part|imaginary]]
$$
Re([a,b,c]) = 0
\,.
$$
=--
+-- {: .proof}
###### Proof
That the associator vanishes as soon as one argument is real is just the linearity of an algebra
product over the ground ring.
Hence in fact
$$
[a,b,c] = [Im(a), Im(b), Im(c)]
\,.
$$
This implies the second statement by linearity. And so follows the third statement by skew-symmetry:
$$
[a,a^\ast,b] = -[a,a,b] = 0
\,.
$$
The fourth statement finally follows by this computation:
$$
\begin{aligned}
\,[ a, b, c]^\ast
& =
-[c^\ast, b^\ast, a^\ast]
\\
& =
-[c,b,a]
\\
& =
-[a,b,c]
\end{aligned}
\,.
$$
Here the first equation follows by inspection and using that $(a b)^\ast = b^\ast a^\ast$,
the second follows from the first statement above, and the third is the anti-symmetry of the associator.
=--
It is immediate to check that:
+-- {: .num_prop #RCHOAreAlternative}
###### Proposition
**($\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ are real [[alternative algebras]])**
The real algebras of [[real numbers]], [[complex numbers]], def. \ref{TheComplexNumbers},[[quaternions]] def. \ref{TheQuaternions} and
[[octonions]] def. \ref{TheOctonions} are [[alternative algebras]] (def. \ref{AlternativeAlgebra}).
=--
+-- {: .proof}
###### Proof
Since the [[real numbers]], [[complex numbers]] and [[quaternions]] are [[associative algebras]],
their [[associator]] vanishes identically. It only remains to see that the associator of the
[[octonions]] is skew-symmetric. By linearity it is sufficient to check this on generators.
So let $e_i \to e_j \to e_k$ be a circle or a cyclic permutation of an edge in the [[Fano plane]].
Then by definition of the octonion multiplication we have
$$
\begin{aligned}
(e_i e_j) e_j
&=
e_k e_j
\\
&=
- e_j e_k
\\
& =
-e_i
\\
& =
e_i (e_j e_j)
\end{aligned}
$$
and similarly
$$
\begin{aligned}
(e_i e_i ) e_j
&=
- e_j
\\
&=
- e_k e_i
\\
&=
e_i e_k
\\
&=
e_i (e_i e_j)
\end{aligned}
\,.
$$
=--
The analog of the [[Hurwitz theorem]] (prop. \ref{HurwitzTheorem}) is now this:
+-- {: .num_prop #ZornTheorem}
###### Proposition
**($\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ are precisely the [[alternative algebra|alternative]] [[real number|real]] [[division algebras]])**
The only [[division algebras]] over the [[real numbers]] which are also [[alternative algebras]] (def. \ref{AlternativeAlgebra}) are the [[real numbers]] themselves, the [[complex numbers]], the [[quaternions]] and the [[octonions]] from prop. \ref{RCHOAreAlternative}.
=--
This is due to ([Zorn 30](normed+division#algebra#Zorn30)).
For the following, the key point of alternative algebras is this equivalent characterization:
+-- {: .num_prop #ArtinTheorem}
###### Proposition
**([[alternative algebra]] detected on subalgebras spanned by any two elements)**
A [[nonassociative algebra]] is alternative, def. \ref{AlternativeAlgebra}, precisely if the [[subalgebra]] generated by any two elements is an [[associative algebra]].
=--
This is due to [[Emil Artin]], see for instance ([Schafer 95, p. 18](alternative+algebra#Schafer95)).
Proposition \ref{ArtinTheorem} is what allows to carry over a minimum of [[linear algebra]] also to the [[octonions]]
such as to yield a representation of the [[Clifford algebra]] on $\mathbb{R}^{9,1}$. This happens in the proof
of prop. \ref{SpinorRepsByNormedDivisionAlgebra} below.
So we will be looking at a [[fragment]] of [[linear algebra]] over these four [[normed division algebras]].
To that end, fix the following notation and terminology:
+-- {: .num_defn #MatrixNotation}
###### Definition
**([[hermitian matrices]] with values in real [[normed division algebras]])**
Let $\mathbb{K}$ be one of the four real [[normed division algebras]] from prop. \ref{HurwitzTheorem}, hence equivalently one of the four
real [[alternative algebra|alternative]] [[division algebras]] from prop. \ref{ZornTheorem}.
Say that an $n \times n$ [[matrix]] with [[coefficients]] in $\mathbb{K}$
$$
A\in Mat_{n\times n}(\mathbb{K})
$$
is a _[[hermitian matrix]]_ if the [[transpose matrix]] $(A^t)_{i j} \coloneqq A_{j i}$ equals the componentwise [[complex conjugation|conjugated]] matrix (def. \ref{Conjugation}):
$$
A^t = A^\ast
\,.
$$
Hence with the notation
$$
(-)^\dagger \coloneqq ((-)^t)^\ast
$$
we have that $A$ is a [[hermitian matrix]] precisely if
$$
A = A^\dagger
\,.
$$
We write $Mat_{2 \times 2}^{her}(\mathbb{K})$ for the [[real vector space]] of hermitian matrices.
=--
+-- {: .num_defn #TraceReversal}
###### Definition
**([[trace]] reversal)**
Let $A \in Mat_{2 \times 2}^{her}(\mathbb{K})$ be a hermitian $2 \times 2$ matrix as in def. \ref{MatrixNotation}.
Its _trace reversal_ is the result of subtracting its [[trace]] times the identity matrix:
$$
\tilde A \;\coloneqq\; A - (tr A) 1_{n\times n}
\,.
$$
=--
$\,$
**Minkowski spacetime in dimensions 3,4,6 and 10**
{#MinkowskiSpacetimeInCriticalDimensions}
We now discover [[Minkowski spacetime]] of dimension 3,4,6 and 10, in terms of the
real [[normed division algebras]] $\mathbb{K}$ from prop. \ref{HurwitzTheorem}, equivalently the
real [[alternative algebra|alternative]] [[division algebras]] from prop. \ref{ZornTheorem}:
this is prop./def. \ref{SpacetimeAsMatrices} and def. \ref{MinkowskiSpacetime} below.
+-- {: .num_prop #SpacetimeAsMatrices}
###### Proposition/Definition
**([[Minkowski spacetime]] as [[real vector space]] of [[hermitian matrices]] in [[real number|real]] [[normed division algebras]])**
Let $\mathbb{K}$ be one of the four real [[normed division algebras]] from prop. \ref{HurwitzTheorem}, hence one of the four
real [[alternative algebra|alternative]] [[division algebras]] from prop. \ref{ZornTheorem}.
Then the [[real vector space]] of $2 \times 2$ [[hermitian matrices]] over $\mathbb{K}$ (def. \ref{MatrixNotation})
equipped with the [[inner product]] $\eta$ whose [[quadratic form]] ${\vert -\vert^2_\eta}$ is the negative of the [[determinant]] operation on matrices is _[[Minkowski spacetime]]_:
$$
\label{MinkowskiSpacetimeFromHermitianMatricesWithDeterminant}
\begin{aligned}
\mathbb{R}^{dim_{\mathbb{R}}(\mathbb{K})+1,1}
&
\coloneqq
\left(
\mathbb{R}^{dim_{\mathbb{R}(\mathbb{K})}+2}
,
{\vert -\vert^2_\eta}
\right)
& \coloneqq
\left(Mat_{2 \times 2}^{her}(\mathbb{K}), -det \right)
\end{aligned}
\,.
$$
hence
1. $\mathbb{R}^{2,1}$ for $\mathbb{K} = \mathbb{R}$;
1. $\mathbb{R}^{3,1}$ for $\mathbb{K} = \mathbb{C}$;
1. $\mathbb{R}^{5,1}$ for $\mathbb{K} = \mathbb{H}$;
1. $\mathbb{R}^{9,1}$ for $\mathbb{K} = \mathbb{O}$.
Here we think of the [[vector space]] on the left as $\mathbb{R}^{p,1}$
with
$$
p \coloneqq dim_{\mathbb{R}}(\mathbb{K})+1
$$
equipped with the canonical coordinates labeled $(x^\mu)_{\mu = 0}^p$.
As a [[linear map]] the identification is given by
$$
(x^0, x^1, \cdots, x^{d-1})
\;\mapsto\;
\left(
\array{
x^0 + x^1 & y
\\
y^\ast & x^0 - x^1
}
\right)
\;\;\;
\text{with}\;
y \coloneqq x^2 1 + x^3 e_1 + x^4 e_2 + \cdots + x^{2 + dim_{\mathbb{R}(\mathbb{K})}} \,e_{dim_{\mathbb{R}}(\mathbb{K})-1}
\,.
$$
This means that the [[quadratic form]] ${\vert - \vert^2_\eta}$ is given on an element $v = (v^\mu)_{\mu = 0}^p$ by
$$
{\vert v \vert}^2_{\eta}
\;=\;
- (v^0)^2 + \underoverset{j = 1}{p}{\sum} (x^j)^2
\,.
$$
By the [[polarization identity]] the [[quadratic form]] ${\vert - \vert^2_\eta}$ induces a [[bilinear form]]
$$
\eta \;\colon\; \mathbb{R}^{p,1}\otimes \mathbb{R}^{p,1} \longrightarrow \mathbb{R}
$$
given by
$$
\begin{aligned}
\eta(v_1, v_2)
& =
\eta_{\mu \nu} v_1^\mu v_1^\nu
\\
& \coloneqq
- v_1^0 v_2^0 + \underoverset{j = 1}{p}{\sum} v_1^j v_2^j
\end{aligned}
\,.
$$
This is called the _[[Minkowski metric]]_.
Finally, under the above identification the operation of trace reversal from def. \ref{TraceReversal}
corresponds to _time reversal_ in that
$$
\widetilde{
\left(
\array{
x^0 + x^1 & y
\\
y^\ast & x^0 - x^1
}
\right)
}
\;=\;
\left(
\array{
-x^0 + x^1 & y
\\
y^\ast & -x^0 - x^1
}
\right)
\,.
$$
=--
+-- {: .proof}
###### Proof
We need to check that under the given identification, the Minkowski norm-square is indeed given by minus the determinant on the
corresponding hermitian matrices.
This follows from the nature of the conjugation operation $(-)^\ast$ from def. \ref{Conjugation}:
$$
\begin{aligned}
-
det
\left(
\array{
x^0 + x^1 & y
\\
y^\ast & x^0 - x^1
}
\right)
& =
-(x^0 + x^1)(x^0 - x^1)
+
y y^\ast
\\
& =
-(x^0)^2 + \underoverset{i = 1}{p}{\sum} (x^i)^2
\end{aligned}
\,.
$$
=--
+-- {: .num_remark #MinkowskiMetricAndPhysicalUnitOfLength}
###### Remark
**([[physical units]] of [[length]])**
As the term "[[metric]]" suggests, in application to [[physics]], the [[Minkowski metric]] $\eta$ in prop./def. \ref{SpacetimeAsMatrices} is regarded as a _measure of [[length]]_: for $v \in \Gamma_x(T \mathbb{R}^{p,1})$ a [[tangent vector]] at a point $x$ in Minkowski spacetime, interpreted as a displacement from [[event]] $x$ to event $x + v$, then
1. if $\eta(v,v) \gt 0$ then
$$
\sqrt{\eta(v,v)} \in \mathbb{R}
$$
is interpreted as a measure for the _[[spacelike|spatial]] [[distance]]_ between $x$ and $x + v$;
1. if $\eta(v,v) \lt 0$ then
$$
\sqrt{-\eta(v,v)} \in \mathbb{R}
$$
is interpreted as a measure for the _[[timelike|time]] [[distance]]_ between $x$ and $x + v$.
But for this to make physical sense, an operational prescription needs to be specified that tells the experimentor how the _[[real number]]_ $\sqrt{\eta(v,v)}$ is to be translated into an physical distance between actual [[events]] in the [[observable universe]].
Such an operational prescription is called a _[[physical unit]] of [[length]]_.
For example "[[centimeter]]" $cm$ is a physical unit of length, another one is "[[femtometer]]" $fm$.
The combined information of a [[real number]] $\sqrt{\eta(v,v)} \in \mathbb{R}$ and a [[physical unit]] of [[length]] such as [[meter]], jointly written
$$
\sqrt{\eta(v,v)} \, cm
$$
is a prescription for finding actual distance in the [[observable universe]]. Alternatively
$$
\sqrt{\eta(v,v)} \, fm
$$
is another prescription, that translates the same [[real number]] $\sqrt{\eta(v,v)}$ into another physical distance.
But of course they are related, since [[physical units]] form a [[torsor]] over the [[group]] $\mathbb{R}_{\gt 0}$ of [[non-negative number|non-negative]] [[real numbers]], meaning that any two are related by a unique rescaling. For example
$$
fm = 10^{-13} cm
\,,
$$
with $10^{-13} \in \mathbb{R}_{\gt 0}$.
This means that once any one prescription of turning real numbers into spacetime distances is specified, then any other such prescription is obtained from this by rescaling these real numbers. For example
$$
\begin{aligned}
\sqrt{\eta(v,v)} \, fm
& = \left( 10^{-13} \sqrt{\eta(v,v)}\right) \,cm
\\
& =
\sqrt{ 10^{-26} \eta(v,v) } \, cm
\end{aligned}
\,.
$$
The point to notice here is that, via the last line, we may think of this as
_rescaling the [[metric]]_ from $\eta$ to $10^{-30} \eta$.
In [[quantum field theory]] [[physical units]] of [[length]] are typically expressed in terms of a
[[physical unit]] of "[[action]]", called "[[Planck's constant]]" $\hbar$, via the
combination of units called the _[[Compton wavelength]]_
$$
\label{ComptonWavelength}
\ell_m = \frac{2\pi \hbar}{m c}
\,.
$$
parameterized, in turn, by a [[physical unit]] of [[mass]] $m$. For the mass of the [[electron]],
the [[Compton wavelength]] is
$$
\ell_e = \frac{2\pi \hbar}{m_e c} \sim 386 \, fm
\,.
$$
Another [[physical unit]] of [[length]] parameterized by a [[mass]] $m$ is the _[[Schwarzschild radius]]_ $r_m \coloneqq 2 m G/c^2$, where $G$ is the [[gravitational constant]]. Solving the [[equation]]
$$
\array{
& \ell_m &=& r_m
\\
\Leftrightarrow & 2\pi\hbar / m c &=& 2 m G / c^2
}
$$
for $m$ yields the _[[Planck mass]]_
$$
m_{P} \coloneqq \tfrac{1}{\sqrt{\pi}} m_{\ell = r} = \sqrt{\frac{\hbar c}{G}}
\,.
$$
The corresponding [[Compton wavelength]] $\ell_{m_{P}}$ is given by the _[[Planck length]]_ $\ell_P$
$$
\ell_{P} \coloneqq \tfrac{1}{2\pi} \ell_{m_P} = \sqrt{ \frac{\hbar G}{c^3} }
\,.
$$
=--
+-- {: .num_defn #MinkowskiSpacetime}
###### Definition
**([[Minkowski spacetime]] as a [[pseudo-Riemannian manifold|pseudo-Riemannian]] [[Cartesian space]])**
Prop./def. \ref{SpacetimeAsMatrices} introduces [[Minkowski spacetime]] $\mathbb{R}^{p,1}$
for $p+1 \in \{3,4,6,10\}$ as a
a [[vector space]] $\mathbb{R}^{p,1}$ equipped with a [[norm]] ${\vert - \vert_\eta}$. The genuine [[spacetime]] corresponding to this
is this vector space regaded as a [[Cartesian space]], i.e. with [[smooth functions]] (instead of just [[linear maps]])
to it and from it (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}).
This still carries one copy of $\mathbb{R}^{p,1}$ over each point $x \in \mathbb{R}^{p,1}$, as its [[tangent space]] (example \ref{TangentVectorFields})
$$
T_x \mathbb{R}^{p,1} \simeq \mathbb{R}^{p,1}
$$
and the [[Cartesian space]] $\mathbb{R}^{p,1}$ equipped with the Lorentzian inner product from
prop./def. \ref{SpacetimeAsMatrices} on each [[tangent space]] $T_x \mathbb{R}^{p,1}$
(a "[[pseudo-Riemannian manifold|pseudo-Riemannian]] [[Cartesian space]]") is _[[Minkowski spacetime]]_ as such.
We write
$$
\label{MinkowskiVolume}
dvol_\Sigma
\;\coloneqq\;
d x^0 \wedge d x^1 \wedge \cdots \wedge d x^p
\in \Omega^{p+1}(\mathbb{R}^{p,1})
$$
for the canonical [[volume form]] on Minkowski spacetime.
We use the [[Einstein summation convention]]: Expressions with repeated indices indicate [[sum|summation]] over the
range of indices.
For example a [[differential 1-form]] $\alpha \in \Omega^1(\mathbb{R}^{p,1})$ on Minkowski spacetime may be expanded as
$$
\alpha = \alpha_\mu d x^\mu
\,.
$$
Moreover we use square brackets around indices to indicate skew-symmetrization. For example a [[differential 2-form]]
$\beta \in \Omega^2(\mathbb{R}^{p,1})$ on Minkowski spacetime may be expanded as
$$
\begin{aligned}
\beta
& = \beta_{\mu \nu} d x^\mu \wedge d x^\nu
\\
& = \beta_{[\mu \nu]} d x^\mu \wedge d x^\nu
\end{aligned}
$$
=--
$\,$
The identification of [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) in the exceptional dimensions with the
generalized [[Pauli matrices]] (prop./def. \ref{SpacetimeAsMatrices}) has some immediate useful implications:
+-- {: .num_prop #DeterminantViaProductWithTraceReversal}
###### Proposition
**([[Minkowski spacetime|Minkowski metric]] in terms of [[trace]] reversal)**
In terms of the trace reversal operation $\widetilde{(-)}$ from def. \ref{TraceReversal},
the [[determinant]] operation on [[hermitian matrices]] (def. \ref{MatrixNotation}) has the following alternative expression
$$
\begin{aligned}
-det(A) & = A \tilde A
\\
& = \tilde A A
\end{aligned}
\,.
$$
and the Minkowski inner product from prop. \ref{SpacetimeAsMatrices} has the alternative expression
$$
\begin{aligned}
\eta(A,B) & = \tfrac{1}{2}Re(tr(A \tilde B))
\\
& = \tfrac{1}{2} Re(tr(\tilde A B))
\end{aligned}
\,.
$$
=--
([Baez-Huerta 09, prop. 5](geometry+of+physics+--+supersymmetry#BaezHuerta09))
+-- {: .num_prop #SLGrupOnPaulimatrices}
###### Proposition
**([[special linear group]] $SL(2,\mathbb{K})$ acts by linear [[isometries]] on [[Minkowski spacetime]] )**
For $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}\}$ one of the four real [[normed division algebras]]
(prop. \ref{HurwitzTheorem}) the [[special linear group]] $SL(2,\mathbb{K})$ [[action|acts]] on [[Minkowski spacetime]] $\mathbb{R}^{p,1}$
in dimension $p+1 \in \{2+1, \,3+1, \, 5+1. \, 9+1\}$ (def. \ref{MinkowskiSpacetime})
by [[linear map|linear]] [[isometries]] given under the identification with the [[Pauli matrices]] in prop./def. \ref{SpacetimeAsMatrices} by [[conjugation]]:
$$
\array{
SL(2,\mathbb{K}) \times \mathbb{R}^{dim(\mathbb{K}+1,1)}
& \simeq &
SL(2, \mathbb{K}) \times Mat^{herm}_{2 \times 2}(\mathbb{K})
&\overset{}{\longrightarrow}&
Mat^{herm}_{2 \times 2}(\mathbb{K})
& \simeq &
\mathbb{R}^{dim(\mathbb{K}+1,1)}
\\
&&
(G, A) &\mapsto& G \, A \, G^\dagger
}
$$
=--
+-- {: .proof}
###### Proof
For $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$ this is immediate from [[matrix calculus]], but we spell it out now.
While the argument does not directly apply to the case $\mathbb{K} = \mathbb{O}$ of the [[octonions]], one can check that it
still goes through, too.
First we need to see that the [[action]] is well defined. This follows from the [[associativity]] of [[matrix multiplication]]
and the fact that forming [[conjugate transpose matrices]] is an [[antihomomorphism]]: $(G_1 G_2)^\dagger = G_2^\dagger G_1^\dagger$.
In particular this implies that the action indeed sends [[hermitian matrices]] to hermitian matrices:
$$
\begin{aligned}
\left(
G \, A \, G^\dagger
\right)^\dagger
& =
\underset{= G}{\underbrace{\left( G^\dagger \right)}} \, \underset{= A}{\underbrace{A^\dagger}} \, G^\dagger
\\
& =
G \, A \, G^\dagger
\end{aligned}
\,.
$$
By prop./def. \ref{SpacetimeAsMatrices} such an action is an [[isometry]] precisely if it preserves the
[[determinant]]. This follows from the multiplicative property of determinants: $det(A B) = det(A) det(B)$
and their compativility with conjugate transposition: $det(A^\dagger) = det(A^\ast)$, and
finally by the assumption that $G \in SL(2,\mathbb{K})$ is an element of the [[special linear group]], hence
that its determinant is $1 \in \mathbb{K}$:
$$
\begin{aligned}
det\left(
G \, A \, G^\dagger
\right)
& =
\underset{ = 1}{\underbrace{det(G)}} \, det(A) \, \underset{= 1^\ast = 1}{\underbrace{det(G^\dagger)}}
\\
& = det(A)
\end{aligned}
\,.
$$
=--
In fact the [[special linear groups]] of [[linear map|linear]] [[isometries]] in prop. \ref{SLGrupOnPaulimatrices}
are the [[spin groups]] (def. \ref{SpinGroup} below) in these dimensions.
[[!include exceptional spinors and division algebras -- table]]
This we explain now.
$\,$
**Lorentz group and spin group**
{#LorentzGroupAndSpinGroup}
+-- {: .num_defn #LorentzGroup}
###### Definition
**([[Lorentz group]])**
For $d \in \mathbb{N}$, write
$$
O(d-1,1) \hookrightarrow GL(\mathbb{R}^d)
$$
for the [[subgroup]] of the [[general linear group]] on those [[linear maps]] $A$ which preserve this bilinear form
on [[Minkowski spacetime]] (def \ref{MinkowskiSpacetime}), in that
$$
\eta(A(-),A(-)) = \eta(-,-)
\,.
$$
This is the **[[Lorentz group]]** in dimension $d$.
The elements in the Lorentz group in the image of the [[special orthogonal group]] $SO(d-1) \hookrightarrow O(d-1,1)$ are _[[rotations]]_ in space. The further elements in the special Lorentz group $SO(d-1,1)$, which mathematically are "hyperbolic rotations" in a space-time plane, are called _[[boosts]]_ in [[physics]].
One distinguishes the following further [[subgroups]] of the [[Lorentz group]] $O(d-1,1)$:
* the _[[proper Lorentz group]]_
$$
SO(d-1,1) \hookrightarrow O(d-1,1)
$$
is the subgroup of elements which have [[determinant]] +1 (as elements $SO(d-1,1)\hookrightarrow GL(d)$ of the [[general linear group]]);
* the _[[proper orthochronous Lorentz group|proper orthochronous]]_ (or _restricted_) Lorentz group
$$
SO^+(d-1,1) \hookrightarrow SO(d-1,1)
$$
is the further [[subgroup]] of elements $A$ which preserve the time orientation of vectors $v$ in that
$(v^0 \gt 0) \Rightarrow ((A v)^0 \gt 0)$.
=--
+-- {: .num_prop #ConnectedComponentsOfLorentzGroup}
###### Proposition
**([[connected component]] of [[Lorentz group]])**
As a [[smooth manifold]], the [[Lorentz group]] $O(d-1,1)$ (def. \ref{LorentzGroup})
has four [[connected components]]. The connected component of the identity is the
[[proper orthochronous Lorentz group]] $SO^+(3,1)$ (def. \ref{LorentzGroup}). The other three components are
1. $SO^+(d-1,1)\cdot P$
1. $SO^+(d-1,1)\cdot T$
1. $SO^+(d-1,1)\cdot P T$,
where, as [[matrices]],
$$
P \coloneqq diag(1,-1,-1, \cdots, -1)
$$
is the operation of point reflection at the origin in space, where
$$
T \coloneqq diag(-1,1,1, \cdots, 1)
$$
is the operation of reflection in time and hence where
$$
P T = T P = diag(-1,-1, \cdots, -1)
$$
is point reflection in spacetime.
=--
The following concept of the [[Clifford algebra]] (def. \ref{CliffordAlgebra}) of [[Minkowski spacetime]]
encodes the structure of the [[inner product space]] $\mathbb{R}^{d-1,1}$
in terms of algebraic operation ("[[geometric algebra]]"), such that the action of the
[[Lorentz group]] becomes represented by a [[conjugation action]] (example \ref{CliffordConjugtionReflectionAndRotation} below).
In particular this means that every element of the proper orthochronous Lorentz group may be "split in half"
to yield a [[double cover]]: the [[spin group]] (def. \ref{SpinGroup} below).
+-- {: .num_defn #CliffordAlgebra}
###### Definition
**([[Clifford algebra]])**
For $d \in \mathbb{N}$, we write
$$
Cl(\mathbb{R}^{d-1,1})
$$
for the $\mathbb{Z}/2$-[[graded algebra|graded]] [[associative algebra]] over $\mathbb{R}$
which is generated from $d$ generators $\{\Gamma_0, \Gamma_1, \Gamma_2, \cdots, \Gamma_{d-1}\}$
in odd degree ("Clifford generators"), subject to the [[generators and relations|relation]]
$$
\label{RelationCliffordAlgebra}
\Gamma_{a} \Gamma_b + \Gamma_b \Gamma_a = - 2\eta_{a b}
$$
where $\eta$ is the [[inner product]] of [[Minkowski spacetime]] as in def. \ref{MinkowskiSpacetime}.
These relations say equivalently that
$$
\begin{aligned}
& \Gamma_0^2 = +1
\\
& \Gamma_i^2 = -1 \;\; \text{for}\; i \in \{1,\cdots, d-1\}
\\
& \Gamma_a \Gamma_b = - \Gamma_b \Gamma_a \;\;\; \text{for}\; a \neq b
\end{aligned}
\,.
$$
We write
$$
\Gamma_{a_1 \cdots a_p}
\;\coloneqq\;
\frac{1}{p!}
\underset{{permutations \atop \sigma}}{\sum} (-1)^{\vert \sigma\vert } \Gamma_{a_{\sigma(1)}} \cdots \Gamma_{a_{\sigma(p)}}
$$
for the antisymmetrized product of $p$ Clifford generators.
In particular, if all the $a_i$ are pairwise distinct, then this is simply the plain product of generators
$$
\Gamma_{a_1 \cdots a_n}
=
\Gamma_{a_1} \cdots \Gamma_{a_n}
\;\;\;
\text{if}
\;
\underset{i,j}{\forall} (a_i \neq a_j)
\,.
$$
Finally, write
$$
\overline{(-)}
\;\colon\;
Cl(\mathbb{R}^{d-1,1})
\longrightarrow
Cl(\mathbb{R}^{d-1,1})
$$
for the algebra [[antihomomorphism|anti-]][[automorphism]] given by
$$
\overline{\Gamma_a}
\coloneqq
\Gamma_a
$$
$$
\overline{\Gamma_a \Gamma_b}
\coloneqq
\Gamma_b \Gamma_a
\,.
$$
=--
+-- {: .num_remark #VectorsInsideCliffordAlgebra}
###### Remark
**([[vectors]] inside [[Clifford algebra]])**
By construction, the [[vector space]] of [[linear combinations]] of the generators
in a [[Clifford algebra]] $Cl(\mathbb{R}^{d-1,1})$ (def. \ref{CliffordAlgebra}) is canonically identified
with [[Minkowski spacetime]] $\mathbb{R}^{d-1,1}$ (def. \ref{MinkowskiSpacetime})
$$
\widehat{(-)}
\;\colon\;
\mathbb{R}^{d-1,1}
\hookrightarrow
Cl(\mathbb{R}^{d-1,1})
$$
via
$$
x_a \mapsto \Gamma_a
\,,
$$
hence via
$$
v = v^a x_a \mapsto \hat v = v^a \Gamma_a
\,,
$$
such that the defining [[quadratic form]] on $\mathbb{R}^{d-1,1}$ is identified with
the [[anti-commutator]] in the Clifford algebra
$$
\eta(v_1,v_2) = -\tfrac{1}{2}( \hat v_1 \hat v_2 + \hat v_2 \hat v_1)
\,,
$$
where on the right we are, in turn, identifying $\mathbb{R}$ with the linear span of the unit in $Cl(\mathbb{R}^{d-1,1})$.
=--
The key point of the [[Clifford algebra]] (def. \ref{CliffordAlgebra}) is that it realizes
spacetime [[reflections]], [[rotations]] and [[boosts]] via [[conjugation actions]]:
+-- {: .num_example #CliffordConjugtionReflectionAndRotation}
###### Example
**(Clifford conjugation)**
For $d \in \mathbb{N}$ and $\mathbb{R}^{d-1,1}$ the [[Minkowski spacetime]] of def. \ref{MinkowskiSpacetime},
let $v \in \mathbb{R}^{d-1,1}$ be any [[vector]], regarded as an element $\hat v \in Cl(\mathbb{R}^{d-1,1})$ via remark \ref{VectorsInsideCliffordAlgebra}.
Then
1. the [[conjugation action]] $\hat v \mapsto -\Gamma_a^{-1} \hat v \Gamma_a$
of a single Clifford generator $\Gamma_a$ on $\hat v$ sends $v$ to its
[[reflection]] at the hyperplane $x_a = 0$;
1. the [[conjugation action]]
$$
\hat v \mapsto \exp(- \tfrac{\alpha}{2} \Gamma_{a b}) \hat v \exp(\tfrac{\alpha}{2} \Gamma_{a b})
$$
sends $v$ to the result of [[rotation|rotating]] it in the $(a,b)$-plane through an angle $\alpha$.
=--
+-- {: .proof}
###### Proof
This is immediate by inspection:
For the first statement, observe that conjugating the Clifford generator $\Gamma_b$ with $\Gamma_a$
yields $\Gamma_b$ up to a sign, depending on whether $a = b$ or not:
$$
- \Gamma_a^{-1} \Gamma_b \Gamma_a
=
\left\{
\array{
-\Gamma_b & \vert \text{if}\, a = b
\\
\Gamma_b & \vert \text{otherwise}
}
\right.
\,.
$$
Therefore for $\hat v = v^b \Gamma_b$ then $\Gamma_a^{-1} \hat v \Gamma_a$ is the result of
multiplying the $a$-component of $v$ by $-1$.
For the second statement, observe that
$$
-\tfrac{1}{2}[\Gamma_{a b}, \Gamma_c]
=
\Gamma_a \eta_{b c} - \Gamma_b \eta_{a c}
\,.
$$
This is the canonical action of the Lorentzian [[special orthogonal Lie algebra]] $\mathfrak{so}(d-1,1)$.
Hence
$$
\exp(-\tfrac{\alpha}{2} \Gamma_{ab}) \hat v \exp(\tfrac{\alpha}{2} \Gamma_{ab})
=
\exp(\tfrac{1}{2}[\Gamma_{a b}, -])(\hat v)
$$
is the rotation action as claimed.
=--
+-- {: .num_remark #AmbiguityInCliffordConjugation}
###### Remark
Since the [[reflections]], [[rotations]] and [[boosts]] in example \ref{CliffordConjugtionReflectionAndRotation} are given by [[conjugation actions]], there is a crucial ambiguity in the Clifford elements that induce them:
1. the conjugation action by $\Gamma_a$ coincides precisely with the conjugation action by $-\Gamma_a$;
1. the [[conjugation action]] by $\exp(\tfrac{\alpha}{4} \Gamma_{a b})$ coincides precisely with the
conjugation action by $-\exp(\tfrac{\alpha}{2}\Gamma_{a b})$.
=--
+-- {: .num_defn #SpinGroup}
###### Definition
**([[spin group]])**
For $d \in \mathbb{N}$, the **[[spin group]]** $Spin(d-1,1)$ is the group of
even graded elements of the Clifford algebra $Cl(\mathbb{R}^{d-1,1})$ (def. \ref{CliffordAlgebra})
which are [[unitary operator|unitary]] with respect to the conjugation operation $\overline{(-)}$
from def. \ref{CliffordAlgebra}:
$$
Spin(d-1,1)
\;\coloneqq\;
\left\{
A \in Cl(\mathbb{R}^{d-1,1})_{even}
\;\vert\;
\overline{A} A = 1
\right\}
\,.
$$
=--
+-- {: .num_prop #SpinDoubleCover}
###### Proposition
The [[function]]
$$
Spin(d-1,1)
\longrightarrow
GL(\mathbb{R}^{d-1,1})
$$
from the [[spin group]] (def. \ref{SpinGroup}) to the [[general linear group]] in $d$-dimensions
given by sending $A \in Spin(d-1,1) \hookrightarrow Cl(\mathbb{R}^{d-1,1})$ to the
[[conjugation action]]
$$
\overline{A}(-) A
$$
(via the identification of Minkowski spacetime as the subspace of the [[Clifford algebra]]
containing the [[linear combinations]] of the generators, according to remark \ref{VectorsInsideCliffordAlgebra})
is
1. a [[group]] [[homomorphism]] onto the [[proper orthochronous Lorentz group]] (def. \ref{LorentzGroup}):
$$
Spin(d-1,1) \longrightarrow SO^+(d-1,1)
$$
1. exhibiting a $\mathbb{Z}/2$-[[central extension]].
=--
+-- {: .proof}
###### Proof
That the function is a group homomorphism into the [[general linear group]], hence that it acts by [[linear transformations]]
on the generators follows by using that it clearly lands in [[automorphisms]] of the Clifford algebra.
That the function lands in the [[Lorentz group]] $O(d-1,1) \hookrightarrow GL(d)$ follows from remark \ref{VectorsInsideCliffordAlgebra}:
$$
\begin{aligned}
\eta(\overline{A}v_1A , \overline{A} v_2 A)
&=
\tfrac{1}{2}
\left(
\left(\overline{A} \hat v_1 A\right) \left(\overline{A}\hat v_2 A\right)
+
\left(\overline{A} \hat v_2 A\right) \left(\overline{A} \hat v_1 A\right)
\right)
\\
& =
\tfrac{1}{2}
\left(
\overline{A}(\hat v_1 \hat v_2 + \hat v_2 \hat v_1) A
\right)
\\
& =
\overline{A} A \tfrac{1}{2}\left( \hat v_1 \hat v_2 + \hat v_2 \hat v_1\right)
\\
& =
\eta(v_1, v_2)
\end{aligned}
\,.
$$
That it moreover lands in the [[proper Lorentz group]] $SO(d-1,1)$ follows from
observing (example \ref{CliffordConjugtionReflectionAndRotation}) that every reflection is given by the [[conjugation action]] by
a linear combination of generators, which are excluded from the group $Spin(d-1,1)$
(as that is defined to be in the even subalgebra).
To see that the homomorphism is surjective, use that all elements of $SO(d-1,1)$
are products of [[rotations]] in hyperplanes. If a hyperplane is spanned by the [[bivector]]
$(\omega^{a b})$, then such a rotation is given, via example \ref{CliffordConjugtionReflectionAndRotation}
by the conjugation action by
$$
\exp(\tfrac{\alpha}{2} \omega^{a b}\Gamma_{a b})
$$
for some $\alpha$, hence is in the image.
That the [[kernel]] is $\mathbb{Z}/2$ is clear from the fact that the only even Clifford elements which commute
with all vectors are the multiples $a \in \mathbb{R} \hookrightarrow Cl(\mathbb{R}^{d-1,1})$ of the identity.
For these $\overline{a} = a$ and hence
the condition $\overline{a} a = 1$ is equivalent to $a^2 = 1$. It is clear that these two elements
$\{+1,-1\}$ are in the [[center]] of $Spin(d-1,1)$. This
kernel reflects the ambiguity from remark \ref{AmbiguityInCliffordConjugation}.
=--
$\,$
**Spinors in dimensions 3, 4, 6 and 10**
{#InTermsOfNormedDivisionAlgebraInDimension3To10}
We now discuss how [[real spin representations]] (def. \ref{SpinGroup}) in
spacetime dimensions 3,4, 6 and 10 are naturally induced from
[[linear algebra]] over the four real [[alternative algebras|alternative]] [[division algebras]] (prop. \ref{HurwitzTheorem}).
+-- {: .num_defn #CliffordAlgebraInTermsOfNormedDivisionAlgebra}
###### Definition
**([[Clifford algebra]] via [[normed division algebra]])**
Let $\mathbb{K}$ be one of the four real [[normed division algebras]] from prop. \ref{HurwitzTheorem}, hence one of the four
real [[alternative algebra|alternative]] [[division algebras]] from prop. \ref{ZornTheorem}.
Define a [[real numbers|real]] [[linear map]]
$$
\Gamma \;\colon\; \mathbb{R}^{dim_{\mathbb{R}}(\mathbb{K})+1,1} \longrightarrow End_{\mathbb{R}}(\mathbb{K}^4)
$$
from (the real vector space underlying) [[Minkowski spacetime]] to real [[linear maps]] on $\mathbb{K}^4$
$$
\Gamma(A)
\left(
\array{
\psi
\\
\phi
}
\right)
\;\coloneqq\;
\left(
\array{
- \tilde A \phi
\\
A \psi
}
\right)
\,.
$$
Here on the right we are using the isomorphism from prop. \ref{SpacetimeAsMatrices} for identifying
a spacetime vector with a $2 \times 2$-matrix, and we are using the trace reversal $\widetilde(-)$
from def. \ref{TraceReversal}.
=--
+-- {: .num_remark}
###### Remark
**([[Clifford algebra|Clifford multiplication]] via [[octonion]]-valued matrices)**
Each operation of $\Gamma(A)$ in def. \ref{CliffordAlgebraInTermsOfNormedDivisionAlgebra} is clearly a [[linear map]],
even for $\mathbb{K}$ being the non-associative [[octonions]]. The only point to beware of is that for $\mathbb{K}$
the octonions, then the composition of two such linear maps is not in general given by the
usual matrix product.
=--
+-- {: .num_prop #SpinorRepsByNormedDivisionAlgebra}
###### Proposition
**([[real spin representations]] via [[normed division algebras]])**
The map $\Gamma$ in def. \ref{CliffordAlgebraInTermsOfNormedDivisionAlgebra} gives a [[representation]] of the [[Clifford algebra]]
$Cl(\mathbb{R}^{dim_{\mathbb{R}}(\mathbb{K})+1,1} )$
([this def.](geometry+of+physics+--+supersymmetry#CliffordAlgebra)), i.e of
1. $Cl(\mathbb{R}^{2,1})$ for $\mathbb{K} = \mathbb{R}$;
1. $Cl(\mathbb{R}^{3,1})$ for $\mathbb{K} = \mathbb{C}$;
1. $Cl(\mathbb{R}^{5,1})$ for $\mathbb{K} = \mathbb{H}$;
1. $Cl(\mathbb{R}^{9,1})$ for $\mathbb{K} = \mathbb{O}$.
Hence this Clifford representation induces [[spin representations|representations]]
of the [[spin group]] $Spin(dim_{\mathbb{R}}(\mathbb{K})+1,1)$ on
the real vector spaces
$$
S_{\pm } \coloneqq \mathbb{K}^2
\,.
$$
and hence on
$$
S \coloneqq S_+ \oplus S_-
\,.
$$
=--
([Baez-Huerta 09, p. 6](geometry+of+physics+--+supersymmetry#BaezHuerta09))
+-- {: .proof}
###### Proof
We need to check that the Clifford relation
$$
\begin{aligned}
(\Gamma(A))^2
& = -\eta(A,A)1
\\
& = + det(A)
\end{aligned}
$$
is satisfied (where we used (eq:RelationCliffordAlgebra) and (eq:MinkowskiSpacetimeFromHermitianMatricesWithDeterminant)). Now by definition, for any $(\phi,\psi) \in \mathbb{K}^4$ then
$$
(\Gamma(A))^2
\left(
\array{
\phi
\\
\psi
}
\right)
\;=\;
-
\left(
\array{
\tilde A(A \phi)
\\
A(\tilde A \psi)
}
\right)
\,,
$$
where on the right we have in each component ordinary matrix product expressions.
Now observe that both expressions on the right are sums of triple products that involve either
one real factor or two factors that are conjugate to each other:
$$
\begin{aligned}
A (\tilde A \psi)
& =
\left(
\array{
x_0 + x_1 & y
\\
y^\ast & x_0 - x_1
}
\right)
\cdot
\left(
\array{
(-x_0 + x_1) \phi_1 + y \phi_2
\\
y^\ast \phi_1 - (x_0 + x_1)\phi_2
}
\right)
\\
& =
\left(
\array{
(-x_0^2 + x_1^2) \phi_1 + (x_0 + x_1)(y \phi_2) + y (y^\ast \phi_1) - y( (x_0 + x_1) \phi_2 )
\\
\cdots
}
\right)
\end{aligned}
\,.
$$
Since the [[associators]] of triple products that involve a real factor and those involving both an element and its
conjugate vanish by prop. \ref{PropertiesOfAssociatorInAlternativeAlgebra} (hence ultimately by Artin's theorem, prop. \ref{ArtinTheorem}).
In conclusion all associators involved vanish, so that we may rebracket to obtain
$$
(\Gamma(A))^2
\left(
\array{
\phi
\\
\psi
}
\right)
\;=\;
-
\left(
\array{
(\tilde A A) \phi
\\
(A \tilde A) \psi
}
\right)
\,.
$$
This implies the statement via the equality $-A \tilde A = -\tilde A A = det(A)$ (prop. \ref{DeterminantViaProductWithTraceReversal}).
=--
+-- {: .num_prop #RealSpinorPairingsViaDivisionAlg}
###### Proposition
**([[spinor]] [[bilinear map|bilinear]] pairings)**
Let $\mathbb{K}$ be one of the four real [[normed division algebras]] and $S_\pm \simeq_{\mathbb{R}}\mathbb{K}^2$
the corresponding [[spin representation]] from prop. \ref{SpinorRepsByNormedDivisionAlgebra}.
Then there are [[bilinear maps]] from two [[spinors]] (according to prop. \ref{SpinorRepsByNormedDivisionAlgebra})
to the [[real numbers]]
$$
\overline{(-)}(-) \;\colon\; S_+ \otimes S_-\longrightarrow \mathbb{R}
$$
as well as to $\mathbb{R}^{dim(\mathbb{K}+1,1)}$
$$
\overline{(-)}\Gamma (-) \;\colon\; S_\pm \otimes S_{\pm}\longrightarrow \mathbb{R}^{dim(\mathbb{K}+1,1)}
$$
given, respectively, by forming the [[real part]] (def. \ref{Conjugation}) of the canonical $\mathbb{K}$-[[inner product]]
$$
\overline{(-)}(-) \colon S_+\otimes S_- \longrightarrow \mathbb{R}
$$
$$
(\psi,\phi)\mapsto \overline{\psi} \phi \coloneqq Re(\psi^\dagger \cdot \phi)
$$
and by forming the product of a column vector with a row vector to produce a matrix, possibly up to trace reversal (def. \ref{TraceReversal})
under the identification $\mathbb{R}^{dim(\mathbb{K})+1,1} \simeq Mat^{her}_{2 \times 2}(\mathbb{K})$ from prop. \ref{SpacetimeAsMatrices}:
$$
S_+ \otimes S_+ \longrightarrow \mathbb{R}^{dim(\mathbb{K})+1,1}
$$
$$
(\psi , \phi) \mapsto \overline{\psi}\Gamma \phi \coloneqq
\widetilde{\psi \phi^\dagger + \phi \psi^\dagger}
$$
and
$$
S_- \otimes S_- \longrightarrow \mathbb{R}^{dim(\mathbb{K}+1,1)}
$$
$$
(\psi , \phi) \mapsto {\psi \phi^\dagger + \phi \psi^\dagger}
$$
For $A \in Mat^{her}_{2 \times 2}(\mathbb{K})$ the $A$-component of this map is
$$
\eta(\overline{\psi}\Gamma \phi, A)
=
Re (\psi^\dagger (A\phi))
\,.
$$
These pairings have the following properties
1. both are $Spin(dim(\mathbb{K})+1,1)$-equivalent;
1. the pairing $\overline{(-)}\Gamma(-)$ is _[[symmetric bilinear form|symmetric]]_:
$$
\label{SpinorToVectorPairingIsSymmetric}
\overline{\psi_1} \,\Gamma\, \psi_2 = + \overline{\phi_2}\, \Gamma\, \psi_1
\phantom{AAAA}
\text{for} \phantom{AA} \psi_1, \psi_2 \in S_+ \oplus S_-
$$
=--
([Baez-Huerta 09, prop. 8, prop. 9](geometry+of+physics+--+supersymmetry#BaezHuerta09)).
+-- {: .num_remark #TwoComponentSpinorNotation}
###### Remark
**(two-component [[spinor]] notation)**
In the [[physics]]/[[QFT]] literature the expressions for [[spin representations]]
given by prop. \ref{SpinorRepsByNormedDivisionAlgebra} are traditionally written in
_two-component spinor notation_ as follows:
* An element of $S_+$ is denoted $(\chi_a \in \mathbb{K})_{a = 1,2}$
and called a _[[Weyl spinor|left handed spinor]]_;
* an element of $S_-$ is denoted $(\xi^{\dagger \dot a})_{\dot a = 1,2}$
and called a _[[Weyl spinor|right handed spinor]]_;
* an element of $S = S_+ \oplus S_-$ is denoted
$$
\label{TwoComponentNotationForDiracSpinor}
(\psi^\alpha) = \left( (\chi_a), (\xi^{\dagger \dot a}) \right)
$$
and called a _[[Dirac spinor]]_;
and the Clifford action of prop. \ref{CliffordAlgebraInTermsOfNormedDivisionAlgebra} corresponds to the generalized "[[Pauli matrices]]":
* a hermitian matrix $A \in Mat^{her}_{2\times 2}(\mathbb{K})$ as in prop \ref{SpacetimeAsMatrices} regarded as a linear map $S_- \to S_+$ via def. \ref{CliffordAlgebraInTermsOfNormedDivisionAlgebra} is denoted
$$
\left(x_\mu \sigma^\mu_{a \dot a}\right)
\;\coloneqq\;
\left( \array{ x_0 + x_1 & y \\ y^\ast & x_0 - x_1 } \right) \,;
$$
* the negative of the trace-reversal (def. \ref{TraceReversal}) of such a hermitian matrix, regarded as a linear map $S_+ \to S_-$, is denoted
$$
\left(
x_\mu \widetilde \sigma^{\mu \dot a a}
\right)
\;\coloneqq\;
-
\left( \array{ -x_0 + x_1 & y \\ y^\ast & -x_0 - x_1 } \right)
\,.
$$
* the corresponding Clifford generator $\Gamma(A) \;\colon\; S_+ \oplus S_- \to S_+ \oplus S_-$ (def. \ref{CliffordAlgebraInTermsOfNormedDivisionAlgebra})
is denoted
$$
x_\mu (\gamma^\mu)_{\alpha \beta}
\;\coloneqq\;
\left(
\array{
0 & x_\mu \sigma^\mu_{a \dot b}
\\
x_\mu \widetilde \sigma^{\mu \dot a b}
}
\right)
$$
* the bilinear spinor-to-vector pairing from prop. \ref{RealSpinorPairingsViaDivisionAlg}
is written as the [[matrix multiplication]]
$$
\left( \overline{\psi} \, \gamma^\mu \, \phi\right)
\;\coloneqq\;
\overline{\psi}\,\Gamma \,\phi
\,,
$$
where the _[[Dirac conjugate]]_ $\overline{\psi}$ on the left is given
on $(\psi_\alpha) = (\chi_a, \xi^{\dagger \dot c})$ by
$$
\label{DiracConjugate}
\begin{aligned}
\overline{\psi}
& \coloneqq
\psi^\dagger \gamma^0
\\
& =
( \xi^a, \chi^\dagger_{\dot a} )
\end{aligned}
$$
hence, with (eq:TwoComponentNotationForDiracSpinor):
$$
\label{TwoComponentNotationForSpinorToVectorPairing}
\begin{aligned}
\overline{\psi_1} \,\gamma^\mu\, \psi_2
& =
\psi_1^\dagger \, \gamma^0 \gamma^\mu \, \psi_2
\\
& =
(\xi_1)^a \, \sigma^\mu_{a \dot c}\, (\xi_2)^{\dagger \dot c}
+
(\chi_1)^\dagger_{\dot a} \, \widetilde \sigma^{\mu \dot a c} \, (\chi_2)_c
\end{aligned}
$$
Finally, it is common to abbreviate contractions with the [[Clifford algebra]] generators $(\gamma^\mu)$
by a slash, as in
$$
k\!\!\!/\, \;\coloneqq\; \gamma^\mu k_\mu
$$
or
$$
\label{FeynmanSlashNotationForMasslessDiracOperator}
i \partial\!\!\!/\,
\;\coloneqq\;
i \gamma^\mu \frac{\partial}{\partial x^\mu}
\,.
$$
This is called the _[[Feynman slash notation]]_.
=--
(e.g. [Dermisek I-8](Dirac+field#DermisekI8), [Dermisek I-9](Dirac+field#DermisekI9))
Below we spell out the example of the [[Lagrangian field theory]] of the [[Dirac field]]
in detail (example \ref{LagrangianDensityForDiracField}). For discussion of
_massive chiral spinor fields_ one also needs the following, here we just mention this for
completeness:
+-- {: .num_defn #ChiralSpinorMassPairing}
###### Proposition
**(chiral spinor mass pairing)**
In dimension 2+1 and 3+1, there exists a non-trivial skew-symmetric pairing
$$
\epsilon
\;\colon\;
S \wedge S
\longrightarrow
\mathbb{R}
$$
which may be normalized such that in the two-component spinor basis of
remark \ref{TwoComponentSpinorNotation} we have
$$
\label{ConjugationOfLeftRightCliffordGeneratorToRightLeft}
\tilde \sigma^{\mu \dot a a} = \epsilon^{a b} \epsilon^{\dot a \dot b} \sigma^\mu_{b \dot b}
\,.
$$
=--
+-- {: .proof}
###### Proof
Take the non-vanishing components of $\epsilon$ to be
$$
\epsilon^{1 2} = \epsilon^{\dot 1 \dot 2} = \epsilon_{21} = \epsilon_{\dot 2 \dot 1} = 1
$$
and
$$
\epsilon^{2 1} = \epsilon^{\dot 2 \dot 1} = \epsilon_{1 2} = \epsilon_{\dot 1 \dot 2} = -1
\,.
$$
With this equation (eq:ConjugationOfLeftRightCliffordGeneratorToRightLeft) is checked explicitly.
It is clear that $\epsilon$ thus defined is skew symmetric as long as the component algebra
is commutative, which is the case for $\mathbb{K}$ being $\mathbb{R}$ or $\mathbb{C}$.
=--
$\,$
**Causal structure**
{#CausalStructure}
We need to consider the following concepts and constructions related to the [[causal structure]] of [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}).
+-- {: .num_defn #SpacelikeTimelikeLightlike}
###### Definition
**([[spacelike]], [[timelike]], [[lightlike]] [[direction of a vector|directions]]; [[past]] and [[future]])**
Given two points $x,y \in \Sigma$ in [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}),
write
$$
v \coloneqq y - x \in \mathbb{R}^{p,1}
$$
for their difference, using the [[vector space]] structure underlying [[Minkowski spacetime]].
Recall the Minkowski [[inner product]] $\eta$ on $\mathbb{R}^{p,1}$, given by prop./def. \ref{SpacetimeAsMatrices}.
Then via remark \ref{MinkowskiMetricAndPhysicalUnitOfLength} we say that the difference vector $v$ is
1. _[[spacelike]]_ if $\eta(v,v) \gt 0$,
1. _[[timelike]]_ if $\eta(v,v) \lt 0$,
1. _[[lightlike]]_ if $\eta(v,v) = 0$.
If $v$ is [[timelike]] or [[lightlike]] then we say that
1. $y$ is in the _[[future]]_ of $x$ if $y^0 - x^0 \geq 0$;
1. $y$ is in the _[[past]]_ of $x$ if $y^0 - x^0 \leq 0$.
=--
+-- {: .num_defn #CausalPastAndFuture}
###### Definition
**([[causal cones]])**
For $x \in \Sigma$ a point in spacetime (an [[event]]), we write
$$
V^+(x), V^-(x) \subset \Sigma
$$
for the [[subsets]] of [[events]] that are in the [[timelike]] [[future]] or in the [[timelike]] [[past]] of $x$, respectively (def. \ref{SpacelikeTimelikeLightlike})
called the _[[open future cone]]_ and _[[open past cone]]_, respectively, and
$$
\overline{V}^+(x), \overline{V}^-(x) \subset \Sigma
$$
for the [[subsets]] of [[events]] that are in the [[timelike]] or [[lightlike]] [[future]] or [[past]], respectivel,
called the _[[closed future cone]]_ and _[[closed past cone]]_, respectively.
The [[union]]
$$
J(x) \coloneqq \overline{V}^+(x) \cup \overline{V}^-(x)
$$
of the closed [[future cone]] and [[past cone]] is called the full _[[causal cone]]_ of the [[event]] $x$.
Its [[boundary]] is the _[[light cone]]_.
More generally for $S \subset \Sigma$ a [[subset]] of [[events]] we write
$$
\overline{V}^\pm(S)
\;\coloneqq\;
\underset{x \in S}{\cup} \overline{V}^{\pm}(x)
$$
for the [[union]] of the future/past closed cones of all events in the subset.
=--
+-- {: .num_defn #CompactlySourceCausalSupport}
###### Definition
**(compactly sourced causal support)
Consider a [[vector bundle]] $E \overset{}{\to} \Sigma$ (def. \ref{VectorBundle}) over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}).
Write $\Gamma_{\Sigma}(E)$ for the [[space of sections|spaces of smooth sections]] (def. \ref{Sections}), and write
$$
\begin{aligned}
\Gamma_{cp}(E) & \,\text{compact support}
\\
\Gamma_{\Sigma,\pm cp}(E) & \,\text{compactly sourced future/past support}
\\
\Gamma_{\Sigma,scp}(E) & \,\text{spacelike compact support}
\\
\Gamma_{\Sigma,(f/p)cp}(E) & \,\text{future/past compact support}
\\
\Gamma_{\Sigma,tcp}(E) & \,\text{timelike compact support}
\end{aligned}
$$
for the subsets on those smooth sections whose [[support]] is
1. ($cp$) inside a [[compact subset]],
1. ($\pm cp$) inside the [[closed future cone]]/[[closed past cone]], respectively, of a [[compact subset]],
1. ($scp$) inside the [[closed causal cone]] of a [[compact subset]], which equivalently means that the [[intersection]] with every ([[spacelike]]) [[Cauchy surface]] is compact ([Sanders 13, theorem 2.2](#Sanders12)),
1. ($fcp$) inside the past of a Cauchy surface ([Sanders 13, def. 3.2](#Sanders12)),
1. ($pcp$) inside the future of a Cauchy surface ([Sanders 13, def. 3.2](#Sanders12)),
1. ($tcp$) inside the future of one Cauchy surface and the past of another ([Sanders 13, def. 3.2](#Sanders12)).
=--
([Bär 14, section 1](space+of+sections#Baer14), [Khavkine 14, def. 2.1](Green+hyperbolic+differential+operator#Khavkine14))
+-- {: .num_defn #CausalOrdering}
###### Definition
**([[causal order]])**
Consider the [[relation]] on the set $P(\Sigma)$ of [[subsets]] of [[spacetime]]
which says a [[subset]] $S_1 \subset \Sigma$ is _not prior_ to a subset $S_2 \subset \Sigma$,
denoted $S_1 {\vee\!\!\!\wedge} S_2$, if $S_1$ does not [[intersection|intersect]] the [[causal past]] of $S_2$ (def. \ref{CausalPastAndFuture}),
or equivalently that $S_2$ does not intersect the [[causal future]] of $S_1$:
$$
\begin{aligned}
S_1 {\vee\!\!\!\wedge} S_2
& \;\;\coloneqq\;\;
S_1 \cap \overline{V}^-(S_2) = \emptyset
\\
& \;\;\Leftrightarrow\;\; S_2 \cap \overline{V}^+(S_1) = \emptyset
\end{aligned}
\,.
$$
(Beware that this is just a [[relation]], not an [[ordering]], since it is not [[transitive relation|relation]].)
If $S_1 {\vee\!\!\!\wedge} S_2$ and $S_2 {\vee\!\!\!\wedge} S_1$ we say that the two subsets are _[[spacelike]] separated_ and write
$$
S_1 {\gt\!\!\!\!\lt} S_2
\;\;\;\coloneqq\;\;\;
S_1 {\vee\!\!\!\wedge} S_2 \;\text{and}\; S_2 {\vee\!\!\!\wedge} S_1
\,.
$$
=--
+-- {: .num_example #CausalComplementOfSubsetOfLorentzianManifold}
###### Definition
**([[causal complement]] and [[causal closure]] of subset of [[spacetime]])**
For $S \subset X$ a [[subset]] of [[spacetime]], its _[[causal complement]]_ $S^\perp$ is the [[complement]] of the [[causal cone]]:
$$
S^\perp \;\coloneqq\; S \setminus J_X(S)
\,.
$$
The causal complement $S^{\perp \perp}$ of the causal complement $S^\perp$ is called the _[[causal closure]]_. If
$$
S = S^{\perp \perp}
$$
then the subset $S$ is called a _[[causally closed subset]]_.
Given a [[spacetime]] $\Sigma$, we write
$$
CausClsdSubsets(\Sigma) \;\in\; Cat
$$
for the [[partially ordered set]] of causally closed subsets, partially ordered by
inclusion $\mathcal{O}_1 \subset \mathcal{O}_2$.
=--
+-- {: .num_defn #CutoffFunctions}
###### Definition
**([[adiabatic switching]])**
For a [[causally closed subset]] $\mathcal{O} \subset \Sigma$ of [[spacetime]] (def. \ref{CausalComplementOfSubsetOfLorentzianManifold}) say that an _[[adiabatic switching]] function_ or _[[infrared cutoff]] function_ for $\mathcal{O}$
is a [[smooth function]] $g_{sw}$ of [[compact support]] (a [[bump function]]) whose
restriction to some [[neighbourhood]] $U$ of $\mathcal{O}$ is the [[constant function]]
with value $1$:
$$
Cutoffs(\mathcal{O})
\;\coloneqq\;
\left\{
g_{sw} \in C^\infty_c(\Sigma)
\;\vert\;
\underset{ {U \supset \mathcal{O}} \atop { \text{neighbourhood} } }{\exists}
\left(
g_{sw}\vert_U = 1
\right)
\right\}
\,.
$$
Often we consider the vector space space $C^\infty(\Sigma)\langle g \rangle $ spanned by a formal
variable $g$ (the [[coupling constant]]) under multiplication with smooth functions, and consider as adiabatic switching functions
the corresponding images in this space,
$$
\array{
C_c^\infty(\Sigma) &\overset{\simeq}{\longrightarrow}& C_c^\infty(X)\langle g\rangle
}
$$
which are thus bump functions constant over a neighbourhood $U$ of $\mathcal{O}$ not on 1 but on
the formal parameter $g$:
$$
g_{sw}\vert_U = g
\,
$$
In this sense we may think of the adiabatic switching as _being_ the spacetime-depependent coupling "constant".
=--
The following lemma \ref{CausalPartition} will be key in the derivation (proof of prop. \ref{CausalLocalityOfThePerturbativeSMatrix} below) of the [[causal locality]] of [[algebra of quantum observables]] in [[perturbative quantum field theory]]:
+-- {: .num_lemma #CausalPartition}
###### Lemma
**(causal partition)**
Let $\mathcal{O} \subset \Sigma$ be a [[causally closed subset]] (def. \ref{CausalComplementOfSubsetOfLorentzianManifold})
and let $f \in C^\infty_{cp}(\Sigma)$ be a [[compact support|compactly supported]] [[smooth function]] which vanishes
on a [[neighbourhood]] $U \supset \mathcal{O}$, i.e. $f\vert_U = 0$.
Then there exists a _causal partition_ of $f$ in that there
exist compactly supported smooth functions $a,r \in C^\infty_{cp}(\Sigma)$
such that
1. they sum up to $f$:
$$
f = a + r
$$
1. their [[support]] satisfies the following causal ordering (def. \ref{CausalOrdering})
$$
supp(a) {\vee\!\!\!\wedge} \mathcal{O} {\vee\!\!\!\wedge} supp(r)
\,.
$$
=--
+-- {: .proof}
###### Proof idea
By assumption $\mathcal{O}$ has a [[Cauchy surface]]. This may be
extended to a Cauchy surface $\Sigma_p$ of $\Sigma$, such that this is one [[leaf]]
of a [[foliation]] of $\Sigma$ by Cauchy surfaces, given by a [[diffeomorphism]]
$\Sigma \simeq (-1,1) \times \Sigma_p$ with the original $\Sigma_p$ at zero.
There exists then $\epsilon \in (0,1)$ such that
the restriction of $supp(f)$ to the interval $(-\epsilon, \epsilon)$ is
in the [[causal complement]] $\overline{\mathcal{O}}$ of the given region (def. \ref{CausalComplementOfSubsetOfLorentzianManifold}):
$$
supp(f) \cap (-\epsilon, \epsilon) \times \Sigma_p
\;\subset\;
\overline{\mathcal{O}}
\,.
$$
Let then $\chi \colon \Sigma \to \mathbb{R}$ be any [[smooth function]] with
1. $\chi\vert_{(-1,0] \times \Sigma_p} = 1$
1. $\chi\vert_{(\epsilon,1) \times \Sigma_p} = 0$.
Then
$$
r \coloneqq \chi \cdot f
\phantom{AAA}
\text{and}
\phantom{AAA}
a \coloneqq (1-\chi) \cdot f
$$
are smooth functions as required.
=--
$\,$
This concludes our discussion of [[spin]] and [[spacetime]]. In the [next chapter](#Fields) we consider the concept of _[[field (physics)|fields]]_ on [[spacetime]].
|
A first idea of quantum field theory -- Symmetries | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Symmetries |
## Symmetries
{#Symmetries}
In this chapter we discuss these topics:
* _[Infinitesimal symmetries of the Lagrangian density](#InfinitesimalSymmetriesOfTheLagrangianDensity)_
* _[Infinitesimal symmetries of the presymplectic potential current](#InfinitesimalSymmetriesOfThePresymplecticPotentialCurrent)_
$\,$
We have introduced the concept of _[[Lagrangian field theories]]_ $(E,\mathbf{L})$ in terms of a [[field bundle]] $E$ equipped with a
[[Lagrangian density]] $\mathbf{L}$ on its [[jet bundle]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
Generally, given any [[object]] equipped with some [[structure]], it is of paramount interest to determine
the [[symmetries]], hence the [[isomorphisms]]/[[equivalences]] of the object that preserve the given [[structure]]
(this is the "[[Erlanger program]]", [Klein 1872](Erlangen+program#Klein1872)).
The [[infinitesimal symmetries of the Lagrangian density]] (def. \ref{SymmetriesAndConservedCurrents} below)
send one [[field history]] to an [[infinitesimal|infinitesimally]] nearby one which is
"[[equivalence|equivalent]]" for all purposes of [[field theory]]. Among these are the _[[infinitesimal gauge symmetries]]_
which will be of concern [below](#GaugeSymmetries).
A central theorem of [[variational calculus]] says that [[infinitesimal symmetries of the Lagrangian]]
correspond to _[[conserved currents]]_, this is [[Noether theorem|Noether's theorem I]], prop. \ref{NoethersFirstTheorem} below.
These conserved currents constitute an [[Lie algebra extension|extension]] of the [[Lie algebra]]
of symmetries, called the _[[Dickey bracket]]_.
But in (eq:DerivativeOfLepageForm) we have seen that the [[Lagrangian density]]
of a [[Lagrangian field theory]] is just one component, in [[codimension]] 0, of an inhomogeneous
"[[Lepage form]]" which in [[codimension]] 1 is given by the [[presymplectic potential current]] $\Theta_{BFV}$ (eq:PresymplecticPotential).
(This will be conceptually elucidated, after we have introduced the [[local BV-complex]], in example \ref{DerivedPresymplecticCurrentOfRealScalarField} below.) This means that in [[codimension]] 1 we are to
consider infinitesimal [[on-shell]] symmetries of the [[Lepage form]] $\mathbf{L} + \Theta_{BFV}$. These are
known as _[[Hamiltonian vector fields]]_ (def. \ref{HamiltonianForms} below) and the analog of [[Noether's theorem|Noether's theorem I]]
now says that these correspond to _[[Hamiltonian differential forms]]_. The [[Lie algebra]] of these
infinitesimal symmetries is called the _[[Poisson bracket Lie n-algebra|local Poisson bracket]]_ (prop. \ref{LocalPoissonBracket} below).
**[[Noether theorem]] and [[Hamiltonian Noether theorem]]**
| $\,$ [[variational differential forms|variational form]] $\,$ | $\,$ [[symmetry]] $\,$ | $\,$ [[Cartan's homotopy formula|homotopy formula]] $\,$ | $\,$ physical quantity $\,\,\,$ | $\,$ [[Lie n-algebra|local symmetry algebra]] $\,$ |
|----|----------|----------------------------|---------------------------------|------|
| [[Lagrangian density]] $\mathbf{L}$ <br/> (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) | $\mathcal{L}_v \mathbf{L} = d \tilde J$ | $ d(\underset{= J_v}{\underbrace{\tilde J - \iota_v \Theta_{BFV}}}) = \iota_v \, \delta_{EL}\mathbf{L}$ | [[conserved current]] $J_v$ <br/> (def. \ref{SymmetriesAndConservedCurrents}) | [[Dickey bracket]] |
| [[presymplectic current]] $\Omega_{BFV}$ <br/> (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) | $\mathcal{L}^{var}_v \Theta_{BFV} = \delta \tilde H$ | $\delta(\underset{= H_v}{\underbrace{\tilde H_v - \iota_v \Theta_{BFV}}}) = \iota_v \Omega_{BFV}$ | [[Hamiltonian differential form|Hamiltonian form]] $H_v$ <br/> (def. \ref{HamiltonianForms}) | [[Poisson bracket Lie n-algebra|local Poisson bracket]] <br/> (prop. \ref{LocalPoissonBracket}) |
$\,$
In the chapter _[Phase space](#PhaseSpace)_ below we [[transgression|transgress]] this [[Poisson bracket Lie n-algebra|local Poisson bracket]]
of [[infinitesimal symmetries]] of the [[presymplectic potential current]] to the "global" [[Poisson bracket]] on the _[[covariant phase space]]_ (def. \ref{PoissonBracketOnHamiltonianLocalObservables} below). This is the structure which then [further below](#Quantization) leads over to the
[[quantization]] ([[deformation quantization]]) of the [[prequantum field theory]] to a genuine [[perturbative quantum field theory]].
However, it will turn out that there may be an [[obstruction]] to this construction, namely the existence of
special infinitesimal symmetries of the Lagrangian densities, called _implicit [[gauge symmetries]]_ (discussed [further below](#GaugeSymmetries)).
$\,$
**[[infinitesimal symmetries]] of the [[Lagrangian density]]**
{#InfinitesimalSymmetriesOfTheLagrangianDensity}
+-- {: .num_defn #Variation}
###### Definition
**(variation)**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles}).
A _variation_ is a [[vertical vector field]] $v$ on the [[jet bundle]] $J^\infty_\Sigma(E)$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) hence a vector field which
vanishes when evaluated in the [[horizontal differential forms]].
In the special case that the [[field bundle]] is [[trivial vector bundle]] over [[Minkowski spacetime]] as in example \ref{TrivialVectorBundleAsAFieldBundle},
a variation is of the form
$$
v = v^a \partial_{\phi^a} + v^a_{,\mu} \partial_{\phi^a_{,\mu}} + v^a_{\mu_1 \mu_2} \partial_{\phi^a_{\mu_1 \mu_2}} + \cdots
$$
=--
The concept of variation in def. \ref{Variation} is very general, in that it allows to vary the
field coordinates independently from the corresponding jets. This generality is necessary for
discussion of symmetries of [[presymplectic currents]] in def. \ref{HamiltonianForms} below.
But for discussion of symmetries of [[Lagrangian densities]] we are interested in
explicitly varying just the [[field (physics)|field]] coordinates (def. \ref{EvolutionaryVectorField} below)
and inducing from this the corresponding variations of the field derivatives (prop. \ref{EvolutionaryVectorFieldProlongation}) below.
In order to motivate the following definition \ref{EvolutionaryVectorField} of _[[evolutionary vector fields]]_ we follow remark \ref{ReplacingBundleMorphismsByDifferentialOperators} saying that concepts in [[variational calculus]] are obtained from their analogous concepts in plain [[differential calculus]] by replacing plain [[bundle morphisms]] by morphisms out of the [[jet bundle]]:
Given a [[fiber bundle]] $E \overset{fb}{\to} \Sigma$, then a _[[vertical vector field]]_ on $E$ is
a [[section]] of its [[vertical tangent bundle]] $T_\Sigma E$ (def. \ref{VerticalTangentBundle}), hence is a [[bundle morphism]]
of this form
$$
\array{
E && \overset{\text{vertical vector field}}{\longrightarrow} && T_\Sigma E
\\
& {}_{\mathllap{id}}\searrow && \swarrow
\\
&& E
}
$$
The variational version replaces the vector bundle on the left with its jet bundle:
+-- {: .num_defn #EvolutionaryVectorField}
###### Definition
**([[evolutionary vector fields]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles}). Then an _[[evolutionary vector field]]_ $v$ on $E$ is "variational vertical vector field" on $E$, hence a smooth [[bundle]] [[homomorphism]]
out of the [[jet bundle]] (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})
$$
\array{
J^\infty_\Sigma E
&& \overset{v}{\longrightarrow} &&
T_\Sigma E
\\
& {}_{\mathllap{jb_{\infty,0}}}\searrow && \swarrow_{\mathllap{}}
\\
&& E
}
$$
to the [[vertical tangent bundle]] $T_\Sigma E \overset{}{\to} \Sigma$ (def. \ref{VerticalTangentBundle}) of $E \overset{fb}{\to} \Sigma$.
In the special case that the [[field bundle]] is a [[trivial vector bundle]] over [[Minkowski spacetime]] as in example \ref{TrivialVectorBundleAsAFieldBundle}, this means that an evolutionary vector field is
a [[tangent vector field]] (example \ref{TangentVectorFields}) on $J^\infty_\Sigma(E)$ of the special form
$$
\begin{aligned}
v
& =
v^a \partial_{\phi^a}
\\
& =
v^a\left(
(x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots
\right) \partial_{\phi^a}
\end{aligned}
\,,
$$
where the [[coefficients]] $v^a \in C^\infty(J^\infty_\Sigma(E))$ are general [[smooth functions]]
on the [[jet bundle]] (while the cmponents are [[tangent vectors]] along the field coordinates $(\phi^a)$,
but not along the spacetime coordinates $(x^\mu)$ and not along the jet coordinates $\phi^a_{,\mu_1 \cdots \mu_k}$).
We write
$$
\Gamma_E^{ev}\left( T_\Sigma E \right)
\;\in\;
\Omega^{0,0}_\Sigma(E) Mod
$$
for the space of evolutionary vector fields,
regarded as a [[module]] over the $\mathbb{R}$-[[associative algebra|algebra]]
$$
\Omega^{0,0}_\Sigma(E)
\;=\;
C^\infty\left( J^\infty_\Sigma(E) \right)
$$
of [[smooth functions]] on the [[jet bundle]].
=--
An [[evolutionary vector field]] (def. \ref{EvolutionaryVectorField}) describes
an infinitesimal change of field values _depending_ on, possibly, the point in spacetime and the values of the field
and all its derivatives (locally to finite order, by prop. \ref{JetBundleIsLocallyProManifold}).
This induces a corresponding infinitesimal change of the derivatives of the fields, called the _prolongation_
of the evolutionary vector field:
+-- {: .num_prop #EvolutionaryVectorFieldProlongation}
###### Proposition
**(prolongation of [[evolutionary vector field]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[fiber bundle]].
Given an [[evolutionary vector field]] $v$ on $E$ (def. \ref{EvolutionaryVectorField})
there is a unique [[tangent vector field]] $\hat v$ (example \ref{TangentVectorFields}) on the [[jet bundle]] $J^\infty_\Sigma(E)$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) such that
1. $\hat v$ agrees on field coordinates (as opposed to jet coordinates) with $v$:
$$
(jb_{\infty,0})_\ast(\hat v) = v
\,,
$$
which means in the special case that $E \overset{fb}{\to} \Sigma$ is a [[trivial vector bundle]] over [[Minkowski spacetime]]
(example \ref{TrivialVectorBundleAsAFieldBundle})
that $\hat v$ is of the form
$$
\label{GenericComponentsForProlongationOfEvolutionaryVectorField}
\hat v
\;=\;
\underset{
= v
}{
\underbrace{
v^a \partial_{\phi^a}
}}
\,+\,
\hat v^a_{\mu} \partial_{\phi^a_{,\mu}}
+
\hat v^a_{\mu_1 \mu_2} \partial_{\phi^a_{,\mu_1 \mu_2}}
+
\cdots
$$
1. contraction with $\hat v$ (def. \ref{ContractionOfFormsWithVectorFields}) anti-commutes with the [[total derivative|total spacetime derivative]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}):
$$
\label{ProlongedEvolutionaryVectorFieldContractionAnticommutedWithHorizontalDerivative}
\iota_{\hat v} \circ d + d \circ \iota_{\hat v} = 0
\,.
$$
In particular [[Cartan's homotopy formula]] (prop. \ref{CartanHomotopyFormula})
for the [[Lie derivative]] $\mathcal{L}_{\hat v}$ holds with respect to the [[variational derivative]] $\delta$:
$$
\label{HomotopyFormulaForLieDerivativeAlongProlongationOfEvolutionaryVectorField}
\mathcal{L}_{\hat v} = \delta \circ \iota_{\hat v} + \iota_{\hat v} \circ \delta
$$
Explicitly, in the special case that the [[field bundle]] is a [[trivial vector bundle]] over [[Minkowski spacetime]] (example \ref{TrivialVectorBundleAsAFieldBundle}) $\hat v$ is given by
$$
\label{ProlongationOfEvolutionaryVectorFieldExplicit}
\hat v = \underoverset{n = 0}{\infty}{\sum} \frac{d^n v^a}{ d x^{\mu_1} \cdots d x^{\mu_n} } \partial_{\phi^a_{\mu_1 \cdots \mu_n}}
\,.
$$
=--
+-- {: .proof}
###### Proof
It is sufficient to prove the coordinate version of the statement.
We prove this by [[induction]] over the maximal jet order $k$. Notice that the coefficient of $\partial_{\phi^a_{\mu_1 \cdots \mu_k}}$
in $\hat v$ is given by the contraction $\iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_k}$ (def. \ref{ContractionOfFormsWithVectorFields}).
Similarly (at "$k = -1$") the component of $\partial_{\mu_1}$ is given by $\iota_{\hat v} d x^{\mu}$. But by the second condition
above this vanishes:
$$
\begin{aligned}
\iota_{\hat v} d x^\mu
& =
d \iota_{\hat v} x^\mu
\\
& =
0
\end{aligned}
$$
Moreover, the coefficient of $\partial_{\phi^a}$ in $\hat v$ is fixed by the first condition above to be
$$
\iota_{\hat v} \delta \phi^a
=
v^a
\,.
$$
This shows the statement for $k = 0$. Now assume that the statement is true up to some $k \in \mathbb{N}$.
Observe that the coefficients of all $\partial_{\phi^a_{\mu_1 \cdots \mu_{k+1}}}$ are fixed by the
contractions with $\delta \phi^a_{\mu_1 \cdots \mu_{k} \mu_{k+1}} \wedge d x^{\mu_{k+1}}$. For this we find again from the second
condition and using $\delta \circ d + d \circ \delta = 0$ as well as the induction assumption that
$$
\begin{aligned}
\iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_{k+1}} \wedge d x^{\mu_{k+1}}
& =
\iota_{\hat v} \delta d \phi^a_{\mu_1 \cdots \mu_k}
\\
& =
d \iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_k}
\\
&
= d \frac{d^k v^a}{d x^{\mu_1} \cdots d x^{\mu_k}}
\\
& =
\frac{d^{k+1}v^a }{d x^{\mu_1} \cdots d x^{\mu_{k+1}}} d x^{\mu_{k+1}}
\,.
\end{aligned}
$$
This shows that $\hat v$ satisfying the two conditions given exists uniquely.
Finally formula (eq:HomotopyFormulaForLieDerivativeAlongProlongationOfEvolutionaryVectorField) for the [[Lie derivative]] follows from the second of the two conditions with [[Cartan's homotopy formula]]
$\mathcal{L}_{\hat v} = \mathbf{d} \circ \iota_{\hat v} + \iota_{\hat v} \circ \mathbf{d}$ (prop. \ref{CartanHomotopyFormula})
together with $\mathbf{d} = \delta + d$ (eq:VariationalDerivative).
=--
+-- {: .num_prop #EvolutionaryVectorFieldLieAlgebra}
###### Proposition
**([[evolutionary vector fields]] form a [[Lie algebra]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[fiber bundle]]. For any two [[evolutionary vector fields]]
$v_1$, $v_2$ on $E$ (def. \ref{EvolutionaryVectorField}) the [[Lie bracket]] of [[tangent vector fields]]
of their prolongations $\hat v_1$, $\hat v_2$ (def. \ref{EvolutionaryVectorFieldProlongation})
is itself the prolongation $\widehat{[v_1, v_2]}$ of a unique evolutionary vector field $[v_1,v_2]$.
This defines the structure of a [[Lie algebra]] on evolutionary vector fields.
=--
+-- {: .proof}
###### Proof
It is clear that $[\hat v_1, \hat v_2]$ is still [[vertical vector field|vertical]], therefore, by
prop. \ref{EvolutionaryVectorFieldProlongation}, it is sufficient to show that contraction $\iota_{[v_1, v_2]}$ with this
vector field (def. \ref{ContractionOfFormsWithVectorFields}) anti-commutes with the [[horizontal derivative]] $d$,
hence that $[d, \iota_{[\hat v_1, \hat v_2]}] = 0$.
Now $[d, \iota_{[\hat v_1, \hat v_2]}]$ is an
operator that sends vertical 1-forms to horizontal 1-forms and vanishes on
horizontal 1-forms. Therefore it is sufficient to see that this operator
in fact also vanishes on all vertical 1-forms. But for this it is
sufficient that it commutes with the vertical derivative.
This we check by [[Cartan calculus]], using $[d,\delta] = 0$ and $[d, \iota_{\hat v_i}]=0$, by assumption:
$$
\begin{aligned}
{[ \delta, [ d,\iota_{[\hat v_1, \hat v_2]}] ]}
& =
- [d, [\delta, \iota_{[\hat v_1, \hat v_2]}]]
\\
& =
- [d, \mathcal{L}_{[\hat v_1, \hat v_2]}]
\\
& =
-[d, [\mathcal{L}_{\hat v_1}, \iota_{\hat v_2}] ]
\\
& =
- [d, [ [\delta, \iota_{\hat v_1}], \iota_{\hat v_2} ]]
\\
& =
0
\,.
\end{aligned}
$$
=--
Now given an evolutionary vector field, we want to consider the [[flow]] that it
induces on the [[space of field histories]]:
+-- {: .num_defn #FlowOfFieldHistoriesAlongEvolutionaryVectorField}
###### Definition
**([[flow]] of [[field histories]] along [[evolutionary vector field]])**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles})
and let $v$ be an [[evolutionary vector field]] (def. \ref{EvolutionaryVectorField})
such that the ordinary [[flow]] of its prolongation $\hat v$ (prop. \ref{EvolutionaryVectorFieldProlongation})
$$
\exp(t \hat v)
\;\colon\; J^\infty_\Sigma(E) \longrightarrow J^\infty_\Sigma(E)
$$
exists on the [[jet bundle]] (e.g. if the order of derivatives of field coordinates that it depends
on is bounded).
For $\Phi_{(-)} \colon U_1 \to \Gamma_\Sigma(E)$ a collection of [[field histories]]
(hence a plot of the [[space of field histories]] (def. \ref{SupergeometricSpaceOfFieldHistories}) ) the _[[flow]]_ of $v$
through $\Phi_{(-)}$ is the [[smooth function]]
$$
U_1 \times \mathbb{R}^1
\overset{\exp(v)(\Phi_{(-)})}{\longrightarrow}
\Gamma_\Sigma(E)
$$
whose unique factorization $\widehat{\exp(v)}(\Phi_{(-)})$ through the space of jets of field histories
(i.e. the [[image]] $im(j^\infty_\Sigma)$ of [[jet prolongation]], def. \ref{JetProlongation})
$$
\array{
&& im(j^\infty_\Sigma) &\hookrightarrow& \Gamma_\Sigma(J^\infty_\Sigma(E))
\\
& {}^{\mathllap{\widehat{\exp(v)}(\Phi_{(-)})}} \nearrow& \downarrow^{\mathrlap{\simeq}}
\\
U_1 \times \mathbb{R}^1
&\underset{ \exp(v)(\Phi) }{\longrightarrow}&
\Gamma_{\Sigma}(E)_{}
}
$$
takes a plot $t_{(-)} \;\colon\; U_2 \to \mathbb{R}^1$ of the [[real line]]
(regarded as a [[super formal smooth set|super smooth set]] via example \ref{SuperSmoothSetSuperCartesianSpaces}), to the plot
$$
\label{LocalDataForFlowOfImplicitInfinitesimalGaugeSymmetry}
(\exp(t(-) \hat v) \circ j^\infty_\Sigma(\Phi_{(-)})
\;\colon\:
U_1 \times U_2 \longrightarrow \Gamma_\Sigma\left( J^\infty_\Sigma(E) \right)
$$
of the [[smooth space|smooth]] [[space of sections]] of the [[jet bundle]].
(That $\exp(t(-) \hat v)$ indeed flows jet prolongations $j^\infty_\Sigma(\Phi(-))$ again to jet prolongations
is due to its defining relation to the [[evolutionary vector field]] $v$ from prop. \ref{EvolutionaryVectorFieldProlongation}.)
=--
+-- {: .num_defn #SymmetriesAndConservedCurrents}
###### Definition
**([[infinitesimal symmetries of the Lagrangian]] and [[conserved currents]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
Then
1. an _[[infinitesimal symmetry of the Lagrangian]]_ is an [[evolutionary vector field]] $v$ (def. \ref{EvolutionaryVectorField})
such that the [[Lie derivative]] of the [[Lagrangian density]] along its
prolongation $\hat v$ (prop. \ref{EvolutionaryVectorFieldProlongation}) is a [[total spacetime derivative]]:
$$
\mathcal{L}_{\hat v} \mathbf{L} \;=\; d \tilde J_{\hat v}
$$
1. an _[[on-shell]] [[conserved current]]_ is a horizontal $p$-form $J \in \Omega^{p,0}_\Sigma(E)$ (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime})
whose [[total derivative|total spacetime derivative]] vanishes on the [[prolonged shell]] (eq:ShellInJetBundle)
$$
d J\vert_{\mathcal{E}^\infty} \;=\; 0
\,.
$$
=--
+-- {: .num_prop #NoethersFirstTheorem}
###### Proposition
**([[Noether's theorem]] I)**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
If $v$ is an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents})
with $\mathcal{L}_{\hat v} \mathbf{L} = d \tilde J_{\hat v}$, then
$$
\label{NoetherCurrent}
J_{\hat v} \coloneqq \tilde J_{\hat v} - \iota_{\hat v} \Theta_{BFV}
$$
is an [[on-shell]] [[conserved current]] (def. \ref{SymmetriesAndConservedCurrents}), for $\Theta_{BFV}$
a presymplectic potential (eq:PresymplecticPotential) from def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}.
=--
([[Noether's theorem|Noether's theorem II]] is prop. \ref{NoetherIdentities} below.)
+-- {: .proof}
###### Proof
By [[Cartan's homotopy formula]] for the [[Lie derivative]] (prop. \ref{CartanHomotopyFormula})
and the decomposition of the variational derivative $\delta \mathbf{L}$ (eq:dLDecomposition)
and the fact that contraction $\iota_{\hat v}$ with the prolongtion of an evolutionary vector field vanishes on horizontal differential forms
(eq:GenericComponentsForProlongationOfEvolutionaryVectorField) and anti-commutes with the horizontal differential (eq:ProlongedEvolutionaryVectorFieldContractionAnticommutedWithHorizontalDerivative), by def. \ref{EvolutionaryVectorField},
we may re-express the defining equation for the symmetry as follows:
$$
\begin{aligned}
d \tilde J_{\hat v}
& =
\mathcal{L}_{\hat v} \mathbf{L}
\\
& =
\iota_{\hat v} \underset{= \delta_{EL}\mathbf{L} - d \Theta_{BFV}}{\underbrace{\mathbf{d} \mathbf{L}}}
+
\mathbf{d} \underset{= 0}{\underbrace{\iota_v \mathbf{L}}}
\\
& =
\iota_{\hat v} \delta_{EL} \mathbf{L}
+ d \iota_{\hat v} \Theta_{BFV}
\end{aligned}
$$
which is equivalent to
$$
\label{CurrentNoetherConservation}
d(\underset{= J_{\hat v}}{\underbrace{\tilde J_{\hat v} - \iota_{\hat v} \Theta_{BFV}}})
\;=\;
\iota_{\hat v} \delta_{EL}\mathbf{L}
$$
Since, by definition of the [[shell]] $\mathcal{E}$, the differential form on the right
vanishes on $\mathcal{E}$ this yields the claim.
=--
+-- {: .num_example #ScalarFieldEnergyMomentum}
###### Example
**([[energy-momentum]] of the [[scalar field]])**
Consider the [[Lagrangian field theory]] of the [[free field|free]] [[scalar field]] from def. \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}:
$$
\mathbf{L}
\;=\;
\tfrac{1}{2}
\left(
\eta^{\mu \nu}\phi_{,\mu} \phi_{,\nu}
-
m^2 \phi^2
\right)
dvol_\Sigma
\,.
$$
For $\nu \in \{0, 1, \cdots, p\}$ consider the vector field on the jet bundle given by
$$
v_\nu
\;\coloneqq\;
\phi_{,\nu} \partial_{\phi}
+
\phi_{,\mu \nu} \partial_{\phi_{,\mu}}
+
\cdots
\,.
$$
This describes infinitesimal translations of the fields in the direction of $\partial_\nu$.
And this is an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents}), since
$$
\iota_{v_\nu} \mathbf{d}\mathbf{L}
=
d L \wedge \iota_{\partial_\nu} dvol_\Sigma
\,.
$$
With the formula (eq:PresymplecticPotentialOfFreeScalarField) for the presymplectic potential
$$
\Theta_{BFV}
=
\eta^{\mu \nu} \phi_{,\mu} \delta \phi \iota_{\partial_{\nu}} dvol_\Sigma
$$
it hence follows from [[Noether's theorem]] (prop. \ref{NoethersFirstTheorem}) that the corresponding
[[conserved current]] (def. \ref{SymmetriesAndConservedCurrents}) is
$$
\begin{aligned}
T_\nu
& =
L \, \iota_{\partial_\nu} dvol_\Sigma
-
\iota_{v_\nu}\Theta_{BFV}
\\
&
=
L \, \iota_{\partial_\nu} dvol_\Sigma
-
\eta^{\rho \mu} \phi_{,\rho} \phi_{,\nu}
\,
\iota_{\partial_\mu} dvol_\Sigma
\\
& =
(
\underset{=: T^\mu_\nu}{
\underbrace{
\delta^\mu_\nu L
-
\eta^{\rho \mu} \phi_{,\rho} \phi_{,\nu}
}
}
)
\,
\iota_{\partial_\mu} dvol_\Sigma
\end{aligned}
\,.
$$
This [[conserved current]] is called the _[[energy-momentum tensor]]_.
=--
+-- {: .num_example #DiracCurrent}
###### Example
**([[Dirac current]])**
Consider the [[Lagrangian field theory]] of the [[free field theory|free]] [[Dirac field]] on [[Minkowski spacetime]] in spacetime dimension $p + 1 = 3+1$ (example \ref{LagrangianDensityForDiracField})
$$
\mathbf{L} = i \overline{\psi} \gamma^\mu \psi_{,\mu} \, dvol_\Sigma
\,.
$$
Then the prolongation (prop. \ref{EvolutionaryVectorFieldProlongation}) of the [[evolutionary vector field]] (def. \ref{EvolutionaryVectorField})
$$
v \;\coloneqq\; i \psi_\alpha \partial_{\psi_\alpha}
$$
is an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents}). The [[conserved current]] that corresponds to this under [[Noether's theorem|Noether's theorem I]] (prop. \ref{NoethersFirstTheorem}) is
$$
i
\overline{\psi} \gamma^\mu \psi \, \iota_{\partial_\mu} dvol_\Sigma
\;\in\;
\Omega^{p,0}_{\Sigma}(E)
\,.
$$
This is called the _[[Dirac current]]_.
=--
+-- {: .proof}
###### Proof
By equation (eq:ProlongationOfEvolutionaryVectorFieldExplicit) the prolongation of $v$ is
$$
\hat v
=
i \psi_\alpha \partial_{\psi_\alpha}
+
i \psi_{\alpha,\mu} \partial_{\psi_{\alpha,\mu}}
+
\cdots
\,.
$$
Therefore the [[Lagrangian density]] is strictly invariant under the [[Lie derivative]] along $\hat v$
$$
\begin{aligned}
\mathcal{L}_{\hat v} \left(
i \overline{\psi} \gamma^\mu \psi_{,\mu}
\right)
dvol_\Sigma
& =
\underset{
= i \cdot (-i) \overline{\psi} \gamma^\mu \psi_{,\mu}
}{
\underbrace{
i \overline{i \psi} \gamma^\mu \psi_{,\mu}
}
}
dvol_\Sigma
+
\underset{
i \cdot i \overline{\psi} \gamma^\mu \psi_{,\mu}
}{
\underbrace{
i \overline{\psi} \gamma^\mu (i \psi_{,\mu})
}
}
dvol_\Sigma
\\ & =
0
\,.
\end{aligned}
$$
and so the formula for the corresponding conserved current (eq:NoetherCurrent) is
$$
\begin{aligned}
J_v
& =
- \iota_{\hat v}
\left(
\underset{
-
\overline{\psi}
\gamma^\mu
\delta \psi
\,
\iota_{\partial_\mu} dvol_\Sigma }{
\underbrace{
\Theta_{BFV}
}
}
\right)
\\
&
= + i \overline{\psi}\gamma^\mu \psi \, \iota_{\partial_\mu} dvol_\Sigma
\end{aligned}
\,,
$$
where under the brace we used example \ref{PresymplecticCurrentDiracField} to identify the [[presymplectic potential]] for the [[free field theory|free]] [[Dirac field]].
=--
$\,$
Since an [[infinitesimal symmetry of a Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents}) by definition changes the Lagrangian only up to a [[total spacetime derivative]], and since the [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] by construction depend on the [[Lagrangian density]] only up to a [[total spacetime derivative]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}), it is plausible that and [[infinitesimal symmetry of the Lagrangian]] preserves the [[equations of motion]] (eq:EulerLagrangeEquationGeneral), hence the [[shell]] (eq:ProlongedShellInJetBundle). That this is indeed the case is the statement of prop. \ref{InfinitesimalSymmetriesOfLagrangianAreAlsoSymmetriesOfTheEquationsOfMotion} below.
To make the proof transparent, we now first introduce the concept of the _[[evolutionary derivative]]_ (def. \ref{FieldDependentDifferentialOperatorDerivative}) below and then observe that in terms of these the [[Euler-Lagrange derivative]] is in fact a [[derivation]] (prop. \ref{EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}).
+-- {: .num_defn #FieldDependentSections}
###### Definition
**([[field (physics)|field]]-dependent [[sections]])**
For
$$
E \overset{fb}{\longrightarrow} \Sigma
$$
a [[fiber bundle]] (def. \ref{FiberBundle}), regarded as a [[field bundle]] (def. \ref{FieldsAndFieldBundles}), and for
$$
E' \overset{fb'}{\longrightarrow} \Sigma
$$
any other [[fiber bundle]] over the same base space ([[spacetime]]), we write
$$
\Gamma_{J^\infty_\Sigma(E)}(E')
\;\coloneqq\;
\Gamma_{J^\infty_\Sigma(E)}( jb^\ast E' )
\;=\;
Hom_\Sigma(J^\infty_\Sigma(E), E')
\;\simeq\;
DiffOp(E,E')
$$
for the [[space of sections]] of the [[pullback of bundles]] of $E'$ to the [[jet bundle]] $J^\infty_\Sigma(E) \overset{jb}{\longrightarrow} \Sigma$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) along $jb$.
$$
\Gamma_{J^\infty_\Sigma(E)}(E')
\;=\;
\left\{
\array{
&& E'
\\
& {}^{\mathllap{}}\nearrow & \downarrow \mathrlap{fb'}
\\
J^\infty_\Sigma(E)
&\underset{jb}{\longrightarrow}&
\Sigma
}
\phantom{A}\,\,
\right\}
\,.
$$
(Equivalently this is the space of [[differential operators]] from sections of $E$ to sections of $E'$, according to prop. \ref{DifferentialOperator}. )
=--
In ([Olver 93, section 5.1, p. 288](evolutionary+derivative#Olver93)) the field dependent sections of def. \ref{FieldDependentSections}, considered in [[local coordinates]], are referred to as [[tuples]] of _differential functions_.
+-- {: .num_example #EvolutionaryVectorFieldsAsFieldDependentSections}
###### Example
**([[source forms]] and [[evolutionary vector fields]] are field-dependent sections)**
For $E \overset{fb}{\to} \Sigma$ a [[field bundle]], write $T_\Sigma E$ for its [[vertical tangent bundle]] (example \ref{VerticalTangentBundle}) and $T_\Sigma^\ast E$ for its [[dual vector bundle]] (def. \ref{DualVectorBundle}), the [[vertical cotangent bundle]].
Then the field-dependent sections of these bundles according to def. \ref{FieldDependentSections} are identified as follows:
* the space $\Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)$ contains the space of [[evolutionary vector fields]] $v$ (def. \ref{EvolutionaryVectorField}) as those bundle morphism which respect not just the projection to $\Sigma$ but also its factorization through $E$:
$$
\left(
\array{
&& T_\Sigma E
\\
& {}^{\mathllap{v}}\nearrow & \downarrow^{\mathrlap{tb_\Sigma}}
\\
J^\infty_\Sigma(E) &\underset{jb_{\infty,0}}{\longrightarrow}& E & \underset{fb}{\longrightarrow}& \Sigma
}
\right)
\;\in\;
\Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)
$$
* $\Gamma_{J^\infty_\Sigma(E)}( T^\ast_\Sigma E) \otimes \wedge^{p+1}_\Sigma(T^\ast \Sigma)$ contains the space of [[source forms]] $E$ (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) as those bundle morphisms which respect not just the projection to $\Sigma$ but also its factorization through $E$:
$$
\left(
\array{
&& T^\ast_\Sigma E
\\
& {}^{E}\nearrow & \downarrow^{\mathrlap{ctb_\Sigma}}
\\
J^\infty_\Sigma(E) &\underset{jb_{\infty,0}}{\longrightarrow}& E & \underset{fb}{\longrightarrow}& \Sigma
}
\right)
\;\in\;
\Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)
$$
This makes manifest the duality pairing between [[source forms]] and [[evolutionary vector fields]]
$$
\array{
\Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)
\otimes
\Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)
&\longrightarrow&
C^\infty(J^\infty_\Sigma(E))
}
$$
which in local coordinates is given by
$$
(v^a \partial_{\phi^a} \,,\, \omega_a \delta \phi^a)
\mapsto
v^a \omega_a
$$
for $v^a, \omega_a \in C^\infty(J^\infty_\Sigma(E))$ [[smooth functions]] on the [[jet bundle]] (as in prop. \ref{JetBundleIsLocallyProManifold}).
=--
+-- {: .num_defn #FieldDependentDifferentialOperatorDerivative}
###### Definition
**([[evolutionary derivative of field-dependent section]])**
Let
$$
E \overset{fb}{\to} \Sigma
$$
be a [[fiber bundle]] regarded as a [[field bundle]] (def. \ref{FieldsAndFieldBundles}) and let
$$
V \overset{vb}{\to} \Sigma
$$
be a [[vector bundle]] (def. \ref{VectorBundle}). Then for
$$
P \in \Gamma_{J^\infty_\Sigma(E)}(V)
$$
a field-dependent section of $E$ according to def. \ref{FieldDependentSections}, its _evolutionary derivative_ is the morphism
$$
\array{
\Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)
& \overset{ \mathrm{D}P }{\longrightarrow} &
\Gamma_{J^\infty_\Sigma(E)}(V)
\\
v &\mapsto& \hat v(P)
}
$$
which, under the identification of example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}, sense an [[evolutionary vector field]] $v$ to the [[derivative]] of $P$ (example \ref{TangentVectorFields}) along the prolongation [[tangent vector field]] $\hat v $ of $v$ (prop. \ref{EvolutionaryVectorFieldProlongation}).
In the case that $E$ and $V$ are [[trivial vector bundles]] over [[Minkowski spacetime]] with coordinates $((x^\mu), (\phi^a))$ and $((x^\mu), (\rho^b))$, respectively (example \ref{TrivialVectorBundleAsAFieldBundle}), then by (eq:ProlongationOfEvolutionaryVectorFieldExplicit) this is given by
$$
((\mathrm{D}P)(v))^b
\;=\;
\left(
v^a \frac{\partial P^b}{\partial \phi^a}
+
\frac{d v^a}{d x^\mu}
\frac{\partial P^b}{\partial \phi^a_{,\mu}}
+
\frac{d^2 v^a}{d x^\mu d x^\nu}
\frac{\partial P^b}{\partial \phi^a_{,\mu \nu}}
+
\cdots
\right)
$$
This makes manifest that $\mathrm{D}P$ may equivalently be regarded as a $J^\infty_\Sigma(E)$-dependent [[differential operator]] (def. \ref{DifferentialOperator}) from the [[vertical tangent bundle]] $T_\Sigma E$ (def. \ref{VerticalTangentBundle}) to $V$, namely a [[bundle homomorphism]] over $\Sigma$ of the form
$$
\mathrm{D}_P
\;\colon\;
J^\infty_\Sigma(E) \times_\Sigma J^\infty_\Sigma T_\Sigma E
\longrightarrow
V
$$
in that
$$
\label{FrechetDerivativeAsDifferentialOperatorEquality}
\mathrm{D}_P(-,v)
=
\mathrm{D}P(v)
=
\hat v (P)
\,.
$$
=--
([Olver 93, def. 5.24](evolutionary+derivative#Olver93))
+-- {: .num_example #DifferentialOperatorDerivativeOfLagrangianFunction}
###### Example
**([[evolutionary derivative]] of [[Lagrangian function]])**
Over [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}), let $\mathbf{L} = L dvol \in \Omega^{p+1,0}_\Sigma(E)$ be a [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}), with coefficient function regarded as a field-dependent section (def. \ref{FieldDependentSections}) of the [[trivial bundle|trivial]] [[real line bundle]]:
$$
L \;\in \; \Gamma_{J^\infty_\Sigma}(\Sigma \times \mathbb{R})
\,,
$$
Then the [[formally adjoint differential operator]] (def. \ref{FormallyAdjointDifferentialOperators})
$$
(\mathrm{D}_L)^\ast
\;\colon\;
J^\infty_\Sigma(E)\times_\Sigma (\Sigma \times \mathbb{R})^\ast
\longrightarrow
T_\Sigma^\ast E
$$
of its [[evolutionary derivative]], def. \ref{FieldDependentDifferentialOperatorDerivative}, regarded as a $J^\infty_\Sigma(E)$-dependent differential operator $\mathrm{D}_P$ from $T_\Sigma$ to $V$ and applied to the constant section
$$
1 \in \Gamma_\Sigma(\Sigma \times \mathbb{R}^\ast)
$$
is the [[Euler-Lagrange derivative]] (eq:EulerLagrangeEquationGeneral)
$$
\delta_{EL}\mathbf{L}
\;=\;
\left(\mathrm{D}_{L}\right)^\ast(1)
\;\in\;
\Gamma_{J^\infty_\Sigma(E)}(T_\Sigma^\ast)
\simeq
\Omega^{p+1,1}_\Sigma(E)_{source}
$$
via the identification from example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}.
=--
+-- {: .num_prop #EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}
###### Proposition
**([[Euler-Lagrange derivative]] is [[derivation]] via [[evolutionary derivatives]])**
Let $V \overset{vb}{\to} \Sigma$ be a [[vector bundle]] (def. \ref{VectorBundle}) and write $V^\ast \overset{}{\to} \Sigma$ for its [[dual vector bundle]] (def. \ref{DualVectorBundle}).
For field-dependent sections (def. \ref{FieldDependentSections})
$$
\alpha \in \Gamma_{J^\infty_\Sigma(E)}(V)
$$
and
$$
\beta^\ast \in \Gamma_{J^\infty_\Sigma(E)}(V^\ast)
$$
we have that the [[Euler-Lagrange derivative]] (eq:EulerLagrangeEquationGeneral) of their canonical pairing to a [[smooth function]] on the [[jet bundle]] (as in prop. \ref{JetBundleIsLocallyProManifold}) is the sum of the derivative of either one via the [[formally adjoint differential operator]] (def. \ref{FormallyAdjointDifferentialOperators}) of the [[evolutionary derivative]] (def. \ref{FieldDependentDifferentialOperatorDerivative}) of the other:
$$
\delta_{EL}( \alpha \cdot \beta^\ast )
\;=\;
(\mathrm{D}_\alpha)^\ast(\beta^\ast)
+
(\mathrm{D}_{\beta^\ast})^\ast(\alpha)
$$
=--
+-- {: .proof}
###### Proof
It is sufficient to check this in [[local coordinates]]. By the [[product law]] for [[differentiation]] we have
$$
\begin{aligned}
\frac{
\delta_{EL} \left(\alpha \cdot \beta^\ast \right)
}
{
\delta \phi^a
}
& =
\frac{\partial \left(\alpha \cdot \beta^\ast \right)}{\partial \phi^a}
-
\frac{d}{d x^\mu}
\left(
\frac{\partial \left( \alpha \cdot \beta^\ast \right)}{\partial \phi^a_{,\mu}}
\right)
+
\frac{d}{d x^\mu d x^\nu}
\left(
\frac{\partial \left( \alpha \cdot \beta^\ast \right) }{\partial \phi^a_{,\mu \nu}}
\right)
-
\cdots
\\
& =
\phantom{+}
\frac{\partial \alpha }{\partial \phi^a}
\cdot \beta^\ast
-
\frac{d}{d x^\mu}
\left(
\frac{\partial \alpha }{\partial \phi^a_{,\mu}}
\cdot
\beta^\ast
\right)
+
\frac{d}{d x^\mu d x^\nu}
\left(
\frac{\partial \alpha }{\partial \phi^a_{,\mu \nu}}
\cdot
\beta^\ast
\right)
-
\cdots
\\
& \phantom{=}
+
\frac{\partial \beta^\ast }{\partial \phi^a}
\cdot \alpha
-
\frac{d}{d x^\mu}
\left(
\frac{\partial \beta^\ast }{\partial \phi^a_{,\mu}}
\cdot
\alpha
\right)
+
\frac{d}{d x^\mu d x^\nu}
\left(
\frac{\partial \beta^\ast }{\partial \phi^a_{,\mu \nu}}
\cdot
\alpha
\right)
-
\cdots
\\
& =
(\mathrm{D}_\alpha)_a^\ast(\beta^\ast)
+
(\mathrm{D}_{\beta^\ast})_a^\ast(\alpha)
\end{aligned}
$$
=--
+-- {: .num_prop #EvolutionaryDerivativeOfEulerLagrangeFormIsFormallySelfAdjoint}
###### Proposition
**([[evolutionary derivative]] of [[Euler-Lagrange forms]] is [[formally self-adjoint differential operator|formally self-adjoint]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) and regard the [[Euler-Lagrange derivative]]
$$
\delta_{EL}\mathbf{L}
\;=\;
\delta_{EL}L \wedge dvol_\Sigma
$$
(from prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) as a field-dependent section of the [[vertical cotangent bundle]]
$$
\delta_{EL}L
\;\in\;
\Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)
$$
as in example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}. Then the corresponding [[evolutionary derivative]] field-dependent [[differential operator]] $D_{\delta_{EL}L}$ (def. \ref{FieldDependentDifferentialOperatorDerivative}) is [[formally self-adjoint differential operator|formally self-adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}):
$$
(D_{\delta_{EL}L})^\ast
\;=\;
D_{\delta_{EL}L}
\,.
$$
(In terms of the [[Euler-Lagrange complex]], remark \ref{EulerLagrangeComplex}, this says that the [[Helmholtz operator]] vanishes on the image of the [[Euler-Lagrange operator]].)
=--
([Olver 93, theorem 5.92](evolutionary+derivative#Olver93)) The following proof is due to [[Igor Khavkine]].
+-- {: .proof}
###### Proof
By definition of the [[Euler-Lagrange form]] (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) we have
$$
\frac{\delta_{EL} L }{\delta \phi^a}
\delta \phi^a
\,
\wedge
dvol_\Sigma
\;=\;
\delta L \,\wedge dvol_\Sigma \;+\; d(...)
\,.
$$
Applying the [[variational derivative]] $\delta$ (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}) to both sides of this equation yields
$$
\left(\delta \frac{\delta_{EL} L }{\delta \phi^a}\right)
\wedge
\delta \phi^a
\,
\wedge
dvol_\Sigma
\;=\;
\underset{= 0}{\underbrace{\delta \delta L}} \wedge dvol_\Sigma
\;+\;
d(...)
\,.
$$
It follows that for $v,w$ any two [[evolutionary vector fields]] the contraction (def. \ref{ContractionOfFormsWithVectorFields}) of their prolongations $\hat v$ and $\hat w$ (def. \ref{EvolutionaryVectorFieldProlongation}) into the [[variational differential form|differential 2-form]] on the left is
$$
\left(
\delta \frac{\delta_{EL} L }{\delta \phi^a}
\wedge
\delta \phi^a
\right)(v,w)
=
w^a
(\mathrm{D}_{\delta_{EL}})_a(v)
-
v^b(\mathrm{D}_{\delta_{EL}})_b(w)
\,,
$$
by inspection of the definition of the [[evolutionary derivative]] (def. \ref{FieldDependentDifferentialOperatorDerivative}).
Moreover, their contraction into the differential form on the right is
$$
\iota_{\hat v}
\iota_{\hat w}
d(...)
\;=\;
d(...)
$$
by the fact (prop. \ref{EvolutionaryVectorFieldProlongation}) that contraction with prolongations of evolutionary vector fields anti-commutes with the [[total spacetime derivative]] (eq:ProlongedEvolutionaryVectorFieldContractionAnticommutedWithHorizontalDerivative).
Hence the last two equations combined give
$$
w^a
(\mathrm{D}_{\delta_{EL}})_a(v)
-
v^b(\mathrm{D}_{\delta_{EL}})_b(w)
\;=\;
d(...)
\,.
$$
This is the defining condition for $\mathrm{D}_{\delta_{EL}}$ to be [[formally self-adjoint differential operator]] (def. \ref{FormallyAdjointDifferentialOperators}).
=--
$\,$
Now we may finally prove that an [[infinitesimal symmetry of the Lagrangian]] is also an infinitesimal symmetry of the [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]]:
+-- {: .num_prop #InfinitesimalSymmetriesOfLagrangianAreAlsoSymmetriesOfTheEquationsOfMotion}
###### Proposition
**([[infinitesimal symmetries of the Lagrangian]] are also [[infinitesimal symmetries]] of the [[equations of motion]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]]. If an [[evolutionary vector field]] $v$ is an [[infinitesimal symmetry of the Lagrangian]] then the [[flow]] along its prolongation $\hat v$ preserves the [[prolonged shell]] $\mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E)$ (eq:ProlongedShellInJetBundle) in that the [[Lie derivative]] of the [[Euler-Lagrange form]] $\delta_{EL}\mathbf{L}$ along $\hat v$ vanishes on $\mathcal{E}^\infty$:
$$
\mathcal{L}_{\hat v}\mathbf{L} = d(...)
\phantom{AAA}
\Rightarrow
\phantom{AAA}
\mathcal{L}_{\hat v} \, \delta_{EL}\mathbf{L}\vert_{\mathcal{E}^\infty} = 0
\,.
$$
=--
+-- {: .proof}
###### Proof
Notice that for any vector field $\hat v$ the [[Lie derivative]] (prop. \ref{CartanHomotopyFormula})$\mathcal{L}_{\hat v}$ of the [[Euler-Lagrange form]] $\delta_{EL}\mathbf{L} = \frac{\delta_{EL}L}{\delta \phi^a} \delta \phi^a \wedge dvol_\Sigma$ differs from that of its component functions $\frac{\delta_{EL}L}{\delta \phi^a} dvol_\Sigma$ by a term proportional to these component functions, which by definition vanishes on-shell:
$$
\mathcal{L}_{\hat v} \left( \frac{\delta_{EL} L}{\delta \phi^a} \delta \phi^a \wedge dvol_\Sigma \right)
\;=\;
\underset{
= \hat v\left( \frac{\delta_{EL}L}{\delta \phi^a} \right)
}{
\underbrace{
\left(
\mathcal{L}_{\hat v}
\frac{\delta_{EL}L}{\delta \phi^a}
\right)
}
}
\delta \phi^a
\wedge
dvol_\Sigma
+
\underset{
= 0 \, \text{on} \, \mathcal{E}^\infty
}{
\underbrace{
\frac{\delta_{EL}L}{\delta \phi^a}
}
}
\left(
\mathcal{L}_{\hat v} \delta \phi^a
\right)
\wedge dvol_\Sigma
$$
But the Lie derivative of the component functions is just their plain derivative. Therefore it is sufficient to show that
$$
\hat v
\left(
\frac{\delta_{EL} L}{\delta \phi^a}
\right)
\vert_{\mathcal{E}^\infty}
\;=\;
0
\,.
$$
Now by [[Noether's theorem|Noether's theorem I]] (prop. \ref{NoethersFirstTheorem}) the condition $\mathcal{L}_{\hat v} = d \tilde J_{\hat v}$ for an [[infinitesimal symmetry of the Lagrangian]] implies that the contraction (def. \ref{ContractionOfFormsWithVectorFields}) of the [[Euler-Lagrange form]] with the corresponding [[evolutionary vector field]] is a [[total spacetime derivative]]:
$$
\iota_{\hat v} \, \delta_{EL}\mathbf{L}
\;=\;
d J_{\hat v}
\,.
$$
Since the [[Euler-Lagrange derivative]] vanishes on [[total spacetime derivative]] (example \ref{TrivialLagrangianDensities}) also its application on the contraction on the left vanishes. But via example \ref{EvolutionaryVectorFieldsAsFieldDependentSections} that contraction is a pairing of field-dependent sections as in prop. \ref{EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}. Hence we use this proposition to compute:
$$
\label{TowardsProofThatSymmetriesPreserveTheShell}
\begin{aligned}
0
& =
\frac{\delta_{EL} \left( v \cdot \delta_{EL} L\right) }{ \delta \phi^a }
\\
& =
(\mathrm{D}_{v})^\ast_a( \delta_{EL}L )
+
(\mathrm{D}_{\delta_{EL}L})^\ast_a(v)
\\
& =
(\mathrm{D}_{v})^\ast_a( \delta_{EL}L )
+
(\mathrm{D}_{\delta_{EL}L})_a(v)
\\
& =
(\mathrm{D}_{v})^\ast_a( \delta_{EL}L )
+
\hat v\left( \frac{\delta_{EL}L}{\delta \phi^a} \right)
\,.
\end{aligned}
$$
Here the first step is by prop. \ref{EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}, the second step is by prop. \ref{EvolutionaryDerivativeOfEulerLagrangeFormIsFormallySelfAdjoint} and the third step is
(eq:FrechetDerivativeAsDifferentialOperatorEquality).
Hence
$$
\begin{aligned}
\hat v(\delta_{EL}L) \vert_{\mathcal{E}^\infty}
& =
-
(\mathrm{D}_{v})^\ast( \delta_{EL}L ) \vert_{\mathcal{E}^\infty}
\\
& =
0
\end{aligned}
\,,
$$
where in the last line we used that on the [[prolonged shell]] $\delta_{EL}L$ and all its horizontal derivatives vanish, by definition.
=--
As a corollary we obtain:
+-- {: .num_prop #FlowAlongInfinitesimalSymmetryOfLagrangianPreservesOnShellSpaceOfFieldHistories}
###### Proposition
**([[flow]] along [[infinitesimal symmetry of the Lagrangian]] preserves [[on-shell]] [[space of field histories]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
For $v$ an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents})
the [[flow]] on the [[space of field histories]] (example \ref{DiffeologicalSpaceOfFieldHistories}) that it induces by def. \ref{FlowOfFieldHistoriesAlongEvolutionaryVectorField}
preserves the space of [[on-shell]] field histories (from prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}):
$$
\array{
\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
&\hookrightarrow&
\Gamma_\Sigma(E)
\\
{\mathllap{\exp(\hat v)\vert_{\delta_{EL}\mathbf{L} = 0} }}
\uparrow
&&
\uparrow {\mathrlap{\exp(\hat v)}}
\\
\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
&\hookrightarrow&
\Gamma_\Sigma(E)
}
$$
=--
+-- {: .proof}
###### Proof
By def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime} a field history
$\Phi \in \Gamma_\Sigma(E)$ is [[on-shell]] precisely if its [[jet prolongation]] $j^\infty_\Sigma(E)$ (def. \ref{JetProlongation})
factors through the [[shell]] $\mathcal{E} \hookrightarrow J^\infty_\Sigma(E)$ (eq:ShellInJetBundle).
Hence by def. \ref{FlowOfFieldHistoriesAlongEvolutionaryVectorField} the statement is equivalently that
the ordinary flow (prop. \ref{CartanHomotopyFormula}) of $\hat v$ (def. \ref{EvolutionaryVectorFieldProlongation})
on the [[jet bundle]] $J^\infty_\Sigma(E)$ preserves the [[shell]]. This in turn means that it preserves the
vanishing locus of the [[Euler-Lagrange form]] $\delta_{EL} \mathbf{L}$, which is the case by prop. \ref{InfinitesimalSymmetriesOfLagrangianAreAlsoSymmetriesOfTheEquationsOfMotion}.
=--
$\,$
**[[infinitesimal symmetries]] of the [[presymplectic potential current]]**
{#InfinitesimalSymmetriesOfThePresymplecticPotentialCurrent}
Evidently [[Noether's theorem|Noether's theorem I]] in [[variational calculus]] (prop. \ref{NoethersFirstTheorem})
is the special case for horizontal $p+1$-forms of a more general phenomenon relating
symmetries of variational forms to forms that are closed up to a contraction. The same phenomenon applied instead
to the [[presymplectic current]] yields the following:
+-- {: .num_defn #LieDerivativeVariational}
###### Definition
**(variational Lie derivative)**
Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles})
with [[jet bundle]] $J^\infty_\Sigma(E)$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
For $v$ a vertical [[tangent vector field]] on the [[jet bundle]] (a variation def. \ref{Variation})
write
$$
\label{LieDerivativeVariational}
\mathcal{L}^{var}_{v}
\;\coloneqq\;
\delta \circ \iota_v + \iota_v \circ \delta
$$
for the _variational Lie derivative_ along $v$, analogous to [[Cartan's homotopy formula]] (prop. \ref{CartanHomotopyFormula})
but defined in terms of the variational derivative $\delta$ (eq:VariationalDerivative) as opposed to the full [[de Rham differential]].
Then for $v_1$ and $v_2$ two vertical vector fields, write
$$
[v_1, v_2]^{var} \;\in \; \Gamma( T_{vert} J^\infty_\Sigma(E) )
$$
for the vector field whose contraction operator (def. \ref{ContractionOfFormsWithVectorFields}) is given by
$$
\begin{aligned}
\iota_{[v_1,v_2]^{var}}
& =
\left[
\mathcal{L}^{var}_{v_1}, \iota_{v_2}
\right]
\\
& \coloneqq
\mathcal{L}^{var}_{v_1} \circ \iota_{v_2}
-
\iota_{v_2} \circ \mathcal{L}^{var}_{v_1}
\end{aligned}
\,,
$$
=--
+-- {: .num_defn #HamiltonianForms}
###### Definition
**([[Hamiltonian vector fields|infinitesimal symmetry of the presymplectic potential]] and [[Hamiltonian differential forms]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) with [[presymplectic potential current]] $\Theta_{BFV}$ (eq:PresymplecticPotential). Write $\mathcal{E} \hookrightarrow J^\infty_\Sigma(E)$ for the [[shell]] (eq:ShellInJetBundle).
Then:
1. An [[on-shell]] variation $v$ (def. \ref{Variation}) is an _[[infinitesimal symmetry]] of the [[presymplectic current]]_
or _[[Hamiltonian vector field]]_ if [[on-shell]] (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime})
its variational Lie derivative along $v$ (def. \ref{LieDerivativeVariational}) is a [[variational derivative]]:
$$
(\delta \circ \iota_v + \iota_v \circ \delta) \Theta_{BFV} = \delta \tilde H_v
\phantom{AAA}
\text{on}\, \mathcal{E}
$$
for some variational form $\tilde H_v$.
1. A _[[Hamiltonian differential form]]_ $H$ (or _local Hamiltonian current_) is a variational form on the shell such that there
exists a variation $v$ with
$$
\delta H = \iota_v \Omega_{BFV}
\phantom{AA}
\,
\text{on}\, \mathcal{E}
\,.
$$
We write
$$
\Omega^{p,0}_{\Sigma, Ham}(E)
\;\coloneqq\;
\left\{
(H,v)
\;\vert\;
v \, \text{is a variation and}\,
\iota_v \Omega_{BFV} = \delta H
\right\}
$$
for the space of pairs consisting of a Hamiltonian differential forms [[on-shell]] and a corresponding variation.
=--
+-- {: .num_prop #HamiltonianDifferentialForms}
###### Proposition
**([[Hamiltonian Noether's theorem]])**
A variation $v$ is an infinitesimal symmetry of the presymplectic potential (def. \ref{HamiltonianForms})
with $\mathcal{L}^{var}_v ( \Theta_{BFV} ) = \delta \tilde H_v$ precisely if
$$
H_v \coloneqq \tilde H_v - \iota_v \Theta_{BFV}
$$
is a [[Hamiltonian differential form]] for $v$.
=--
+-- {: .proof}
###### Proof
From the definition (eq:LieDerivativeVariational) of $\mathcal{L}^{var}_v$ we have
$$
\begin{aligned}
& \mathcal{L}^{var}_v \Theta_{BFV}
=
\delta \tilde H_v
\\
\Leftrightarrow\;\;
&
\delta \iota_v \Theta_{BFV} + \iota_v \underset{= \Omega_{BFV}}{\underbrace{\delta \Theta_{BFV}}} = \delta \tilde H_v
\\
\Leftrightarrow\;\;
&
\delta \left(
\tilde H_v - \iota_v \Theta_{BFV}
\right)
=
\iota_v \Omega_{BFV}
\,,
\end{aligned}
$$
where we used the definition (eq:PresymplecticCurrent) of $\Omega_{BFV}$ .
=--
$\,$
Since therefore both the [[conserved currents]] from [[Noether's theorem]] as well as
the [[Hamiltonian differential forms]] are generators of infinitesimal [[symmetries]] of certain variational forms
(namely of the [[Lagrangian density]] and of the [[presymplectic current]], respectively) they
form a [[Lie algebra]]. For the conserved currents this is sometimes known as the _[[Dickey bracket]] Lie algebra_.
For the Hamiltonian forms it is the _[[Poisson bracket Lie n-algebra|Poisson bracket Lie p+1-algebra]]_. Since here for simplicity
we are considering just vertical variations, we have just a plain [[Lie algebra]].
The [[transgression of variational differential forms|transgression]] of this Lie algebra of Hamiltonian forms
on the jet bundle to [[Cauchy surfaces]] yields a [[presymplectic structure]] on [[phase space]], this we discuss
[below](#PhaseSpace).
+-- {: .num_prop #LocalPoissonBracket}
###### Proposition
**([[Poisson bracket Lie n-algebra|local Poisson bracket]])**
Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
On the space $\Omega^{p,0}_{\Sigma,Ham}(E)$ pairs $(H,v)$ of [[Hamiltonian differential forms]] $H$ with compatible variation $v$ (def. \ref{HamiltonianForms}) the following operation constitutes a [[Lie bracket]]:
$$
\label{LocalPoissonLieBracket}
\left\{(H_1, v_1),\, (H_2, v_2)\right\}
\;\coloneqq\;
(\iota_{v_1} \iota_{v_2} \Omega_{BFV},\, [v_1,v_2]^{var})
\,,
$$
where $[v_1, v_2]^{var}$ is the variational Lie bracket from def. \ref{LieDerivativeVariational}.
We call this the _local Poisson Lie bracket_.
=--
+-- {: .proof}
###### Proof
First we need to check that the bracket is well defined in itself. It
is clear that it is linear and skew-symmetric, but what needs proof is that
it does indeed land in $\Omega^{p,0}_{\Sigma,Ham}(E)$, hence that the following
equation holds:
$$
\delta \iota_{v_2} \iota_{v_1} \Omega_{BFV}
\;=\;
\iota_{[v_1, v_2]^{var}} \Omega_{BFV}
\,.
$$
With def. \ref{LieDerivativeVariational} for $\mathcal{L}^{var}$ and $[-,-]^{var}$ we compute this as follows:
$$
\begin{aligned}
\delta \iota_{v_1} \iota_{v_2} \Omega_{BFV}
& =
\tfrac{1}{2} \delta \iota_{v_1} \iota_{v_2} \Omega_{BFV}
-
\tfrac{1}{2} (v_1 \leftrightarrow v_2)
\\
& =
\tfrac{1}{2}
\left(
\mathcal{L}^{var}_{v_1} \iota_{v_2} \Omega_{BFV}
-
\iota_{v_1} \delta \iota_{v_2} \Omega_{BFV}
\right)
-
\tfrac{1}{2} (v_1 \leftrightarrow v_2)
\\
& =
\tfrac{1}{2}
\left(
\mathcal{L}^{var}_{v_1} \iota_{v_2} \Omega_{BFV}
-
\iota_{v_1} \mathcal{L}^{var}_{v_2} \Omega_{BFV}
+
\iota_{v_1} \iota_{v_2} \underset{= 0}{\underbrace{\delta \Omega_{BFV}}}
\right)
-
\tfrac{1}{2} (v_1 \leftrightarrow v_2)
\\
& =
[\mathcal{L}^{var}_{v_2}, \iota_{v_1}] \Omega_{BFV}
\\
& =
\iota_{[v_1, v_2]^{var}} \Omega_{BFV}
\,.
\end{aligned}
$$
This shows that the bracket is well defined.
It remains to see that the bracket satifies the [[Jacobi identity]]:
$$
\left\{ (H_1, v_1), \left\{ (H_2, v_2), (H_3,v_3) \right\} \right\}
\;+\; (cyclic)
\;=\;
0
$$
hence that
$$
\left( \iota_{v_1} \iota_{[v_2,v_3]^{var}} \Omega_{BFV} ,\, [v_1, [v_2, v_2]^{var}]^{var} \right)
\;+\; (cyclic)
\;=\; 0
\,.
$$
Here $ [v_1, [v_2, v_3]^{var}]^{var} + (cyclic) = 0 $ holds because by def. \ref{LieDerivativeVariational} $[v_1,-]^{var}$
acts as a derivation,
and hence what remains to be shown is that
$$
\iota_{v_1} \iota_{\left([v_2, v_3]^{var}\right)} \Omega_{BFV} + (cyclic) = 0
$$
We check this by repeated uses of def. \ref{LieDerivativeVariational}, using in addition
that
1. $\delta \iota_{v_i} \Omega_{BFV} = 0$
(since $\iota_{v_i} \Omega_{BFV} = \delta H_i$ by $v_i$ being Hamiltonian)
1. $\mathcal{L}^{var}_{v_i} \Omega_{BFV} = 0$
(since in addition $\delta \Omega_{BFV} = 0$)
1. $\iota_{v_1} \iota_{v_2} \iota_{v_3} \Omega_{BFV} = 0$
(since $\Omega_{BFV} \in \Omega^{p,2}_\Sigma(E)$ is of vertical degree 2, and since all variations $v_i$ are vertical by assumption).
So we compute as follows (a special case of [FRS 13b, lemma 3.1.1](Poisson+bracket+Lie+n-algebra#FRS13b)):
$$
\begin{aligned}
0
& =
\delta \iota_{v_1} \iota_{v_2} \iota_{v_3} \Omega_{BFV}
\\
& =
\mathcal{L}^{var}_{v_1} \iota_{v_2} \iota_{v_3} \Omega_{BFV}
-
\iota_{v_1} \delta \iota_{v_2} \iota_{v_3} \Omega_{BFV}
\\
& =
\iota_{[v_1, v_2]^{var}} \iota_{v_3} \Omega_{BFV}
+
\iota_{v_2} \mathcal{L}^{var}_{v_1} \iota_{v_3} \Omega_{BFV}
-
\iota_{v_1} \mathcal{L}^{var}_{v_2} \iota_{v_3} \Omega_{BFV}
+
\iota_{v_1} \iota_{v_2} \delta \iota_{v_3} \Omega_{BFV}
\\
& =
\iota_{[v_1, v_2]^{var}} \iota_{v_3} \Omega_{BFV}
+
\iota_{v_2} \iota_{[v_1,v_3]^{var}} \Omega_{BFV}
-
\iota_{v_1} \iota_{[v_2, v_3]^{var}} \Omega_{BFV}
\\
& =
-
\iota_{v_1} \iota_{[v_2, v_3]^{var}} \Omega_{BFV}
-
\iota_{v_2} \iota_{[v_3, v_1]^{var}} \Omega_{BFV}
-
\iota_{v_3} \iota_{[v_1, v_2]^{var}} \Omega_{BFV}
\,.
\end{aligned}
$$
=--
$\,$
The [[Poisson bracket Lie n-algebra|local Poisson bracket]] [[Lie algebra]] $(\Omega^{p,0}_{\Sigma,Ham}(E), [-,-]^{var})$ from prop.
\ref{LocalPoissonBracket} is but the lowest stage of
a [[higher Lie theory|higher Lie theoretic]] structure called the _[[Poisson bracket Lie n-algebra|Poisson bracket Lie p-algebra]]_.
Here we will not go deeper into this [[schreiber:Higher Structures|higher structure]] (see at _[[schreiber:Higher Prequantum Geometry]]_
for more), but below we will need the following simple shadow of it:
+-- {: .num_lemma #HorizontallyExactFormsDropOutOfLocalLieBracket}
###### Lemma
The horizontally exact Hamiltonian forms constitute a [[Lie ideal]] for the
local Poisson Lie bracket (eq:LocalPoissonLieBracket).
=--
+-- {: .proof}
###### Proof
Let $E$ be a horizontally exact Hamiltonian form, hence
$$
E = d K
$$
for some $K$. Write $e$ for a [[Hamiltonian vector field]] for $E$.
Then for $(H,v)$ any other pair consisting of a Hamiltonian form and a corresponding
Hamiltonian vector field, we have
$$
\begin{aligned}
\iota_v \, \iota_e \, \Omega_{BFV}
& =
\phantom{-}\iota_v \, \delta E
\\
& =
\phantom{-}\iota_v \, \delta \, d \, K
\\
& =
- \iota_v \, d \, \delta K
\\
& =
\phantom{-}d \, \iota_v \, \delta \, K
\,.
\end{aligned}
$$
Here we used that the horizontal derivative anti-commutes with the
vertical one by construction of the [[variational bicomplex]], and
that $\iota_e$ anti-commutes with the horizontal derivative $d$ since the variation $e$ (def. \ref{Variation}) is
by definition vertical.
=--
+-- {: .num_example #LocalPoissonBracketForRealScalarField}
###### Example
**([[Poisson bracket Lie n-algebra|local Poisson bracket]] for [[real scalar field]])**
Consider the [[Lagrangian field theory]] for the [[free field|free]] [[real scalar field]] from example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}.
By example \ref{FreeScalarFieldEOM} its [[presymplectic current]] is
$$
\Omega_{BFV}
=
\eta^{\mu \nu} \delta \phi_{,\mu} \wedge \delta \phi \wedge \iota_{\partial_\mu} dvol_\Sigma
\,
$$
The corresponding [[Poisson bracket Lie n-algebra|local Poisson bracket algebra]] (prop. \ref{LocalPoissonBracket}) has in degree 0
[[Hamiltonian forms]] (def. \ref{HamiltonianDifferentialForms}) such as
$$
Q \;\coloneqq\; \phi \,\iota_{\partial_0} dvol_\Sigma \in \Omega^{p,0}(E)
$$
and
$$
P \;\coloneqq\; \eta^{\mu \nu} \phi_{,\mu} \, \iota_{\partial_\nu} dvol_{\Sigma} \in \Omega^{p,0}(E)
\,.
$$
The corresponding [[Hamiltonian vector fields]] are
$$
v_Q = -\partial_{\phi_{,0}}
$$
and
$$
v_P = - \partial_{\phi}
\,.
$$
Hence the corresponding [[Poisson bracket Lie n-algebra|local Poisson bracket]] is
$$
\{P,Q\} = \iota_{v_P} \iota_{v_Q} \omega = \iota_{\partial_0} dvol_\Sigma
\,.
$$
More generally for $b_1, b_2 \in C^\infty_{cp}(\Sigma)$ two [[bump functions]] then
$$
\{ b_1 P, b_2 Q \} = b_1 b_2 \iota_{\partial_0} dvol_\Sigma
\,.
$$
=--
+-- {: .num_example #LocalPoissonBracketForDiracField}
###### Example
**([[Poisson bracket Lie n-algebra|local Poisson bracket]] for [[free field theory|free]] [[Dirac field]])**
Consider the [[Lagrangian field theory]] of the [[free field theory|free]] [[Dirac field]] on [[Minkowski spacetime]] (example \ref{LagrangianDensityForDiracField}), whose [[presymplectic current]] is, according to example \ref{PresymplecticCurrentDiracField}, given by
$$
\label{RecallPresymplecticCurrentOfDiracField}
\Omega_{BFV}
\;=\;
(\overline{\delta \psi}) \, \gamma^\mu \, (\delta \psi) \, \iota_{\partial_\mu} dvol_\Sigma
\,.
$$
Consider this specifically in [[spacetime]] [[dimension]] $p + 1 = 4$ in which case the components $\psi_\alpha$ are [[complex number]]-valued (by prop./def. \ref{SpacetimeAsMatrices}), so that the [[tuple]] $(\psi_\alpha)$ amounts to 8 real-valued coordinate functions. By changing complex coordinates, we may equivalently consider $(\psi_\alpha)$ as four coordinate functions, and $(\overline{\psi}^\alpha)$ as another four independent coordinate functions.
Using this coordinate transformation, it is immediate to find the following [[pairs]] of [[Hamiltonian vector fields]] and their [[Hamiltonian differential forms]] from def. \ref{HamiltonianForms} applied to (eq:RecallPresymplecticCurrentOfDiracField)
| [[Hamiltonian vector field]] | [[Hamiltonian differential form]] |
|------------------------------|-----------------------------------|
| $\phantom{AA} \partial_{\psi_\alpha}$ | $\phantom{AA}\left(\overline{\delta \psi}\gamma^\mu\right)^\alpha\, \iota_{\partial_\mu} dvol_\Sigma$ |
| $\phantom{AA} \partial_{\overline{\psi}_\alpha}$ | $\phantom{AA}\left( \gamma^\mu \psi \right)_\alpha \, \iota_{\partial_\mu} dvol_\Sigma$ |
and to obtain the following non-trivial [[Poisson bracket Lie n-algebra|local Poisson brackets]] (prop. \ref{LocalPoissonBracket}) (the other possible brackets vanish):
$$
\left\{
\left( \gamma^\mu \psi \right)_\alpha \, \iota_{\partial_\mu} dvol_\Sigma
\,,\,
\left(\overline{\psi}\gamma^\mu\right)^\beta\, \iota_{\partial_\mu} dvol_\Sigma
\right\}
\;=\;
\left(\gamma^\mu\right)_\alpha{}^{\beta} \, \iota_{\partial_\mu} dvol_\Sigma
\,.
$$
Notice the signs: Due to the odd-grading of the field coordinate function $\psi$,
its variational derivative $\delta \psi$ has bi-degree $(1,odd)$ and the
contraction operation $\iota_{\psi}$ has bi-degree $(-1,odd)$, so that commuting it
past $\overline{\psi}$ picks up _two_ minus signs, a "cohomological" sign due to the differential form
degrees, and a "supergeometric" one (def. \ref{DifferentialFormOnSuperCartesianSpaces}):
$$
\iota_{\partial_\psi} \overline{\delta \psi} \cdots
=
(-1) (-1) \overline{\delta \psi} \,\iota_{\partial_\psi} \cdots
\,.
$$
For the same reason, the [[Poisson bracket Lie n-algebra|local Poisson bracket]]
is a _[[super Lie algebra]]_ with _symmetric_ super Lie bracket:
$$
\left\{
\left( \gamma^\mu \psi \right)_\alpha \, \iota_{\partial_\mu} dvol_\Sigma
\,,\,
\left(\overline{\psi}\gamma^\mu\right)^\beta\, \iota_{\partial_\mu} dvol_\Sigma
\right\}
\;=\;
+
\left\{
\left(\overline{\psi}\gamma^\mu\right)^\beta\, \iota_{\partial_\mu} dvol_\Sigma
\,,\,
\left( \gamma^\mu \psi \right)_\alpha \, \iota_{\partial_\mu} dvol_\Sigma
\right\}
\,.
$$
=--
$\,$
This concludes our discussion of general [[infinitesimal symmetries of a Lagrangian]]. We pick this up again in the discussion of
_[Gauge symmetries](#GaugeSymmetries)_ below. First, in the [next chapter](#Observables) we discuss
the concept of [[observables]] in [[field theory]].
|
A first idea of quantum field theory > history | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+%3E+history |
see _[[geometry of physics -- A first idea of quantum field theory]]_ |
A History of Western Philosophy | https://ncatlab.org/nlab/source/A+History+of+Western+Philosophy |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--
This page collects material related to
* [[Bertrand Russell]],
_A History of Western Philosophy_
1945
on the history of [[philosophy]].
## Chapter 22 -- Hegel
Chapter 22 is on [[Georg Hegel]] and his philosophy as expressed in the _[[Science of Logic]]_ etc.
> {#MysticYouth} In youth he was much attracted to [[mysticism]], and his later views may be regarded, to some extent, as an intellctualizing of what had first appeared to him as a mystic insight.
## Chapter 31 -- The Philosophy of Logical Analysis
([html text](http://www.personal.kent.edu/~rmuhamma/Philosophy/RBwritings/philoLogicalAnaly.htm))
> In philosophy ever since the time of Pythagoras there has been an opposition between the men whose thought was mainly inspired by mathematics and those who were more influenced by the empirical sciences. Plato, Thomas Aquinas, Spinoza, and Kant belong to what may be called the mathematical party; Democritus, Aristotle, and the modern empiricist from Locke onwards, belong to the opposite party. In our day a school of philosophy has arisen which sets to work to eliminate Pythagoreanism from the principles of mathematics, and to combine empiricism with an interest in the deductive parts of human knowledge. The aims of this school are less spectacular than those of most philosophers in the past, but some of its achievements are as solid as those of the men of science.
> The origin of this philosophy is in the achievements of mathematicians who set to work to purge their subject of fallacies and slipshod reasoning. The great mathematicians of the seventeenth century were optimistic and anxious for quick results; consequently they left the foundations of analytical geometry and the infinitesimal calculus insecure. Leibnitz believed in actual infinitesimals, but although this belief suited his metaphysics it has no sound basis in mathematics. Weierstrass, soon after the middle of the nineteenth century, showed how to establish the calculus without infinitesimals, and thus at last made it logically secure. Next came [[Georg Cantor]], who developed the theory of continuity and infinite number. "Continuity" had been, until he defined it, a vague word, convenient for philosophers like [[Hegel]], who wished to introduce metaphysical muddles into mathematics. Cantor gave a precise significance to the word, and showed that continuity, as he defined it, was the concept needed by mathematicians and physicist. By this means a great deal of mysticism, such as that of Bergson, was rendered antiquated.
(Here it is maybe worth remarking that not much later [[infinitesimal objects]] as anticipated by [[Gottfried Leibniz]] were given a neat formal basis in [[mathematics]], by formalisms such as [[synthetic differential geometry]] and [[nonstandard analysis]].)
## References
* Wikipedia, _[A History of Western Philosophy](http://en.wikipedia.org/wiki/A_History_of_Western_Philosophy)_
category: reference |
A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory | https://ncatlab.org/nlab/source/A+mechanization+of+the+Blakers-Massey+connectivity+theorem+in+Homotopy+Type+Theory |
* [[Kuen-Bang Hou (Favonia)]], [[Eric Finster]], [[Dan Licata]], [[Peter LeFanu Lumsdaine]]:
**A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory**
LICS'16 (2016) 565–574
[arXiv:1605.03227](https://arxiv.org/abs/1605.03227)
[doi:10.1145/2933575.2934545](https://doi.org/10.1145/2933575.2934545)
on a [[formal proof]] of the [[Blakers-Massey theorem]] in [[homotopy type theory]].
> **Abstract.** This paper continues investigations in "[[synthetic homotopy theory]]": the use of [[homotopy type theory]] to give [[proof assistant|machine-checked proofs]] of constructions from [[homotopy theory]]. We present a mechanized proof of the [[Blakers-Massey theorem|Blakers-Massey connectivity theorem]], a result relating the higher-dimensional [[homotopy groups]] of a [[pushout type]] (roughly, a space constructed by gluing two spaces along a shared subspace) to those of the components of the pushout. This theorem gives important information about the pushout type, and has a number of useful corollaries, including the [[Freudenthal suspension theorem]], which has been studied in previous formalizations.
The new proof is more elementary than existing ones in abstract homotopy-theoretic settings, and the mechanization is concise and high-level, thanks to novel combinations of ideas from [[homotopy theory]] and [[type theory]].
## See also
* [[Blakers-Massey theorem]]
* [[connectivity]]
category: reference |
A Not-So-Nice Submanifold | https://ncatlab.org/nlab/source/A+Not-So-Nice+Submanifold | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
=--
=--
## Idea ##
As shown in [[evaluation fibration of mapping spaces]] and [[tubular neighbourhoods of mapping spaces]], if we carve out a submanifold of a mapping space by specifying "coincidences", we often get a tubular neighbourhood. On this page, we shall give an example of a submanifold with no tubular neighbourhood. The example is simple to describe. To make it concrete, we shall fix as our source space the circle, $S^1$. For our target space, we shall take a finite dimensional smooth manifold, $M$. The full smooth mapping space, $C^\infty(S^1, M)$ is known as the [[smooth loop space]]. For simplicity, let us take _based_ loops within this, which we write as $\Omega M$. Within that, we consider the space of based smooth maps $S^1 \to M$ which are infinitely flat at the point $1 \in S^1$. Let us write this as $\Omega_♭ M$. As we are using based loops, we can identify the tangent space of $M$ at the basepoint with $\mathbb{R}^n$ and so we have a sequence, which is exact by [[Borel's theorem]]:
\[
\Omega_♭ M \to \Omega M \to \prod_{j = 1}^\infty \mathbb{R}^n
\]
It is easy to show that this does not admit a tubular neighbourhood. If it did, there would be a splitting of the induced map on tangent spaces:
\[
T_\alpha \Omega_♭ M \to \Omega_\alpha M \to \prod_{i = 1}^\infty \mathbb{R}^n
\]
but as the second map is surjective, this cannot split as a splitting map would induce a continuous injection from $\prod_{i=1}^\infty \mathbb{R}^n$ to a normed vector space and that is impossible. |
A Polarized View of String Topology | https://ncatlab.org/nlab/source/A+Polarized+View+of+String+Topology |
This entry is about the article
* [[Ralph Cohen]], [[Veronique Godin]],
_A polarized view of string topology_,
In:
[[Graeme Segal]], [[Ulrike Tillmann]] (eds.)
_Topology, Geometry and Quantum Field Theory_,
LMS, Lecture Notes Series 308, 2004
[arXiv:math/0303003](http://arxiv.org/abs/math/0303003)
[doi:10.1017/CBO9780511526398.008](https://doi.org/10.1017/CBO9780511526398.008)
on realization of closed-[[string topology]] operations as an [[HQFT]].
See also the followup
* [[Veronique Godin]], _Higher string topology operations_ ([arXiv:0711.4859](http://arxiv.org/abs/0711.4859))
for more details and the extension of open-closed [[HQFT]].
And see example 4.2.16 and remark 4.2.17, in
* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_
## Related entries
* [[Sullivan chord diagram]]
category: reference |
A pseudo metric space, i.e. a set with a metric where the distance between two distinct points can also be 0, need not be Haussdorf | https://ncatlab.org/nlab/source/A+pseudo+metric+space%2C+i.e.+a+set+with+a+metric+where+the+distance+between+two+distinct+points+can+also+be+0%2C+need+not+be+Haussdorf | >This is not really the place to ask what seems to be a question. You could ask us on the n-forum or better on [math.stackexchange](http://math.stackexchange.com/) for questions that are high level but not research questions. |
a Serre fibration between CW-complexes is a Hurewicz fibration | https://ncatlab.org/nlab/source/a+Serre+fibration+between+CW-complexes+is+a+Hurewicz+fibration | ## Statement
\begin{theorem}
Let $X$ and $Y$ be [[CW-complexes]] and $f:X\to Y$ a [[Serre fibration]]. Then $f$ is in fact a [[Hurewicz fibration]] (at least in a [[convenient category of spaces]]).
\end{theorem}
## References
This was originally proven in
* Mark Steinberger and James West, *Covering homotopy properties of maps between C.W. complexes or ANRs*, Proc. Amer. Math. Soc. 92 (1984), 573-577, [web page with fulltext PDF](https://www.ams.org/journals/proc/1984-092-04/S0002-9939-1984-0760948-6/)
An error was corrected in
* Robert Cauty, *Sur les ouverts des CW-complexes et les fibrés de Serre*, Colloquium Mathematicae (1992) Volume: 63, Issue: 1, page 1-7, [web page with fulltext PDF](https://eudml.org/doc/210130)
|
A string diagram calculus for predicate logic | https://ncatlab.org/nlab/source/A+string+diagram+calculus+for+predicate+logic |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--
This page is to provide links related to the text
* [[Geraldine Brady]], [[Todd Trimble]],
_A string diagram calculus for predicate logic and C. S. Peirce's System Beta_
([[BradyTrimbleString.pdf:file]])
on [[string diagram]] calculus for [[indexed monoidal categories]] with [[poset]] fibers (monoidal [[hyperdoctrines]]). This is related to [[Charles Peirce]]'s "System beta". Based on this work, a [[string diagram]] calculus for general [[indexed monoidal categories]] is given in
* {#PontoShulman12} [[Kate Ponto]], [[Michael Shulman]], _Duality and traces for indexed monoidal categories_, Theory and Applications of Categories, Vol. 26, 2012, No. 23, pp 582-659 ([arXiv:1211.1555](http://arxiv.org/abs/1211.1555))
category: reference |
A study in derived algebraic geometry | https://ncatlab.org/nlab/source/A+study+in+derived+algebraic+geometry | This entry is about the book (in progress)
* [[D. Gaitsgory]], [[N. Rozenblyum]], _A study in derived algebraic geometry_, [web](http://www.math.harvard.edu/~gaitsgde/GL/).
It develops some aspects of the theory of [[derived algebraic geometry]] with the [[categorical geometric Langlands conjecture]] in mind.
## See also
* [[Dennis Gaitsgory]], _[[Notes on geometric Langlands]]_, [web](http://www.math.harvard.edu/~gaitsgde/GL/).
* [[D. Gaitsgory]] et. al., _Geometric representation theory_, graduate seminar, Fall 2009--Spring 2010, [web](http://www.math.harvard.edu/~gaitsgde/grad_2009/).
## Contents
* Preface
* Introduction to Part I: Preliminaries
* Chapter I.1: Some higher algebra
An introduction to $\infty$-categories and review of Lurie's books
* Chapter I.2: Basics of derived algebraic geometry
Introduces (derived) [[prestacks]], [[stacks]], [[schemes]] and [[Artin stacks]].
* Chapter I.3: Quasi-coherent sheaves on prestacks
Introduces and studies the basic properties of the category of quasi-coherent sheaves on a prestack.
* Introduction to Part II: Ind-coherent sheaves
* Chapter II.1: _Ind-coherent sheaves on schemes_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/IndCohSch.pdf)
Introduces and studies elementary properties of the [[stable (infinity,1)-category]] of [[ind-coherent sheaves]] on [[derived schemes]].
* Chapter II.2: _The !-pullback and base change_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/corrtext.pdf).
Discusses how the system of [[stable (infinity,1)-categories]] of [[ind-coherent sheaves]] satisfies the formalism of [[six operations]].
* Chapter II.3: _Relation between QCoh and IndCoh_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/QCohandIndCoh.pdf).
Discusses the relationship between the [[stable (infinity,1)-categories]] of [[quasi-coherent sheaves]] and [[ind-coherent sheaves]].
* Introduction to Part III: Inf-schemes
* Chapter III.1: _Deformation theory_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/deftext.pdf).
Sets up [[derived deformation theory]].
* Chapter III.2: _(Ind)-inf-schemes_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/InfSchtext.pdf).
Introduces [[inf-schemes]], which are algebro-geometric objects that include [[derived schemes]] and de Rham [[prestacks]] of schemes
* Chapter III.3: _Ind-coherent sheaves on (ind)-inf-schemes_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/InfSchtext.pdf).
Extends the formalism of [[ind-coherent sheaves]] to [[inf-schemes]] and discusses functoriality.
* Chapter III.4: _An application: crystals_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/crystext.pdf).
Studies the [[six operations]] for [[D-modules]].
* Introduction to Part IV: Formal geometry
* Chapter IV.1: _Lie algebras and co-commutative co-algebras_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/Lie.pdf).
Sets up the theory of [[Lie algebras]] from the point of view of [[Koszul-Quillen duality]].
* Chapter IV.2: _Formal moduli_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/formalmoduli.pdf).
Reinterprets [[Jacob Lurie]]'s theory of [[formal moduli problems]] using the language of [[inf-schemes]].
* Chapter IV.3: _Formal groups and Lie algebras_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/exp.pdf).
Explains how to pass between [[Lie algebras]] and [[formal groups]] within the framework of [[derived algebraic geometry]].
* Chapter IV.4: _Lie algebroids_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/algebroids.pdf).
Introduces [[Lie algebroids]] and studies various aspects of [[infinitesimal geometry]].
* Chapter IV.5: Infinitesimal differential geometry
Studies various aspects of infinitesimal geometry, such as the n-th infinitesimal neighborhood.
* Introduction to Part V: Categories of correspondences
* Chapter V.1: The $(\infty,2)$-category of correspondences
Introduces the formalism of correspondences.
* Chapter V.2: Extension theorems for category of correspondences
Extends the theory of IndCoh from schemes to inf-schemes.
* Chapter V.3: The (symmetric) monoidal structure on the category of correspondences
Shows how the formalism of correspondences encodes Serre duality
* Introduction to Part A (appendix on $(\infty,2)$-categories)
* Chapter A.1: Basics of $(\infty,2)$-categories
Defines $(\infty,2)$-categories and introduces some basic constructions
* Chapter A.2: Straightening and Yoneda $(\infty,2)$-categories
Constructs the straightening/unstraightening procedures and the Yoneda embedding
* Chapter A.3: Adjunctions in $(\infty,2)$-categories
Studies the procedure of passing to the adjoint 1-morphism
* References |
A Survey of Cohomological Physics | https://ncatlab.org/nlab/source/A+Survey+of+Cohomological+Physics |
* [[Jim Stasheff]], _[[StasheffCohomPhysicsSurvey.pdf:file]]_, 2009
See also
* {#Stasheff16} [[Jim Stasheff]], _Higher homotopy structures, then and now_, talk at _[Opening workshop](https://www.mpim-bonn.mpg.de/node/6356)_ of _[Higher Structures in Geometry and Physics](https://www.mpim-bonn.mpg.de/node/5883)_, MPI Bonn 2016 ([[StasheffHomotopyStructuresReview.pdf:file]])
#Content#
[[Jim Stasheff]] has spent much of his work on identifying and studying _cohomological_ (as in: [[homological algebra]]) and _homotopical_ (as in: [[homotopy theory]]) structures in [[physics]].
This includes notably the invention and study of homotopy-coherent structures such as [[A-infinity operad]]s, [[A-infinity algebra]]s and [[L-infinity-algebra]]s and their application to [[BV theory]], [[string field theory]] (longer list of links should eventually go here).
The above survey lists key concepts and collects references to further literature.
#Higher categorical background#
Essentially using variants of the [[Dold-Kan correspondence]] one may regard many of these [[differential graded algebra]] structures as [[higher category theory|higher categorical]] structures. For instance the [[pretriangulated dg-category|pretriangulated dg-categories]] formed by homotopy coherent structures are [[presentable (infinity,1)-category|presentations]] for [[stable (infinity,1)-category|stable (infinity,1)-categories]].
In this sense "cohomological physics" is understood as part of [[n-categorical physics]].
category: reference |
A Survey of Elliptic Cohomology | https://ncatlab.org/nlab/source/A+Survey+of+Elliptic+Cohomology |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
+-- {: .standout}
This entry is about the text
* [[Jacob Lurie]], _A Survey of Elliptic Cohomology_, in: _Algebraic Topology_, Abel Symposia Volume 4, 2009, pp 219-277 ([pdf](http://www.math.harvard.edu/~lurie/papers/survey.pdf), [doi:10.1007/978-3-642-01200-6_9](https://doi.org/10.1007/978-3-642-01200-6_9))
It
* reviews basics of [[elliptic cohomology]]
* indicates the construction of the [[tmf]]-[[spectrum]]
* and discusses this in the context of [[equivariant cohomology]].
The central theorem is
* the realization of the moduli space of [[elliptic curve]]s as a [[structured (∞,1)-topos]]
* with a structure [[(∞,1)-sheaf]] of [[E-∞ ring]]-valued functions
* such that the [[tmf]]-[[spectrum]] is the [[E-∞ ring]] of global sections of this structure sheaf.
=--
See also _[[Elliptic Cohomology I]]_ and _[[Chromatic Homotopy Theory]]_.
#Table of Contents#
1. [Summary](#Summary)
The following entry has some paragraphs that summarize central ideas.
1. [gluing all elliptic cohomology theories to the tmf spectrum](#gluingallellipticcohomologytheoriestothetmfspectrum)
1. [interpretation in terms of higher topos theory](#interpretationintermsofhighertopostheory)
1. Partial surveys
These links point to pages that contain notes on aspects of the theory that are in the style of and originate from [a seminar on A Survey of Elliptic Cohomology](http://nd.edu/~rgrady/elliptic_seminar.html):
1. [[A Survey of Elliptic Cohomology - cohomology theories]]
1. [[A Survey of Elliptic Cohomology - formal groups and cohomology]]
1. [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]
1. [[A Survey of Elliptic Cohomology - elliptic curves]]
1. [[A Survey of Elliptic Cohomology - equivariant cohomology]]
1. [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]
1. [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]
1. [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves]]
1. [[A Survey of Elliptic Cohomology - towards a proof]]
1. [[A Survey of Elliptic Cohomology - compactifying the derived moduli stack]]
1. [[A Survey of Elliptic Cohomology - descent ss and coefficients]]
1. towards geometric models
These links point to pages that have an exposition of the Stolz-Teichner program for constructing [[geometric models for elliptic cohomology]].
* Outline of the constructions and statements
* [[Axiomatic field theories and their motivation from topology]]
* [[(1,1)-dimensional Euclidean field theories and K-theory]]
* [[(2,1)-dimensional Euclidean field theories and tmf]]
* Definitions
* [[bordism categories following Stolz-Teichner]]
* [[(2,1)-dimensional Euclidean field theories]]
* [[Geometric Models for Elliptic Cohomology|Geometric models for elliptic cohomology -- Axiomatic field theories and their motivation from topology]]
#Contents#
Here is the table of contents of the Survey reproduced. Behind the links are linked keyword lists for relevant terms.
* table of content
{:toc}
## Summary
{#Summary}
The text starts with showing or recalling that
* the collection of all [[elliptic cohomology]] theories
and
* the gluing of their representing [[spectrum|spectra]] into the single [[tmf]] spectrum
is best understood in terms of global sections of the structure sheaf of functions on the refinement of the moduli [[space]] of all elliptic curves to a [[structured (∞,1)-topos]].
Then it uses this [[higher topos theory|higher topos theoretic]] [[derived algebraic geometry]] perspective to analyze further properties of [[elliptic cohomology]] theories, in particular their refinements to [[equivariant cohomology]].
### Gluing all elliptic cohomology theories to the tmf spectrum {#gluingallellipticcohomologytheoriestothetmfspectrum}
The triple of [[generalized (Eilenberg-Steenrod) cohomology]] theories
1. periodic ordinary [[integral cohomology]]
1. complex [[K-theory]]
1. [[elliptic cohomology]]
constitutes the collection of all possible [[generalized (Eilenberg-Steenrod) cohomology]] theories with the extra [[stuff, structure, property|property]] that they are
* [[multiplicative cohomology theory|multiplicative]]
and
* [[periodic cohomology theory|periodic]].
It so happens that all multiplicative periodic generalized Eilenberg-Steenrod cohomology theories $A$ are characterized by the [[formal group]] (an [[infinitesimal space|infinitesimal]] [[group]]) whose [[ring]] of functions is the [[cohomology ring]] $A(\mathbb{C}P^\infty)$ obtained by evaluating $A$ on the complex projective space $\mathbb{C}P^\infty \simeq \mathcal{B} U(1)$ -- the [[classifying space]] for complex [[line bundle]]s -- and whose group product is induced from the morphism $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$ that representes the [[tensor product]] of complex [[line bundle]]s.
There are precisely three different types of such formal groups:
* the additive formal group (a single one)
* the multiplicative formal group (a single one)
* a formal group defined by an [[elliptic curve]] (many).
The first case corresponds to periodic [[integral cohomology]]. The second corresponds to complex [[K-theory]]. Each element in the third family corresponds to one flavor of [[elliptic cohomology]].
It is therefore natural to subsume all elliptic cohomology theories into one single cohomology theory. This is the theory called [[tmf]].
It turns out that the right way to formalize what "subsume" means in the above sentence involves formulating the way in which an [[elliptic cohomology]] theory is associated to a given [[elliptic curve]] in the correct [[higher category theory|higher categorical]] language:
The collection of all 1-dimensional [[elliptic curve]]s forms a generalized [[space]] $M_{1,1}$ -- a [[stack]] -- defined by the property that it is the [[classifying space]] for elliptic curves in that elliptic curves over a ring $R$ correspond to classifying maps $\phi : Spec R \to M_{1,1}$.
Then the classical assignment of an [[elliptic cohomology]] theory to an [[elliptic curve]] is an assignment
$$
\{\phi : Spec R \to M_{1,1}\}
\to
CohomologyTheories
\,.
$$
We may think of maps $Spec R \to M_{1,1}$ as picking certain subsets of the generalized [[space]] $M_{1,1}$ and of morphisms
$$
\array{
Spec(R) &&\to&& Spec(R')
\\
& \searrow && \swarrow
\\
&& M_{1,1}
}
$$
as maps between such subsets. Hence the assignment of cohomology theories to elliptic curves is much like a [[sheaf]] of cohomology theories on the moduli [[space]] ([[stack]]) of [[elliptic curve]]s.
In order to _glue_ all elliptic cohomology theories in some way one would like to take something like the [[category of elements]] of this [[sheaf]], i.e. its [[homotopy limit]]. In order to say what that should mean, one has to specify the suitable nature of the codomain, the collection of "all cohomology theories".
As emphasized at [[generalized (Eilenberg-Steenrod) cohomology]], the best way to do this is to identify a generalized (Eilenberg-Steenrod) cohomology theory with the [[spectrum]] that represents it. It is and was well known how to do this for each elliptic curve separately. What is not so clear is how this can be done coherently for all elliptic curves at once: we need a lift of the above cohomology-theory-valued sheaf to a sheaf of representing [[spectrum|spectra]]
$$
\array{
&& Spectra
\\
& {}^{?}\nearrow & \;\;\;\downarrow^{represent}
\\
\{\phi : Spec R \to M_{1,1}\}
&\to&
CohomologyTheories
}
\,.
$$
In this generality this turns out to be a hard problem. But by definition here we are really interested just in the special case where all cohomology theories in question are [[multiplicative cohomology theory|multiplicative cohomology theories]] and where hence all spectra in question are [[commutative ring spectrum|commutative ring spectra]]
$$
\array{
&& CommRingSpectra
\\
& {}^{O_{M^{der}}}\nearrow & \downarrow
\\
\{\phi : Spec R \to M_{1,1}\}
&\to&
MultiplicativeCohomologyTheories
}
\,.
$$
As indicated, this problem does turn out to have a solution: [[Paul Goerss|Goerss]], [[Mike Hopkins|Hopkins]] and [[Haynes Miller|Miller]] showed that the desired lift denoted $O_{M^{der}}$ above exists -- the _[[Goerss-Hopkins-Miller theorem]]_
Accordingly, one can then obtain the [[tmf]] spectrum as the [[homotopy limit]] of this sheaf of [[E-∞ ring]]s $O_{M^{der}}$. Recall from the discussion at [[limit in a quasi-category]] that such a homotopy limit computes global sections. It is an $\infty$-version of computing sections in a [[Grothendieck construction]], really, as described there.
### Interpretation in terms of higher topos theory {#interpretationintermsofhighertopostheory}
What is noteworthy about the above construction is that, as the notation above suggests, sheaves of [[E-infinity ring]]s generalize sheaves of rings as thery are familiar from the theory of [[ringed space]]s, where they are called **structure sheaves**.
Accordingly, the morphism $O_{M^{der}}$ makes the moduli [[space]] of elliptic curves into a [[structured (∞,1)-topos]].
This perspective embeds the theory of [[elliptic cohomology]] and of the [[tmf]] spectrum as an application into the general context of [[higher topos theory]] and [[derived algebraic geometry]].
## equivariant elliptic cohomology {#equivariantellipticcohomology}
* [[equivariant elliptic cohomology]]
## Contents
{#Contents}
### 1. Elliptic Cohomology {#EllipticCohomology}
#### 1.1 Cohomology Theories {#CohomologyTheories}
* [[cohomology]]
* [[generalized (Eilenberg-Steenrod) cohomology]]
* [[integral cohomology]]
* [[K-theory]]
* [[elliptic cohomology]]
* [[multiplicative cohomology theory]]
* [[periodic cohomology theory]]
* [[topological modular form]]
#### 1.2 Formal Groups from Cohomology Theories {#FormalgroupsFromCohomologyTheories}
* [[formal group]]
#### 1.3 Elliptic Cohomology {#EllipticCohomology}
* [[elliptic curve]]
* [[elliptic cohomology]]
* [[E-∞ ring]]
### 2 Derived Algebraic geometry {#DerivedAlgebraicCohomology}
* [[derived algebraic geometry]]
* [[higher topos theory]]
* [[higher algebra]]
#### 2.1 $E_\infty$ rings
* [[commutative algebra in an (∞,1)-category]]
* [[E-∞ ring]]
* [[commutative ring spectrum]]
#### 2.2 Derived Schemes
* [[structured (∞,1)-topos]]
* [[derived scheme]]
### 3 Derived Group Schemes and Orientations
* [[group scheme]]
#### 3.1 Orientations of the Multiplicative Group
#### 3.2 Orientations of the Additive Group
#### 3.3 The Geometry of Preorientations
#### 3.4 Equivariant $A$-Cohomology for Abelian Groups
#### 3.5 The Nonabelian Case
### 4 Oriented Elliptic Curves
#### 4.1 Construction of the Moduli Stack
#### 4.2 The Proof of Theorem 4.4: The Local Case
#### 4.3 Elliptic Cohomology near $\infty$
### 5 Applications
#### 5.1 2-Equivariant Elliptic Cohomology
* [[equivariant elliptic cohomology]]
#### 5.2 Loop Group Representations
* [[loop group]]
#### 5.3 String Orientation
* [[string structure]]
* [[sigma-orientation]]
#### 5.4 Higher Equivariance
#### 5.5 Elliptic Cohomology and Geometry
## Further references
{#furtherreferences}
Lots of literature on [[modular form]]s is collected at
* [[Nora Ganter]], [Topological modular forms literature list](http://www.math.uiuc.edu/~ganter/talbot/index.html)
An introduction to and survey of the [[Goerss-Hopkins-Miller-Lurie theorem]] is in
* [[Paul Goerss]], _Topological modular forms (after Hopkins, Miller, and Lurie)_ Séminaire BOURBAKI Mars 2009 61ème année, 2008-2009, no 1005(2009)([arXiv](http://arxiv.org/abs/0910.5130))
which has grown out of
* [[Topological Algebraic Geometry - A Workshop]].
A good bit of details is in
* [[David Gepner]], _[[Homotopy topoi and equivariant elliptic cohomology]]_, 1999 ([pdf](http://www.ms.unimelb.edu.au/~nganter/talbot/gepner.thesisformat.pdf))
|
A Survey of Elliptic Cohomology - A-equivariant cohomology | https://ncatlab.org/nlab/source/A+Survey+of+Elliptic+Cohomology+-+A-equivariant+cohomology |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
+-- {: .standout}
This is a sub-entry of
* [[A Survey of Elliptic Cohomology]]
see there for background and context.
This entry contains a basic introduction to getting equivariant cohomology from [[derived group scheme]]s.
=--
Previous:
* [[A Survey of Elliptic Cohomology - cohomology theories]]
* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]
* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]
* [[A Survey of Elliptic Cohomology - elliptic curves]]
* [[A Survey of Elliptic Cohomology - equivariant cohomology]]
* [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]
Next:
* [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves]]
> the following are rough unpolished notes taken more or less verbatim from some seminar talk -- needs attention, meaning: **somebody should go through this and polish** See also at _[[equivariant elliptic cohomology]]_.
***
#Contents#
* toc
{:toc}
#Derived Elliptic Curves#
**Definition** A [[derived elliptic curve]] over an (affine) [[derived scheme]] $\mathrm{Spec} A$ is a commutative [[derived group scheme]] (CDGS) $E /A$ such that $\overline{E} \to \mathrm{Spec} \pi_0 A$ is an [[elliptic curve]].
Let $A$ be an $E_\infty$-ring. Let $E(A)$ denote the $\infty$-groupoid of oriented elliptic curves over $\mathrm{Spec} A$. Note that $E(A)$ is in particular a space (we will return to this point later).
The point is to prove the following due to Lurie.
**Theorem** The functor $A \mapsto E(A)$ is representable by a derived Deligne-Mumford stack $\mathcal{M}^{Der} = (\mathcal{M} , O_\mathcal{M} )$. Further, $\mathcal{M}$ is equivalent to the topos underlying $\mathcal{M}_{1,1}$ and $\pi_0 O_\mathcal{M} = O_{\mathcal{M}_{1,1}}$. Also, restricting to discrete rings, $O_\mathcal{M}$ provides a lift in sense of Hopkins and Miller.
#$\mathbf{G}$-Equivariant $A$-cohomology
{#equivariant}
##The Strategy##
1. Define $A_{S^1 } (*)$;
1. Extend to $A_{S^1} (X)$ where $X$ is a trivial $S^1$-space;
1. Define $A_T (X)$ where $T$ is a compact abelian Lie group where $X$ is again a trivial $T$-space;
1. Extend to $A_T (X)$ for any (finite enough) $T$-space;
1. Define $A_G (X)$ for $G$ any compact Lie group.
##$S^1$-equivariance##
To accomplish (1) we need a map
$$\sigma : \mathrm{Spf} A^{\mathbb{C} P^\infty} \to \mathbf{G}$$
over $\mathrm{Spec} A$. Then we can define $A_{S^1} (*) = O (\mathbf{G})$. Such a map arises from a completion map
$$A_{S^1} \to A^{\mathbb{C}P^\infty}$$
which we may interpret as a preorientation $\sigma \in \mathrm{Map} (BS^1 , \mathbf{G} (A))$. Recall that such a map $\sigma$ is an orientation if the induced map to the formal completion of $\mathbf{G}$ is an isomorphism.
Recall two facts:
1. There is a bijection $\{ BS^1 \to \mathbf{G} (A) \} \leftrightarrow \{ \mathrm{Spf} A^{BS^1} \to \mathbf{G} \}$;
1. Orientations of the multiplicative group $\mathbf{G}_m$ associated to $A$ are in bijection with maps of $E_\infty$-rings $\{ K \to A\}$, where $K$ is the K-theory spectrum.
**Theorem** We can define equivariant $A$-cohomology using $\mathbf{G}_m$ if and only if $A$ is a $K$-algebra.
##The Abelian Lie Group Case for a Point##
Fix $\mathbf{G}/A$ oriented. Now let $T$ be a compact abelian Lie group. We construct a commutative [[derived group scheme]] $M_T$ over $A$ whose global sections give $A_T$ which is equipped with an appropriate completion map.
**Definition** Define the Pontryagin dual, $\hat T$ of $T$ by $\hat T := \mathrm{Hom}_\mathrm{Lie} (T, S^1)$.
**Examples**
1. $T=T^n$, the $n$-fold torus. Then $\hat T = \mathbb{Z}^n$ as
$$\hat T \ni ( \theta_1 , \dots , \theta_n ) \mapsto (k_1 \theta_1 , \dots , k_n \theta_n ).$$
1. If $T = \{ e \}$, then $\hat T = T$.
**Pontryagin Duality** If $T$ is an abelian, locally compact topological group then $\hat \hat T \simeq T$.
**Definition** Let $B$ be an $A$-algebra. Define $M_T$ by
$$M_T (B) := \mathrm{Hom}_\mathrm{AbTop} (\hat T , \mathbf{G} (B)).$$
Further, $M_T$ is representable.
**Examples**
1. $M_{S^1} (B) = \mathrm{Hom} ( \mathbb{Z} , \mathbf{G} (B)) = \mathbf{G} (B).$
1. $M_{T^n} = \mathbf{G} \times_{\mathrm{Spec} A} \dots \times_{\mathrm{Spec} A} \mathbf{G}.$
1. $M_{\mathbb{Z}/n} = \mathrm{hker} (\times n: \mathbf{G}\to \mathbf{G}).$
1. $M_{\{e\}} (B) = \mathrm{Hom} ( \{e \} , \mathbf{G} (B) = \{e\}$, so $M_{\{e\}}$ is final over $\mathrm{Spec} A$, hence it is isomorphic to $\mathrm{Spec} A$.
How do we get a completion map $\sigma_T : BT \to M_T (A)$ for all $T$ given an orientation $\sigma_{S^1} : BS^1 \to \mathbf{G} (A)$? By a composition: define
$$ Bev: BT \to \mathrm{Hom}(\hat T , BS^1), \; p \mapsto (f \mapsto Bf(p))$$
then define
$$\sigma_T := \sigma_{S^1} \circ Bev.$$
**Proposition**
There exists a map $\hat M$ such that the assignment $T \mapsto M_T$ factors as $T \mapsto \hat M (BT)$. That is the functor $M$ factors through the category of classifying spaces of compact Abelian Lie groups $B(CALG)$ (considered as orbifolds). Further, such factorizations are in bijection with the preorientations of $\mathbf{G}$.
_Proof._ That such a factorization exists defines $\hat M$ on objects. Now by choosing a base point in $BT'$ we have
$$ \mathrm{Hom} (BT , BT' ) \simeq BT' \times \mathrm{Hom} (T, T') $$
as spaces. Now we need a map
$$BT' \to \mathrm{Hom} (M_T , M_{T'}) .$$
Because this map must be functorial in $T$ and $T'$ we can restrict to the universal case where $T$ is trivial and then
$$BT' \to \mathrm{Hom} ( M_{\{e\}} , M_{T'} ) = M_{T'} (A)$$
is just a preorientation $\sigma_{T'}$.
##The Abelian Case for General Spaces##
We will see that $A_T (X)$ is the global sections of a quasi-coherent sheaf on $M_T$.
**Theorem** Let $\mathbf{G}$ be preoriented and $X$ a finite $T$-CW complex. There exist a unique family of functors $\{ F_T \} $ from finite $T$-spaces to the category of quasi-coherent sheaves on $M_T$ such that
1. $F_T$ maps $T$-equivariant (weak) homotopy equivalences to equivalence of quasi-coherent sheaves;
1. $F_T$ maps finite homotopy colimits to finite homotopy limits of quasi-coherent sheaves;
1. $F_T (*) = O (M_\mathbf{G})$;
1. If $T \subset T'$ and $X' = (X \times T' )/T$ then $F_{T'} (X') \simeq f_* (F_T (X))$, where $f: M_T \to M_{T'}$ is the induced map;
1. The $F_T$ are compatible under finite chains of inclusions of subgroups $T \subset T' \subset T'' \dots$.
_Proof._ Use (2) to reduce to the case where $X$ is a $T$-equivariant cell, i.e. $X = T/ T_0 \times D^k$ for some subgroup $T_0 \subset T$. Use (1) to reduce to the case where $X = T/T_0$. Use (3) to conclude that $F_T (T/T_0) = f_* F_{T_0} (*)$. Finally, (4) implies that $F_{T_0} (*) = \hat M (*/T_0 )$, where $\hat M$ is specified by the preorientation.
>For trivial actions there is no dependence on the preorientation.
**Remark**
1. $F_T (X)$ is actually a sheaf of algebras.
1. If $X,Y$ are $T$-spaces then we have maps
$$F_T (X) \to F_T (X \times Y) \leftarrow F_T (Y)$$
and
$$F_T (X) \otimes F_T (Y) \to F_T (X \times Y).$$
1. Define relative version for $X_0 \subset X$ by
$$F_T (X, X_0 ) = hker (F_T (X) \to F_T (X_0 ))$$
and for all $T$-spaces $Y$ we have a map
$$F_T (X, X_0 ) \otimes_{A} F_T (Y) \to F_T (X \times Y , X_0 \times Y).$$
**Definition** $A_T (X) = \Gamma (F_T (X))$ as an $E_\infty$-ring (algebra).
We now verify loop maps on $A_T$.
Recall that in the classical setting $A^n (X)$ is represented by a space $Z_n$ and we have suspension maps $Z_0 \to (S^n \to Z_n)$. Now we need to consider all possible $T$-equivariant deloopings, that is $T$-maps from $S^n \to Z_n$.
**Theorem** Let $\mathbf{G}$ be oriented, $V$ a finite dimensional unitary representation of $T$. Denote by $SV \subset BV$ the unit sphere inside of the unit ball. Define $L_V = F_T (BV, SV)$. Then
1. $L_V$ is a line bundle on $M_T$, i.e. invertible;
1. For all (finite) $T$-spaces $X$ the map
$$L_V \otimes F_T (X) \to F_T (X \times BV, X \times SV)$$
is an isomorphism.
_Proof for $T = S^1 = U(1)$ and $V = \mathbb{C}$._ Then
$$L_V = hker ( F_T (BV) \to F_T (SV)) .$$
As $BV$ is contractible $F_T (BV) = O (\mathbf{G})$ and by property (3) above $F_T (SV) = f_* (O (\mathrm{Spec} A))$ for $f: \mathrm{Spec} A \to \mathbf{G}$ is the identity section. As $\mathbf{G}$ is oriented, $\pi_0 \mathbf{G} / \pi_0 \mathrm{Spec} A$ is smooth of relative dimension 1, so $L_V$ can be though of as the invertible sheaf of ideals defining the identity section of $\mathbf{G}$.
Suppose $V$ and $V'$ are representations of $T$ then $L_V \otimes L_{V'} \to L_{V \oplus V'}$ is an equivalence. So if $W$ is a virtual representation (i.e. $W = U - U'$) then
$L_W = L_U \otimes (L_{U'} )^{-1} .$
**Definition** Let $V$ be a virtual representation of $T$ and define
$$A_T^V (X) = \pi_0 \Gamma (F_T (X) \otimes L_V^{-1}) .$$
>The point is that in order to define equivariant cohomology requires functors $A_G^W$ for all representations of $G$, not just the trivial ones. In the derived setting we obtain this once we have an orientation of $\mathbf{G}$.
##The Non-Abelian Lie Group Case##
Let $A$ be an $E_\infty$-ring, $\mathbf{G}$ an orientated commutative [[derived group scheme]] over $\mathrm{Spec} A$, and $T$ a (not necessarily Abelian) compact Lie group.
**Theorem** There exists a functor $A_T$ from (finite?) $T$-spaces to Spectra which is uniquely characterized by the following.
1. $A_T$ preserves equivalence;
1. For $T_0 \subset T$, $A_{T_0} (X) = A_T ((X \times G) / T_0 )$;
1. $A_T$ maps homotopy colimits to homotopy limits;
1. If $T$ is Abelian, then $A_T$ is defined as above;
1. For all spaces $X$ the map
$$ A_T (X) \to A_T (X \times E^{ab} T) $$
where $E^{ab} T$ is a $T$-space characterized by the requirement that for all Abelian subgroups $T_0 \subset T$, $(E^{ab} T )^{T_0}$ is contractible and empty for $T_0$ not Abelian.
Further, for Borel equivariant cohomology we require
1. If $T = \{e \}$, then $A_T(X) = A(X) = A^X$;
1. $A_T (X) \to A_T (X \times ET)$ is an isomorphism.
_Proof._ In the case of ordinary equivariant cohomology we can use property (5) to reduce to the case where $X$ has only Abelian stabilizer groups. Then via (3) we reduce to $X$ being a colimit of $T$-equivariant cells $D^k \times T/T_0$ for $T_0$ Abelian. Via homotopy equivalence (1) we reduce to $X = T / T_0$. Using property (2) we see $A_T (X) = A_{T_0} (*)$, so (4) yields $A_{T_0} (*) = \hat M (*/T_0 )$
|
A Survey of Elliptic Cohomology - cohomology theories | https://ncatlab.org/nlab/source/A+Survey+of+Elliptic+Cohomology+-+cohomology+theories |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
+-- {: .standout}
This is a sub-entry of
* [[A Survey of Elliptic Cohomology]]
see there for background and context.
This entry reviews basics of [[periodic cohomology theory|periodic]] [[multiplicative cohomology theory|multiplicative]] [[cohomology theory|cohomology theories]] and their relation to formal group laws.
=--
Next:
* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]
## rough notes from a talk##
>the following are rough unpolished notes taken more or less verbatim from some seminar talk -- needs attention
A complex oriented [[cohomology]] theory (meant is here and in all of the following a [[generalized (Eilenberg-Steenrod) cohomology]]) is one with a _good notion of [[Thom class]]es, equivalently first [[Chern class]] for complex [[vector bundle]]_
>(this "good notion" will boil down to certain extra assumptions such as multiplicativity and periodicity etc. What one needs is that the [[cohomology ring]] assigned by the cohomology theory to $\mathbb{C}P^\infty \simeq \mathcal{B}U(1)$ is a power series ring. The formal variable of that is then identified with the universal first Chern class as seen by that theory).
ordinary [[Chern class]] lives in [[integral cohomology]]
$
H^*(-,\mathbb{Z})
$
or in [[K-theory]] $K^*(-)$ where for a [[vector bundle]] $V$ we would set $c_1(V) := ([V]-1)\beta$ where $\beta$ is the [[Bott generator]].
In the first case we have that under [[tensor product]] of [[vector bundle]]s the class behaves as
$$
c_1(V\otimes W) = c_1(V) + c_1(W)
$$
whereas in the second case we get
$$
c_1(V \otimes W) = c_1(V)c_1(W)\beta^{-1} + c_1(V) + c_1(W)
\,.
$$
In general we will get that the [[Chern class]] of a tensor product is given by a certain [[power series]] $E^*(pt)$
not all [[formal group law]]s arises this way. the [[Landweber criterion]] gives a condition under which there is a cohomology theory
**definition** of complex-orientation
there is an
$$
x \in \tilde E^2(\mathbb{C}P^\infty)
$$
such that under the map
$$
\tilde E^2(\mathbb{C}P^\infty)
\to
\tilde E^2(\mathbb{C}P^1)
\simeq
\tilde E^2(S^1)
\simeq E^0({*})
$$
induced by
$\mathbb{C}P^1 \to \mathbb{C}P^\infty$
we have $x \mapsto 1$
**remark** this also gives [[Thom class]]es since
$\mathbb{C}P^\infty \to (\mathbb{C}P^\infty)^\gamma$ is a [[homotopy equivalence]]
$$
\tilde E^2((\mathbb{C}P^\infty)^\gamma)
\simeq
\tilde E^2((\mathbb{C}P^\infty))
\ni X
$$
Thom iso $\tilde H^{*+2}(X^\gamma) \simeq H^*(X)$
...
(here and everywhere the tilde sign is for [[reduced cohomology]])
**definition (Bott element and even periodic cohomology theory)**
* An [[even cohomology theory]] is one whose odd cohomology rings vanish: $E^{2k+1}({*}) = 0$.
* A [[periodic cohomology theory]] is one with a [[Bott element]] $\beta \in E^2({*})$ which is invertible (under multiplication in the [[cohomology ring]] of the point)
so that gives an isomorphism $(-)\cdot \beta : E^*({*}) \simeq E^{*+2}({*})$
Periodic cohomology theories are complex-orientable. $E^*(\mathbb{C}P^\infty)$ can be calculated using the [[Atiyah-Hirzebruch spectral sequence]]
$$
H^p(X, E^q({*})) \Rightarrow E^{p+q}(X)
$$
notice that since $\mathbb{C}P^\infty $ is [[homotopy equivalence|homotopy equivalent]] to the [[classifying space]] $\mathcal{B}U(1)$ (which is a topological group) it has a product on it
$$
\mathbb{C}P^\infty \times \mathbb{C}P^\infty
\to
\mathbb{C}P^\infty
$$
which is the one that induces the [[tensor product]] of [[line bundle]]s classified by maps into $\mathbb{C}P^\infty$.
on (at least on even periodic cohomology theories) this induces a map of the form
$$
\array{
\mathbb{C}P^\infty \times \mathbb{C}P^\infty
&\to&
\mathbb{C}P^\infty
\\
E({*})[[x,y]] &\leftarrow& E(*)[[t]]
\\
f(x,y) &\leftarrow |& t
}
$$
this $f$ is called a [[formal group law]] if the following conditions are satisfied
1. **commutativity** $f(x,y) = f(y,x)$
1. **identity** $f(x,0) = x$
1. **associtivity** f(x,f(y,z)) = f(f(x,y),z)
**remark** the second condition implies that the constant term in the power series $f$ is 0, so therefore all these power series are automatically invertible and hence there is no further need to state the existence of inverses in the formal group. So these $f$ always start as
$$
f(x,y) = x + y + \cdots
$$
The [[Lazard ring]] is the "universal formal group law". it can be presented as by generators $a_{i j}$ with $i,j \in \mathbb{N}$
$$
L = \mathbb{Z}[a_{i j}] / (relations 1-3 below)
$$
and relatins as follows
1. $a_{i j} = a_{j i}$
1. $a_{10} = a_{01} = 1$; $\forall i \neq 0: a_{i 0} = 0$
1. the obvious associativity relation
the universal formal group law we get from this is the power series in $x,y$ with coefficients in the [[Lazard ring]]
$$
\ell(x,y) = \sum_{i,j} a_{i j} x^j y^j \in L[[x,y]]
\,.
$$
**remark** the formal group law is not canonically associated to the cohomology theory, only up to a choice of rescaling of the elements $x$. But the underlying [[formal group]] is independent of this choice and well defined.
For any ring $S$ with formal group law $g(x,y) \in power series in x,y with coefficients in S$ there is a unique morphism $L \to S$ that sends $\ell$ to $g$.
**remark** Quillen's theorem says that the Lazard ring is the ring of complex cobordisms
**some universal cohomology theories** $M U$ is the [[spectrum]] for [[complex cobordism cohomology theory]]. The corresponding [[spectrum]] is in degree $2 n$ given by
$$
M U(2n) = Thom
\left(
standard associated bundle to universal bundle
\array{
E U(n) \\ \downarrow \\ B U(n)
}
\right)
$$
periodic [[complex cobordism cohomology theory]] is given by
$$
M P = \vee_{n \in \mathbb{Z}} \Sigma^{2 n} M U
$$
we get a canonical [[oriented cohomology theory|orientation]] from
$$
\omega : \mathbb{C}P^\infty \stackrel{\simeq}{\to}
M U(1)
\;\;\;\;
M U(\mathbb{C}P^\infty)
$$
this is the universal even periodic cohomology theory with orientation
**Theorem (Quillen)** the [[cohomology ring]] $M P(*)$ of periodic [[complex cobordism cohomology theory]] over the point together with its [[formal group law]] is naturally isomorphic to the universal [[Lazard ring]] with its [[formal group law]] $(L,\ell)$
how one might make a [[formal group law]] $(R,f(x,y))$ into a cohomology theory
use the classifying map $M P({*}) \to R$
to build the [[tensor product]]
$$
E^n(X) := M P^n(X) \otimes_{M P({*})} R
$$
this construction could however break the left exactness condition. However, $E$ built this way will be left exact of the ring morphism $M P({*}) \to R$ is a flat morphism. This is the [[Landweber exactness]] condition (or maybe slightly stronger).
[[!redirects A Survey of Eilliptic Cohomology - cohomology theories]] |
A Survey of Elliptic Cohomology - compactifying the derived moduli stack | https://ncatlab.org/nlab/source/A+Survey+of+Elliptic+Cohomology+-+compactifying+the+derived+moduli+stack | [[!redirects A Survey of Elliptic Cohomology - compactifying the derived moduli space]]
<div class="rightHandSide toc">
[[!include higher algebra - contents]]
</div>
>**Abstract** We sketch how to compactify $M^{Der}$ such that the underlying scheme is the Deligne-Mumford compactification of $M_{1,1}$.
+-- {: .standout}
This is a sub-entry of
* [[A Survey of Elliptic Cohomology]]
see there for background and context.
=--
Here are the entries on the previous sessions:
* [[A Survey of Elliptic Cohomology - cohomology theories]]
* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]
* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]
* [[A Survey of Elliptic Cohomology - elliptic curves]]
* [[A Survey of Elliptic Cohomology - equivariant cohomology]]
* [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]
* [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]
* [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves]]
* [[A Survey of Elliptic Cohomology - towards a proof]]
***
# Compactifying $M^{Der}$#
* automatic table of contents goes here
{:toc}
##Introduction##
Let $\mathbf{M}^{Der}$ be the derived Deligne-Mumford moduli stack of oriented elliptic curves. Recall that the underlying Deligne-Mumford stack $\pi_0 \mathbf{M}^{Der}$ is $\mathbf{M}_{1,1}$ the classical Deligne-Mumford stack which is the fine moduli stack of elliptic curves. We would like to construct a derived Deligne-Mumford stack $\overline{\mathbf{M}^{Der}}$ such that $\pi_0 \mathbf{M}^{Der}$ is the classical compactification $\overline{\mathbf{M}_{1,1}}$.
Recall that one can define the $E_\infty$-ring $tmf[\Delta^{-1}]$ as the global sections of $\OM_{\mathbf{M}^{Der}}$. If we take global sections of $\mathbf{O}_{\overline{\mathbf{M}^{Der}}}$ then we get the $E_\infty$-ring $tmf$.
##The Tate Curve##
Let us focus on elliptic curves over $\mathbb{C}$. The coarse moduli space of such elliptic curves is again $\mathbb{C}$ with the classifying map given by the $j$-invariant. $\mathbb{C}$ is not compact, but we can compactify the moduli space by allowing curves with nodal singularities (generalized elliptic curves).
For each $q \in \mathbb{C}, \; \lvert q \rvert \lt 1$, there is an elliptic curve over $\mathbb{C}$ defined by a Weierstrass equation
$$y^2 + xy = x^3 +a_4(q) x + a_6 (q) .$$
If $0 \lt \lvert q \rvert \lt 1$ the elliptic curve is isomorphic to $\mathbb{C}^* /q^\mathbb{Z}$ as a Riemann surface with group structure induced from $\mathbb{C}^*$ (equivalently, this curve corresponds to $\mathbb{C}/ \Lambda_\tau$ where $e^{2\pi i \tau} = q$). As a function of $q$, $a_4$ and $a_6$ are analytic over the open disk and their power series at $q=0$ have integral coefficients hence the Weierstrass equation defines an elliptic curve $T$ over $\mathbb{Z}[ [q] ]$. We really would like to think of $T$ as an elliptic curve over $\mathbb{Z} ( (q) ):= \mathbb{Z} [ [q] ][q^{-1}]$.
It should be noted that this construction (which goes back to Tate) can be extended to more general fields.
The Tate curve defines a cohomology theory $K_{Tate}$ (an elliptic spectrum). As a cohomology theory $K_{Tate}$ is just $K$-theory tensored with $\mathbb{Z}( (q) )$.
##Annular Field Theories##
As shown in Pokman Cheung's thesis, the Tate curve has a connection to supersymmetric field theories as defined by Stolz and Teichner. The main result of is that a subspace of $2|1$ field theories (annular theories) is the $0th$-space of the spectrum $K_{Tate}$.
Let $SAB$ be the subcategory of $2|1$-EB whose morphisms are annuli. Note that $SAB$ does not contain tori. $SAB$ in essence is completely determined by the supergroup $\mathbb{R}^{2|1}$. Let $AFT_n$ be the space of natural transformations analogous to the definition of Stolz and Teichner.
**Theorem (Theorem 2.2.2 of Cheung).** For each $n \in \mathbb{Z}$, we have
$AFT_n \simeq (K_{Tate})_n .$
_Proof._
Let $A_{l,\tau}$ be the cylinder obtained by identifying non-horizontal sides of the parallelogram in the upper-half plane spanned by $l$ and $l\tau$ for $l \in \mathbb{R}_+$ and $\tau \in h$.
Let $E$ be a degree 0 field theory. The family of cylinders $\{A_{l,\tau} \}$ has the properties that $A_{l, \tau+1} = A_{l , \tau}$ and $A_{l, \tau} \circ A_{l, \tau'} = A_{l, \tau + \tau'}$. For a fixed $l \in \mathbb{R}_+$, $\{E(A_{l,\tau}), \tau \in h \}$ is a commuting family of trace class (and hence compact) operators depending only on $\tau$ modulo $\mathbb{Z}$. Therefore, by writing $q= e^{2 \pi i \tau}$ we can write $E (A_{l, \tau}) = q^{L} q^{\overline{L}}$, where $L, \overline{L}$ are unbounded operators with discrete spectrum.
**Lemma.** The spectrum of $L-\overline{L} \subseteq \mathbb{Z}$. Also, $\overline{L} = G^2$, where $G$ is an odd operator.
_Proof of Lemma._ The spectral argument follows from having an $S^1$ action. To see the second claim note that for fixed $l$, $\mathbb{R}^{2|1}_+ /l \mathbb{Z}$ is a super Lie semi-group and the functor $E$ gives a representation $\mathbb{R}^{2|1}_+ \to \mathrm{End} (E(S^1_l))$ which is compatible with the super-semigroup law on $\mathbb{R}^{2|1}$ given by
$$(z_1 , \overline{z}_1 , \theta_1) , (z_2 , \overline{z}_2 , \theta_2) \mapsto (z_1 + z_2 , \overline{z}_1 +\overline{z}_2 + \theta_1 \theta_2 , \theta_1 + \theta_2) .$$
Now $\mathrm{Lie} (\mathbb{R}^{2|1}_+ ) = \langle \partial_z , \partial_{\overline{z}} , Q \rangle$, where $Q = \partial_\theta + \theta \partial_{\overline{z}}$. Under $E$ the vector fields map to $L, \overline{L},$ and $G$ respectively. Further, $[Q,Q] = 2Q^2 = 2 \partial_{\overline{z}},$ hence $G^2 = \overline{L}$.
We see that a degree 0 theory is determined by a pair of operators $(L, G)$. By analyzing the spectral decomposition of the pair $(G, L-G^2)$ one can construct a weak equivalence of categories $AFT \to V$ (a homotopy equivalence of geometric realizations), where $V$ is a certain category of Clifford-modules. Following work of Segal it is proved that $|V| \simeq (K_{Tate})_0$. The nonzero degrees follow a similar line of reasoning.
##$\overline{\mathbf{M}^{Der}}$##
We can extend the standard toric variety construction to derived schemes by replacing $\mathbb{C}$ by a fixed $E_\infty$-ring $R$. That is given a fan $F= \{U_\alpha \}$ we can build a derived scheme $X_F$ where $X_F = \varinjlim \{U_\alpha\}$ and $U_\alpha = \mathrm{Spec} \; R [S_{\sigma_\alpha}]$ is an affine derived scheme. We will define the (derived) Tate curve as the formal completion of the quotient of a toric variety.
Let $F_0 = \{ \{0\} , \mathbb{Z}_{\ge 0} \}$, so $X_{F_0} = \mathrm{Spec} R [\mathbb{Z}_{\ge 0}] = \mathrm{Spec} R[q]$. Also, let $F= \langle \sigma_n \rangle_{n \in \mathbb{Z}}$, where
$$\sigma_n = \{ (a,b) \in \mathbb{Z} \times \mathbb{Z} \; | \; na \le b \le (n+1) a\}.$$
Note that by projection onto the first factor we have a map of fans $F \to F_0$ and consequently an induced map on varieties $f: X_F \to \mathrm{Spec} \; R[q]$.
Consider $f: X_F \to \mathrm{Spec} \; R[q]$, one can show that
1. $f^{-1} (q) \cong \mathbf{G}_m$ for $q \neq 0$;
1. $f^{-1} (0)$ is an infinite chain of rational curves, each intersecting the next in a node.
Consider the automorphism $\tau : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}$ defined by $\tau (a,b) = (a, b+a)$. Note that $\tau (\sigma_n ) = \tau (\sigma_{n+1})$, so $\tau$ preserves the fan $F$ and consequently is an automorphism of the resulting toric variety $X_F$ which we also denote by $\tau$. Then
1. $\tau$ acts on $f^{-1} (q)$ by multiplication by $q$, for $q \neq 0$;
1. $\tau$ acts freely on $f^{-1}$.
Now define $\widehat{X}_F$ to be the formal completion of $X_F$ along $f^{-1} (0)$. Similarly, define $R[ [q] ]$ as the formal completion of $R[q]$ along $q=0$. One can show that $\tau^{\mathbb{Z}}$ which is the multiplicative group generated by $\tau$ acts freely on $\widehat{X}_F$.
Define $\widehat{T}$ to be the formal (derived) scheme $\widehat{X}_F / \tau^\mathbb{Z}$.
**Theorem (Lurie/Grothendieck).**
The formal derived scheme $\widehat{T}$ is the completion of a unique derived scheme over $\mathrm{Spec} \; R[ [q] ]$. That is, there exists a unique derived scheme $T \to \mathrm{Spec} \; R[ [q] ]$ such that $\widehat{T} = \mathrm{Spf} \; T$.
We call the derived scheme $T$ in the theorem above the _Tate curve_.
It is a fact that the restriction of $T$ to the punctured formal disk $\mathrm{Spec} \; R ( (q) )$ is an elliptic curve isomorphic to $\mathbf{G}_m / q^\mathbb{Z}$.
We then see that an orientation of $T$ is equivalent to an orientation of $\mathbf{G}_m$ which is equivalent to working over the complex $K$-theory spectrum $K$. Therefore, the oriented Tate curve is equivalent to a map
$$T: \mathrm{Spec} \; K( (q) ) \to \mathbf{M}^{Der}.$$
Now note that the involution $\alpha : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}$ defined by $\alpha (a,b) = (a, -b)$ preserves the fan $F$ and $(\tau \circ \alpha)^2 = 1$. We allow $\alpha$ to be complex conjugation on $K((q))$ thought of as a group action of $\{\pm 1\}$, so we have a map $\mathrm{Spec} \; K( (q) ) / \{\pm 1\} \to \mathbf{M}^{Der}$.
We can define a new derived Deligne-Mumford stack by forming an appropriate pushout square.
Finally, one can show that the underlying scheme $\pi_0 \overline{\mathbf{M}^{Der}}$ is $\overline{\mathbf{M}_{1,1}}$.
There are many subtleties associated with $\overline{\mathbf{M}^{Der}}$. For instance, we would like to glue the universal curve $\E$ over $\mathbf{M}^{Der}$ with $T$ to obtain a universal elliptic curve $\overline{\E}$ over $\overline{\mathbf{M}^{Der}}$, however the result is only a _generalized_ elliptic curve; it is not a derived group scheme over $\overline{\mathbf{M}^{Der}}$ as it only has a group structure over the smooth locus of the map $\overline{\E} \to \overline{\mathbf{M}^{Der}}$.
Lurie asserts it is possible to construct the necessary geometric objects over $\overline{\mathbf{M}^{Der}}$ (I guess this will show up in DAG VII or VIII). The global sections of the structure sheaf thus constructed is the spectrum $tmf$. |
A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations | https://ncatlab.org/nlab/source/A+Survey+of+Elliptic+Cohomology+-+derived+group+schemes+and+%28pre-%29orientations | <div class="rightHandSide toc">
[[!include cohomology - contents]]
***
[[!include higher algebra - contents]]
</div>
+-- {: .standout}
This is a sub-entry of
* [[A Survey of Elliptic Cohomology]]
see there for background and context.
This entry contains a basic introduction to derived [[group scheme]]s and their orientations.
=--
Previous:
* [[A Survey of Elliptic Cohomology - cohomology theories]]
* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]
* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]
* [[A Survey of Elliptic Cohomology - elliptic curves]]
* [[A Survey of Elliptic Cohomology - equivariant cohomology]]
Next:
* [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]
* [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]
* [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves]]
>the following are rough unpolished notes taken more or less verbatim from some seminar talk -- needs attention, meaning: **somebody should go through this and polish**
***
#Contents#
* toc
{:toc}
# Introduction #
Recall from last time that given $G$ an algebraic group such that the [[formal spectrum]] $Spf A(\mathbb{C}P^\infty)$ is the completion $\hat G$, we could define $A_{S^1}({*}) := \mathcal{o}_{G}$ then passing to germs gave a completion map
$$
A_{S^1}({*}) \to A(\mathbb{C}P^\infty) =
A^{Bor}_{S^1}({*}).
$$
The problem we (begin) to address here is how to extend this equivariant cohomology to other spaces besides the point. This requires derived algebraic geometry.
# Derived group schemes#
Recall that a commutative [[group scheme]] over a scheme $X$ is a functor
$$
G: \mathrm{Sch} /X^{op} \to \mathrm{Ab}
$$
such that composition with the forgetful functor $F: \mathrm{Ab} \to \mathrm{Set}$ is representable.
We would like extend this definition to the world of [[derived scheme]]s. There are two problems
1. Because of the higher categorical nature of derived schemes Hom sets are spaces.
1. Everything should in the $\infty$-setting, that is defined only up to homotopy.
We will not worry about the second concern and address the first by replacing the category $\mathrm{Ab}$ with $\mathrm{TopAb}$ and $\mathrm{Set}$ with $\mathrm{Top}$.
> The following definition is somewhat restrictive and really should incorporate more of the $\infty$-structure.
**Definition** A commutative [[derived group scheme]] over a [[derived scheme]] $X$ is a topological functor
$$
G: \mathrm{DSch} / X^{op} \to \mathrm{TopAb}
$$
such that composition with the forgetful functor $F: \mathrm{TopAb} \to \mathrm{Top}$ is representable (up to weak equivalence) by an object which is [[flat object|flat]] over $X$.
**Examples**
1. Let $X$ be a scheme, then we have an associated [[derived scheme]] $\overline{X}$. The structure sheaf of $\overline{X}$ is obtained by viewing the structure sheaf of $X$ as a presheaf of $E_\infty$-rings and then sheafifying. We then have an equivalence between commutative [[derived group scheme]]s over $\overline{X}$ and commutative [[group scheme]]s which are flat over $X$.
1. For $X$ a [[derived scheme]] we have a map from commutative [[derived group scheme]]s over $X$ to commutative [[group scheme]]s which are flat over $\pi_0 X$.
#Preorientations#
Throughout $A$ will be an $E_\infty$-ring, $X$ the affine [[derived scheme]] $\mathrm{Spec} A$, $G$ a commutative [[derived group scheme]] over $X$, $A_{S^1}$ the $E_\infty$-ring given by $\Gamma (G)$, and $A^{\mathbb{C}P^\infty}$ the $E_\infty$-ring given by
$$
A^{\mathbb{C}P^\infty} = \mathrm{Hom}_{E_\infty} (\mathbb{C}P^\infty , A).
$$
**Definition**(Preliminary) A preorientation of $G$ is a morphism of commutative [[derived group scheme]]s over $X$
$$
\sigma : \mathrm{Spf} A^{\mathbb{C}P^\infty} \to G,
$$
where $\mathrm{Spf} A^{\mathbb{C}P^\infty}$ is the completion wrt the ideal $\mathrm{ker} (A^{\mathbb{C}P^\infty} \to A^{pt} = A)$. A preorientation is an orientation if the induced map
$$
\hat \sigma : \mathrm{Spf} A^{\mathbb{C}P^\infty} \to \hat G
$$
is an isomorphism.
Suppose that $G$ is affine, then a map
$$
\sigma : \mathrm{Spf} A^{\mathbb{C}P^\infty} \to G
$$
corresponds to a map $A_{S^1} \to A^{\mathbb{C}P^\infty}$
which is the same as a map
$$
\mathbb{C}P^\infty \to \mathrm{Hom} (X, G) = G(X).
$$
Hence we are led to the following definition.
**Definition** Let $X$ be a [[derived scheme]] and $G$ a commutative [[derived group scheme]] over $X$. A preorientation of $G$ is a morphism of topological commutative monoids
$$
\mathbb{P} (\mathbb{C} [\alpha]) = \mathbb{C}P^\infty \to G(X).
$$
Notice that $\mathbb{C}P^\infty$ is nearly freely generated. Indeed it follows from the fundamental theorem of algebra that as a topological monoid $\mathbb{C}P^\infty$ is generated by $\mathbb{C}P^1$ subject to the single relation that $\mathrm{pt} = \mathbb{C}P^0 \subset \mathbb{C}P^1$ is the monoidal unit.
**Proposition** A preorientation up to homotopy is a map
$$S^2 \simeq \mathbb{C}P^1 \to G(X) $$
that is an element of $\pi_2 (G(X))$.
Hence, we always have at least one preorientation: the trivial one which corresponds to $0 \in \pi_2 (G(X))$.
#Orientations#
As motivation recall that a map $s: A \to B$ of 1-dim [[formal group]]s is an isomorphism if and only if $s'$ is invertible. We would like to encode this in our derived language (without defining $s'$).
**Definition** Let $A$ be an $E_\infty$-ring, $G$ a commutative [[derived group scheme]] over $\mathrm{Spec} A$ and $\sigma : S^2 \to G(A)$ a preorientation. Then $\sigma$ is an orientation if
1. $\pi_0 G \to \mathrm{Spec} \pi_0 A$ is smooth of relative dimension 1, and
1. The map induced by $\beta : \omega \to \pi_2 A$
$$
\pi_n A \otimes_{\pi_0 A} \omega \to \pi_{n+2} A$$
is an isomorphism for each $n$.
Note that (2) implies that $A$ is weakly periodic. Conversely, if $A$ is weakly periodic then (2) is equivalent to $\beta$ being an isomorphism.
Before defining $\beta$ and $\omega$ we extend the above definition to [[derived group scheme]]s over an arbitrary [[derived scheme]].
**Definition** Let $X$ be a [[derived scheme]], $G$ a commutative [[derived group scheme]] over $X$ and $\sigma: S^2 \to G(X)$ a preorientation. Then $\sigma$ is an orientation if $(G, \sigma) |_U$ is an orientation for all $U \subset X$ affine.
We now define the module $\omega$ and the map $\beta$. Let $A$ be an $E_\infty$-ring, $G$ a commutative [[derived group scheme]] over $\mathrm{Spec} A$ and let $G_0 = \pi_0 G$ a scheme over $\mathrm{Spec} \pi_0 A$. Let $\Omega$ denote the sheaf of differentials on $G_0 / \pi_0 A$. Then define
$$\omega := i^* \Omega, \; i: \mathrm{Spec} \pi_0 A \to G $$
is the identity section.
Now let $U \hookrightarrow G$ be an open affine subscheme containing the identity section, so $U = \mathrm{Spec} B$ for some $E_\infty$-ring $B$. Then $\sigma$ induces a map
$B \to A^{S^2}$ which is the same as a map
$$
\pi_0 B \to A(S^2)= \pi_0 A \oplus \pi_2 A .
$$
It is a fact that the map $\pi_0 B \to \pi_2 A$ is a derivation over $\pi_0 A$ and hence has a classifying map which yields a map
$$\beta : \omega \to \pi_2 A .$$
#The Multiplicative Derived Group Scheme#
The naive guess for $G_m$ is $GL_1$, where $GL_1 (A) = A^x = (\pi_0 A)^x$. It is true that $GL_1$ is a [[derived scheme]] over $\mathrm{Spec} \mathbf{S}$, however it is not flat, nor is $GL_1 (A)$ an Abelian group as $A$ is an $E_\infty$-ring and not an honestly commutative ring.
If $A$ is rational, that is there is a map $H \mathbb{Q} \to A$, then $GL_1 (A)$ can be given an Abelian group structure. Hence, $GL_1$ is a perfectly good [[group scheme]] defined on the category of rational $E_\infty$-rings, however this category is too small; there are too few rational $E_\infty$-rings.
Recall that for a ring $R$, $R^x = \mathrm{Hom}_R (R [ t, t^{-1} ] , R)$. Further, recall that for a group $M$ we can form the group algebra $R[M]$ which is really a [[Hopf algebra]]. Then $\mathrm{Spec} R[M]$ is a [[group scheme]] over $\mathrm{Spec} A$. Further, $R[m]$ is characterized by
$$
\mathrm{Ring} ( R[M] , B) = \mathrm{Ring} (R, B) \times \mathrm{Mon} (M,B)
$$
where $\mathrm{Mon}$ is the category of monoids and the ring $B$ is thought of as a monoid wrt to multiplication. Motivated by these observations we make the following definitions.
**Definition** Let $A$ be an $E_\infty$-ring and $M$ a topological Abelian monoid, then we can define $A[M]$ which is characterized by
$$
\mathrm{Alg}_A (A[M], B) \simeq \mathrm{Alg}_A (A,B) \times \mathrm{TopMon} (M, B^\times ).
$$
Recall that because of the higher categorical nature of things, the hom-sets above are spaces and the symbol $\simeq$ indicates weak equivalence of spaces.
**Definition** Let $A$ be an $E_\infty$-ring. We define the multiplicative group corresponding to $A$ as
$$
G_m = \mathrm{Spec} A[ \mathbb{Z}].
$$
$G_m$ is a derived commutative group scheme over $\mathrm{Spec} A$.
Note that $\pi_* ( A[ \mathbb{Z}]) = (\pi_* A) [ \mathbb{Z}]$. Also, the map $\pi_0 G_m \to \pi_0 \mathrm{Spec} A$ is smooth of relative dimension 1.
##Preorientations of $G_m$##
**Proposition** For any $E_\infty$-ring $A$, we have a bijection (of sets) between preorientations of $G_m$ and maps $\mathbf{S} [ \mathbb{C} P^\infty ] \to A$.
The proof follows from the fact that $\mathbf{S}$ is initial in the category of $E_\infty$-rings and the mapping property of $A [ \mathbb{Z}]$.
**Corollary** $\mathrm{Spec} \mathbf{S} [ \mathbb{C} P^\infty ]$ is the moduli space of preorientations of $G_m$. That is, if $G_m$ is defined over $\mathrm{Spec} A$, then a preorientation of $G_m$ is the same as a map $\mathrm{Spec} A \to \mathrm{Spec} \mathbf{S} [ \mathbb{C} P^\infty ]$.
##Orientations of $G_m$##
We consider the map $\beta : \omega \to \pi_2 A$ where
$$
\omega = i^* \Omega^1_{\pi_0 G / \pi_0 \mathrm{Spec} A}
$$
and $i$ is the identity section. Note that $\pi_0 G_m = (\pi_0 A ) [t, t^{-1}]$, hence it follows that $\omega$ is canonically trivial, so an orientation is just an element $\beta_\sigma \in \pi_2 A$ such that $\beta_\sigma$ is invertible in $\pi_* A$.
Let $\beta$ denote the (universal) orientation of $\mathbf{S} [ \mathbb{C} P^\infty]$. Then we have the following.
**Theorem** $\mathrm{Spec} \mathbf{S} [ \mathbb{C} P^\infty] [ \beta^{-1}]$ is the moduli space of orientations of $G_m$.
It is a theorem of Snaith, that this moduli space has the homotopy type of $KU$ the spectrum of complex K-theory. Note that by considering the homtopy fixed points of a certain action there is a way to recover $KO$ as well.
##Connection to complex orientation##
Let $A$ be an $E_\infty$-ring, so in particular $A$ defines a cohomology theory. An orientation of $G_m$ over $\mathrm{Spec} A$ is a map $KU \to A$. A complex orientation of $A$ is a map $MU \to A$. Recalling that $KU$ is complex oriented, we see that an orientation of $G_m$ gives a complex orientation by precomposing with the map $MU \to KU$.
#The Additive Derived Group Scheme#
The naive definition of $G_a$ is $\mathbf{A}^1$, where $\mathbf{A}^1 (A)$ is the additive group of $A$. It is true that $\mathbf{A}^1$ is a [[derived scheme]] over $\mathrm{Spec} \mathbf{S}$, however it is not flat as for an $E_\infty$-ring $A$
$$
\pi_k \mathbf{A}^1_A = \oplus_{n \ge 0} A^{-k} (B \Sigma_n )
$$
where as if it were flat we would have
$$
\pi_k \mathbf{A}^1_A = \pi_k A [x] .
$$
Also, $\mathbf{A}^1$ is not commutative. $\mathbf{A}^1 (A)$ is an infinite loop space, but not an Abelian monoid. Again $\mathbf{A}^1$ is a derived group scheme when restricted to rational $E_\infty$-rings.
We no restrict to the category of integral $E_\infty$-rings, i.e. those equipped with a map $H \mathbb{Z} \to A$. Note that in this category $H \mathbb{Z}$ is initial.
**Definition** For $A$ an integral $E_\infty$-ring define
$$
G^A_a = \mathrm{Spec} ( A \otimes_\mathbb{Z} \mathbb{Z} [x] ).
$$
It can be shown that $G^A_a$ is flat and has the correct amount of commutativity.
>Why can't we just use $\mathrm{Spec} A [ \mathbb{N}]$?
**Proposition** For all integral $E_\infty$-rings, preorientations of $G_a^A$ are in bijective correspondence with maps $H \mathbb{Z} [ \mathbb{C} P^\infty] \to A$. Consequently, $\mathrm{Spec} H \mathbb{Z} [ \mathbb{C} P^\infty]$ is the moduli space of preorientations of $G_a$.
Now, $\pi_* H \mathbb{Z} [ \mathbb{C} P^\infty] = H_* (\mathbb{C} P^\infty , \mathbb{Z} )$. The right side is a free divided power series on a generator $\beta$ where $\beta \in \pi_2 H \mathbb{Z} [ \mathbb{C} P^\infty]$.
**Proposition** $\mathrm{Spec} H \mathbb{Z} [ \mathbb{C} P^\infty ] [ \beta^{-1}]$ is the moduli space of orientations of $G_a$.
**Proposition** $\mathrm{Spec} H \mathbb{Z} [ \mathbb{C} P^\infty ] [ \beta^{-1}] = KU \otimes \mathbb{Q}$. Hence the Chern character yields an isomorphism with rational periodic cohomology.
|
A Survey of Elliptic Cohomology - descent ss and coefficients | https://ncatlab.org/nlab/source/A+Survey+of+Elliptic+Cohomology+-+descent+ss+and+coefficients | [[!redirects A Survey of Elliptic Cohomology - compactifying the derived moduli space]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
>**Abstract** This entry discusses the [[descent spectral sequence]] and sheaves in homotopy theory. Using said spectral sequence we compute $\pi_* tmf_{(3)}$.
+-- {: .standout}
This is a sub-entry of
* [[A Survey of Elliptic Cohomology]]
see there for background and context.
=--
Here are the entries on the previous sessions:
* [[A Survey of Elliptic Cohomology - cohomology theories]]
* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]
* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]
* [[A Survey of Elliptic Cohomology - elliptic curves]]
* [[A Survey of Elliptic Cohomology - equivariant cohomology]]
* [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]
* [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]
* [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves]]
* [[A Survey of Elliptic Cohomology - towards a proof]]
* [[A Survey of Elliptic Cohomology - compactifying the derived moduli stack]]
***
#The Descent Spectral Sequence#
* automatic table of contents goes here
{:toc}
##The spectral sequence##
We would like to understand the following theorem.
**Theorem.** Let $( X, \mathbf{O})$ be a derived Deligne-Mumford stack. Then there is a spectral sequence
$$H^s (X ; \pi_t \mathbf{O}) \Rightarrow \pi_{t-s} \Gamma (X , \mathbf{O}).$$
###Recalling what is what###
Let $X$ be an $\infty$-topos, heuristically $X$ is ''sheaves of spaces on an $\infty$-category $C$.'' Further $\mathbf{O}$ is a functor $\mathbf{O} : \{ E_\infty \}^{op} \to X$, which for a cover $U$ of $C$ formally assigns
$$A \mapsto (U \mapsto \mathrm{Hom} (A , \mathbf{O} (U))).$$
Via DAG V 2.2.1 we can make sense of global sections and $\Gamma (X , \mathbf{O})$ is an $E_\infty$-ring.
Given an $\infty$-category $C$ we can form the subcategory of $n$ _truncated objects_ $\tau_{\le n} C$ which consists of all objects such that all mapping spaces have trivial homotopy groups above level $n$. Further $\tau_{\le n} : C \to C$ defines a functor which serves the role of the Postnikov decomposition.
Let $X$ be an $\infty$-topos, define $\mathrm{Disc} \; X : = \tau_{\le 0} X$. Further define functors $\pi_n : X_* \to N (\mathrm{Disc} \; X)$ by
$$Y \mapsto \tau_{\le 0} \mathrm{Map} (S^n , Y) = \pi_n (Y) .$$
**Facts.**
1. For $A \in \mathrm{Disc} \; X$ an abelian group object there exists $K(A,n) \in X$, such that
$$ H^n (X, A) := \pi_0 \mathrm{Map} (1_X , K(A,n)) $$
corresponds to sections of $K(A,n)$ along the identity of $C$.
1. If $C$ is an ordinary site, $H^n (X, A)$ corresponds to ordinary sheaf cohomology (HTT 7.2.2.17).
###The non-derived descent ss###
Let us define a mapping space $\mathrm{Tot} \; X = \mathrm{hom}^\Delta (\Delta , X)$, this is the hom-set as simplicial objects. Now
$$\mathrm{Tot} \; X = \lim ( \dots \to \mathrm{Tot}^n \; X \to \mathrm{Tot}^{n-1} \; X \to \dots \to \mathrm{Tot}^0 \to * ) ,$$
where $\mathrm{Tot}^n \; X = \mathrm{Tot} (\mathrm{cosk}_n X )$.
We have a homotopy cofiber sequence
$$F_n \to \mathrm{Tot}^n \; X \to \mathrm{Tot}^{n-1} \; X$$
and it is a fact that
$$F_s \simeq \Omega^s ( \prod_{|I|=s+1} \mathbf{O} (U_I )),$$
for the fibered product $U_I$ corresponding to the cover $\{ U_i \to N\}$ of an object $N$ of the etale site of $M_{1,1}$.
Applying $\pi_*$ to the cofiber sequence we obtain an exact couple and hence a spectral sequence with
$$ E^1_{t,s} = \pi_{t-s} F_s \Rightarrow \lim_n \pi_{t-s} \mathrm{Tot}^n X = \pi_{t-s} \mathrm{Tot} X = \pi_{t-s} \mathbf{O} (N). $$
Note that $\pi_{t-s} F_s$ is the Čech complex of the cover, so the $E^2$-page calculates Čech cohomology. If we choose an affine cover, hence acyclic and $\lim^1 =0$, then
$$E^2_{t,s} \Rightarrow H^s (N, \pi_t \mathbf{O}) .$$
##Stacks and Hopf Algebroids##
Let $X$ be a (non-derived) Deligne-Mumford stack on $\mathrm{Aff}$ and let $\mathrm{Spec} \; A \to X$ be a faithfully flat cover, then
$$ \mathrm{Spec} \; A \times_X \mathrm{Spec} \; A = \mathrm{Spec} \; \Gamma,$$
for some commutative ring $\Gamma$. Via the projection maps (which are both flat) we have a groupoid in $\mathrm{Aff}$, by definition it is a [[commutative Hopf algebroid]] $(A, \Gamma)$.
Now let $(A,\Gamma)$ be a commutative Hopf algebroid, then the collection of principal bundles form a stack $M_{A,\Gamma}$. Here a principal bundle is a map of schemes $P \to X$, a $\mathrm{Spec} \; \Gamma$ equivariant map $P \to \mathrm{Spec} \; A$, where the action is given by a map $P \times_{\mathrm{Spec} A} \mathrm{Spec} \; \Gamma \to P$. In this we have an equivalence of 2-categories
$$ \{DM \; Stacks\} \simeq \{Hopf \; Algebroids, \; bibundles\} .$$
and
$$ \{DM \; stacks \; equipped \; with \; cover\} \simeq \{Hopf \; algebroids, \; functors \; of \; groupoids\}. $$
Let $X$ be a scheme then a sheaf of abelian groups is a functor
$$\mathfrak{I} : \mathrm{Aff}/X^{op} \to \mathrm{Ab} .$$
The structure sheaf $\mathbf{O}_X$ is defined by
$$ \mathbf{O}_X ( \mathrm{Spec} \; A \to X) = A .$$
Let $\mathfrak{I}$ be a sheaf of $\mathbf{O}_X$ modules. $\mathfrak{I}$ is quasi-coherent if for any map $\mathrm{Spec} \; B \to \mathrm{Spec} \; A$ and maps $f: \mathrm{Spec} \; A \to X$, $g: \mathrm{Spec} \; B \to X$ we have
$$ B \otimes_{A} \mathfrak{I} (f) \simeq \mathfrak{I} (g) .$$
We have an equivalence of categories $\mathrm{QCSh}/\mathrm{Spec} \; A \simeq A$-mod via the assignment $\mathfrak{I} \mapsto \mathfrak{I} (1_A)$.
Now consider the stack $M_{A,\Gamma}$ from above. One can show that quasi-coherent sheaves over $M_{A, \Gamma}$ is nothing but a $(A,\Gamma)$ [[comodule]], that is an $A$-module, $M$, and a coaction map of $A$-modules
$$M \to \Gamma \otimes^{d_1}_{A} M$$
where the right hand side is an $A$-module via the map $d_0$.
##Cohomology of Sheaves##
Recall that sheaf cohomology is obtained by deriving the global sections functor. If $X$ is a [[noetherian scheme|noetherian]] scheme/stack then we restrict to deriving
$$ \Gamma (-) : \mathrm{QCSh}/X \to \mathrm{Ab} .$$
Suppose further that $X = \mathrm{Spec} \; A$, so $\Gamma$ lands in $A$-modules, however from above we know $\mathrm{QCSh}/\mathrm{Spec} \; A \simeq A$-mod, hence $\Gamma$ is exact and all higher cohomology groups vanish.
Let $\mathfrak{I}_N$ be a quasi-coherent sheaf on a DM stack $M_{A,\Gamma}$. Then global sections of $\mathfrak{I}_N$ induce global sections $n \in N$ such that the two pullbacks to $\Gamma$ correspond to each other
$$ \Gamma \otimes_A^{d_0} N \to \Gamma \otimes_A^{d_1} N; \; 1 \otimes n \mapsto 1 \otimes n .$$
That is the coaction map $n \mapsto 1 \otimes n$ is well defined and $n: A \to N; \; 1 \mapsto n$ is a map of comodules. This allows us to interpret global sections as
$$ \mathrm{Hom}_{A,\Gamma} (A, -) : \mathrm{Comod}_{A,\Gamma} \to A-\mathrm{mod} ,$$
so a section is a map from the trivial sheaf to the given sheaf. It follows that
$$ H^n ( M_{A,\Gamma} , \mathfrak{I}_N ) = \mathrm{Ext}^n_{A,\Gamma} (A,N) .$$
To simplify notation we write the above as $H^n (A, \Gamma ; N)$ and if the $N$ is suppressed it is assumed that $N=A$. In general we compute these Ext groups via the [[cobar complex]].
###Change of Rings###
Let $(A,\Gamma)$ be a commutative Hopf algebroid and $f: A \to B$ a ring homomorphism. Define
$$ \Gamma_B = B \otimes_A^{d_0} \Gamma \otimes_A^{d_1} B ,$$
so we have a map of Hopf algebroids $f_* : (A, \Gamma) \to (B, \Gamma_B)$ and of stacks
$$ f^* : M_{B,\Gamma_B} \to M_{A,\Gamma} .$$
**Theorem.** If there exists a ring $R$ and a homomorphism $\Gamma \otimes_A B \to R$ such that
$$A \to \Gamma \otimes_A B \to R $$
is faithfully flat, then $f^*$ is an equivalence of stacks.
###The Weierstrass Stack###
Given $C/S$ an [[elliptic curve]], [[Riemann-Roch theorem|Riemann–Roch]] gives us (locally on $S$) sections $x \in \Gamma (C, \mathbf{O} (2e)) , \; y \in \Gamma (C, \mathbf{O} (3e))$ such that $x^3-y^2 \in \Gamma (C, \mathbf{O}(5e))$ and $C \simeq C_{\underline{a}} \subset \mathbb{P}^2$ is given by
$$ y^2 + a_1 xy + a_3 y = x^3 +a_2 x^2 +a_4 X + a_6 $$
for $a_i \in \mathbf{O}_S$ and $e = [0: 1:0]$. Such a curve is said to be in Weierstrass form or simply a Weierstrass curve.
Two Weierstrass curves $(C_\underline{a} , e)$ and $(C_{\underline{a}'},e)$ are isomorphic if and only if they are related by a coordinate change of the form
$$ (X,Y) \mapsto (\lambda^{-2} X + r, \lambda^{-3} y + s \lambda^{-2} x +t ) .$$
For instance, this means that $a_1'= \lambda (a_1 +2s)$. We then build a Hopf algebroid $(A, \Gamma)$ by defining
$$ A = \mathbb{Z} [a_1 , \dots , a_4 , a_6 ] , \; \Gamma = A [r,s,t, \lambda^\pm ] .$$
Further, define the stacks $M_{Weir} = M_{A, \Gamma}$ and $M_{ell} = M_{A[\Delta^{-1}] , \Gamma[\Delta^{-1}]}$. Note that
$$ M_{ell} \subset \overline{M_{ell}} \subset M_{Weir} .$$
Let $\omega_{C/S} = \pi_* \Omega^1_{C/S}$ (which is locally free) and $\pi_{2n} \mathbf{O} = \omega^n$. If $C$ is a Weierstrass curve, then $\omega$ is free with generator of degree 2
$$\eta = \frac{dx}{2y + a_1 x +a_3} .$$
Let $\omega^*$ correspond to the graded comodule
$$ A_* = A [ \eta^\pm ] \to \Gamma [\eta^\pm ] = \Gamma_* ; \; \eta \mapsto \lambda^{-1} \eta .$$
It is classical that
$$ H^{0,*} (A, \Gamma ; A_*) = \mathbb{Z} [c_4 , c_6 , \Delta]/ (12^3 \Delta -c_4^3 +c_6^2 ) $$
that is the ring of modular forms. So we get a map
$$ \pi_* tmf \to \{modular \; forms \} $$
as the edge homomorphism of our spectral sequence
$$ H^{s,t} (A,\Gamma ; A_*) \simeq H^{s,t} (A_* , \Gamma_*) \Rightarrow \pi_{t-s} tmf .$$
It should be noted that we have a comparison map with the Adams-Novikov spectral sequence for $MU$.
##$p$-local Coefficients##
###With 6 inverted###
Note that if 2 is invertible than we can complete the square in the Weierstrass equation to obtain
$$ \overline{y}^2 = x^3 + 1/4 b_2 x^2 + 1/2 b_4 x + 1/4 b_6 $$
and the only automorphisms of the curve are $x \mapsto x+r$. Now if 3 is invertible we complete the cube and have
$$ \overline{y}^2 = \overline{x}^3 -1/48 c_4 \overline{x} -1/864 c_6 $$
and this curve is rigid. Define $C= \mathbb{Z} [c_4 , c_6 ]$ and $\Gamma_C = C$, then
$$ H^{s,*} (A_* , \Gamma_* ) [1/6] \simeq H^{s,*} (C,C) [1/6] = \mathbb{Z} [1/6 , c_4 , c_6 ] $$
if $s=0$ and 0 otherwise.
###Localized at 3###
It is true that $H^{s,t} (A_* , \Gamma_* ) = H^{s,t} (B, \Gamma_B)$ where
$$ B = \mathbb{Z}_{(3)} [ b_2 , b_4, b_6 ] \to \Gamma_B = B[r] ; \; b_2 \mapsto b_2 + 12r $$
and the degree of $r$ is 4.
We have the class $\alpha = [r] \in H^{1,4}$ and $\beta = [ -1/2 (r^2 \otimes r + r \otimes r^2)] \in H^{2,12}$.
Let $I = (3 , b_2 , b_4 )$ and consider the Hopf algebroid $(B/I , \Gamma_B /I)$ which by change of rings theorem is equivalent to $(\mathbf{F}_3 , \mathbf{F}_3 [r]/(r^3))$.
A spectral sequence obtained by filtering by powers of $I$ gives:
**Theorem.** $H^{*,*} (B, \Gamma_B) = \mathbb{Z} [c_4 , c_6 , \Delta , \alpha, \beta ]$ subject to the following relations
1. $12^3 \Delta -c_4^3 +c_6^2 = \alpha^2 = 3\alpha = 3 \beta =0;$
1. $c_4 \alpha = c_6 \alpha = c_4 \beta = c_6 \beta = 0.$
By using the comparison map with the Adams-Novikov spectral sequence one can prove the following theorem.
**Theorem.** The edge homomorphism $\pi_* tmf_{(3)} \to \{Modular Forms\}_{(3)}$ has
1. Cokernel given by $\mathbb{Z}/3\mathbb{Z} [\Delta^n]$ for $n \ge 0$ and not divisble by 3;
1. Kernel consisting a copy of $\mathbb{Z}/3\mathbb{Z}$ in degrees 3,10,13,20,27,30,37,40 modulo 72. This is the 3-torsion in $\pi_* tmf$.
For more see
[[Tilman Bauer]], **Computation of the homotopy of the spectrum tmf**. In _Geom. Topol. Monogr_., 13, 2008. |
A Survey of Elliptic Cohomology - E-infinity rings and derived schemes | https://ncatlab.org/nlab/source/A+Survey+of+Elliptic+Cohomology+-+E-infinity+rings+and+derived+schemes | <div class="rightHandSide toc">
[[!include cohomology - contents]]
***
[[!include higher algebra - contents]]
</div>
+-- {: .standout}
This is a sub-entry of
* [[A Survey of Elliptic Cohomology]]
see there for background and context.
This entry discusses the algebraic/homotopy theoretic prerequisites for [[derived algebraic geometry]].
=--
Previous:
* [[A Survey of Elliptic Cohomology - cohomology theories]]
* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]
Next:
* [[A Survey of Elliptic Cohomology - elliptic curves]]
>the following are rough unpolished notes taken more or less verbatim from some seminar talk -- needs attention, meaning: **somebody should go through this and polish**
# contents #
* table of contents
{: toc}
# part 1 -- the sheaf of elliptic cohomology ring spectra #
We will talk about a lifting problem that will lead to the formulation of [[tmf]]. This requires [[E-infinity ring]]s and [[derived algebraic geometry]].
**Definition**
An $\Omega$-[[spectrum]] is a sequence of pointed [[topological space]]s $\{E_n\}$ and base-point preserving maps $\{\sigma_n : E_n \to \Omega E_{n+1}\}$ that are [[weak homotopy equivalence]]s.
($\Omega E_n$ is the [[loop space]] of $E_n$).
if $\{E_n\}$ is an $\Omega$-spectrum, define $h^{-n}(X) := [X, E_n]$ ([[homotopy]] classes of continuous maps). Then this $h$ is a [[generalized (Eilenberg-Steenrod) cohomology theory]].
It should be noted that all our spaces are based and $h$ is a reduced cohomology theory. Define $\pi_n(E) := [S^0, E_n]$. $\pi_*(E)$ are the coefficients (i.e. the cohomology over the point of the corresponding unreduced theory) of $E$.
**Brown's representability theory**: Any [[reduced cohomology theory]] on [[CW-complex]]es is [[representable functor|represented]] by an $\Omega$-[[spectrum]].
**examples**
1. [[singular cohomology]] with coefficients in $A$: the [[Eilenberg-MacLane spectrum]] $H A$.
1. complex [[K-theory]]: $K_n = \mathbb{Z} \times BU$ for $n$ even and $\cdots = U$ otherwise
Le $M_{1,1}$ be the [[moduli stack]] of all [[elliptic curve]]s, then $Hom(Spec R, M_{1,1}) = \{elliptic curves over Spec R\}$.
(we will construct this more rigorously later)
If $\phi : Spec R \to M_{1,1}$ is a map that is a [[flat morphism]], then we obtain an [[elliptic cohomology]] theory called $A_{\phi}$.
This assignment is a [[presheaf]] of [[cohomology theory|cohomology theories]].
To get a single [[cohomology theory]] from that we want to take global sections, but there is no good way to say what a global section of a cohomology-theoy valued functor would be. One reason is that there is not a good notion to say what a _sheaf_ of [[cohomology theory]]s is.
But if we had an [[(infinity,1)-category]] valued functor, then [[Higher Topos Theory]] would provide all that technology. So that's what we try to get now.
**goal** find lift
$$
\array{
&& Spectra
\\
& {}^{?}\nearrow & \;\;\;\downarrow^{represent}
\\
\{\phi : Spec R \to M_{1,1}\}
&\to&
CohomologyTheories
}
\,.
$$
**Hopkins-Miller**: use the multiplicative nature of cohomology theories to solve this, i.e. instead look for a more refined lift
$$
\array{
&& CommRingSpectra
\\
& {}^{O_{M^{der}}}\nearrow & \downarrow
\\
\{\phi : Spec R \to M_{1,1}\}
&\to&
MultiplicativeCohomologyTheories
}
\,.
$$
**theorem** There exists a [[symmetric monoidal category|symmetric monoidal]] [[model category]] $StTop$ of [[spectrum|spectra]] such that the [[homotopy category]] is the [[stable homotopy category]] as a symmetric monoidal category.
This and the following is described in more detail at [[symmetric monoidal smash product of spectra]].
**Definition** An [[A-infinity ring]] is an ordinary [[monoid]] in $StTop$ and an [[E-infinity ring]] is an ordinray commutative monoid there.
So an $E_\infty$-ring is an honest [[monoid]] with respect to the funny smash product that makes spectra a symmetric monoidal category, but it is just a monoid up to homotopy with respect to the ordinary product of spaces.
For more on this see (for the time being) the literature referenced at [[stable homotopy theory]].
**proposition**
Let $A$ be an [[A-infinity ring]] [[spectrum]].
1. the $\infty$-monoidal structure on the spectrum induces a [[multiplicative cohomology theory]].
1. $\pi_0(A)$ is a commutative ring
1. $\pi_n(A)$ is a [[module]] over $\pi_0(A)$.
**Definition** For $A$ an [[E-infinity ring]], $M$ with a map $A \wedge M \to M$ such that the obvious diagrams commute is a [[module]] for that [[E-infinity ring]].
**Proposition** $\pi_*(M)$ is a graded [[module]] over $\pi_*(A)$.
**Definition** for $A$ an [[E-infinity ring]] and $M$ an $A$-module, we have that $M$ is **flat module** if
1. $\pi_0(M)$ is flat over $\pi_0(A)$ in the ordinary sense
1. $\forall n : \pi_n(A) \otimes_{\pi_0(A)} \pi_0(M) \to \pi_n(M)$ is an isomorphism of $\pi_0(M)$-modules
**definition** a morphism $f : A \to B$ of [[E-infinity ring]]s is flat if $B$ regarded as an $A$-module using this morphism is flat.
**Theorem (Goerss-Hopkins-Miller)**:
A lift $O_{M_{1,1}, der}$ as indicated in the GOAL above (multiplicative version) does exists and is unique up to homotopy equivalence.
The [[tmf]]-[[spectrum]] is the global sections of this:
$$
tmf[\Delta^{-1}] = \Gamma(O_{M_{1,1}, der})
$$
this is not elliptic (its not even nor has period 2), but is a multiplicative spectrum and hence defines a cohomology theory.
The [[spectrum]] [[tmf]] is obtained in the same manner by replacing $M_{1,1}$ by its [[Deligne-Mumford compactification]].
# part 2 - the stable symmetric monoidal $(\infty,1)$-category of spectra#
recall that we want global sections of the [[presheaf]]
$$
\{Spec R \to M_{1,1}\} \to CohomologyTheories
$$
(on the left we have something like the etale [[site]] of the moduli stack $M_{1,1}$ )
but there is no good notion of gluing in CohomologyTheories (lack of colimits) hence no good notion of sheaves with values in cohomology theories. $CohomologyTheories$ is the [[homotopy category]] of some other category, to be identified, and passage to homotopy categories may destroy existence of useful colimits. The category of CohomologyTheories "is" the stable homotopy category.
A simple example:
in the [[(infinity,1)-category]] [[Top]] we have the homotopy pushout
$$
\array{
S^1 &\to& D^2
\\
\downarrow && \downarrow
\\
D^2 &\to& S^2
}
$$
but in the [[homotopy category]] the pushout is instead
$$
\array{
S^1 &\to& D^2
\\
\downarrow && \downarrow
\\
D^2 &\to& *
}
$$
The result is not even homotopy equivalent. In the homotopy category the pushout does not exist.
So we want to refine $CohomologyTheories$ to the cateory of [[spectrum|spectra]] that they come from by the [[Brown representability theorem]].
In fact, we want to lift $MultiplicativeCohomologyTheories$ to that of [[E-infinity ring]]-spectra.
The map
$$
E_\infty Rings \to MultiplicativeCohomologyTheories
$$
should be that of taking the [[homotopy category of an (infinity,1)-category]].
**Approach A** (modern but traditional [[stable homotopy theory]]) choose a [[symmetric monoidal category|symmetric monoidal]] [[simplicial model category]] whose [[homotopy category]] is the [[stable homotopy category]] and whose [[tensor product]] is the [[smash product of spectra]]. For instance use the [[symmetric monoidal smash product of spectra]].
Then define [[E-infinity ring]] spectra to be ordinary [[monoid]] objects in this symmetric monoidal model category of spectra.
**Approach B** ([[Jacob Lurie]]: be serious about working with [[(infinity,1)-category]] instead of just [[model category]] theory) .
1. define [[(infinity,1)-category]] ([[Higher Topos Theory|chapter 1 of HTT]])
in this framework we'll have a [[stable (infinity,1)-category of spectra]], let's call that $Sp$
2. show that $Sp$ is a [[symmetric monoidal (infinity,1)-category]]
3. show that the [[homotopy category of an (infinity,1)-category]] of $Sp$ is the [[stable homotopy category]], where the tensor product goes to the [[smash product of spectra]]
4. define an [[E-infinity ring]] to be a [[commutative monoid in an (infinity,1)-category]] in $Sp$.
These two approaches are equivalent is a suitable sense. See [[Noncommutative Algebra]], page 129 and [[Commutative Algebra]], Remark 0.0.2 and paragraph 4.3.
[[derived algebraic geometry]] categorifies [[algebraic geometry]]
[[E-infinity ring]] categoriefies commutative [[ring]]
[[(infinity,1)-category]] catgeorifies [[category]]
**Definition** An [[(infinity,1)-category]] is (for instance modeled by)
* a [[Top]]-[[enriched category]] (essentially [[simplicially enriched category]])
* a [[quasi-category]]
(see there for details). In fact, see chapter 1 of [[Higher Topos Theory]] for lots of details.
use [[homotopy coherent nerve]] to go from a [[simplicially enriched category]] to its corresponding [[quasi-category]]
**definition** [[homotopy category of an (infinity,1)-category]] (see there)
**definition** morphism of [[(infinity,1)-category|(infinity,1)-categories]] is, when regarded as a [[quasi-category]], just a morphism of [[simplicial set]]s.: this is an [[(infinity,1)-functor]].
There is an [[(infinity,1)-category of (infinity,1)-functors]] between two [[(infinity,1)-categories]]
**why simplicial sets?**
because they provide a convenient calculus for doing [[homotopy coherent category theory]].
suppose some [[(infinity,1)-category]] $C$ and its homotopy category $C \to h C$.
A commutative-up-to-homotopy diagram in $C$ is a functor $I \to h C$
$$
\array{
&& C
\\
&& \downarrow
\\
I &\to& h C
}
$$
for $I$ some diagram category.
to get a **homotopy coherent** diagram instead take the [[nerve]] $N(I)$ of $I$ and then map $N(I) \to C$.
The nerve automatically encodes the homotopy coherence. See [[Higher Topos Theory]] pages 37, 38 (but the general idea is well known from [[simplicial model category]] theory).
Now let $C$ be an [[(infinity,1)-category]]. Suppose that it has a [[zero object]] $0 \in C$, i.e. an object that is both an [[initial object]] and a [[terminal object]].
Assume that $C$ admits [[kernel]]s and [[cokernel]]s, i.e. all [[homotopy pullback]]s and pushouts with $0$ in one corner.
Then from this we get [[loop space object]]s $\Omega X$ and [[delooping]] objects $B X$ in $C$ (called suspension objects $\Sigma X$ in this context).
$$
\array{
X &\stackrel{f}{\to}& Y
\\
\downarrow &\Downarrow& \downarrow
\\
0 &\to& coker f
}
\;\;\;\;
\array{
ker(g) &\stackrel{}{\to}& X
\\
\downarrow &\Downarrow& \downarrow^g
\\
0 &\to& Y
}
$$
in particular a [[loop space object]] $\Omega Y$ is the kernel of the 0-map,while the suspension $\Sigma X$ is the cokernel
$$
\array{
X &\stackrel{f}{\to}& 0
\\
\downarrow &\Downarrow& \downarrow
\\
0 &\to& \Sigma X
}
\;\;\;\;
\array{
\Omega Y &\stackrel{}{\to}& 0
\\
\downarrow &\Downarrow& \downarrow^g
\\
0 &\to& Y
}
$$
One example of this is the [[(infinity,1)-category]] of [[pointed object|pointed]] [[topological space]]s.
**definition** a [[prespectrum object]] in an [[(infinity,1)-category]] $C$ with the properties as above is a [[(infinity,1)-functor]]
$$
X : N(\mathbb{Z} \times \mathbb{Z}) \to C
$$
such that $X(i,j)$ for $i \neq j$ is [[zero object]] 0.
$$
\array{
X(n,n) &\stackrel{}{\to}& X(n,n+1) \simeq 0
\\
\downarrow &\searrow& \downarrow^g
\\
X(n+1,n) \simeq 0 &\to& X(n+1,n+1)
}
$$
(everything filled with 2-cells aka homotopies)
since we have cokernels we get maps from the universal property
$$
\array{
X(n,n) &\stackrel{}{\to}& X(n,n+1) \simeq 0
\\
\downarrow &\searrow& \downarrow^g
\\
0 &\to& \Sigma X(n,n)
\\
&&& \searrow^{\alpha_n}
\\
&&&& X(n+1,n+1)
}
$$
and analogously maps $\beta_n : X(n,n) \to \Omega X(n+1, n+1)$
now $X$ is a **spectrum object** if the $\beta_n$ are equivalences, for all $n$. (We don't require $\alpha_n$ to be equivalences.)
so to each [[(infinity,1)-category]] $C$ we get another [[(infinity,1)-category]] $Sp(C)$, the full subcategory $Fun(N(\mathbb{Z}\times \mathbb{Z}), C)$ on the [[spectrum object]]s.
In particular, we set
$$
Sp := Sp(Top)
$$
the [[stable (infinity,1)-category of spectra]] is the stabilization of the [[(infinity,1)-category]] [[Top]] of [[topological space]]s.
> I think we need pointed topological spaces here?
**Fact**: $Sp$ has an essentially unique structure of a [[symmetric monoidal (infinity,1)-category]].
This monoidal structure $\otimes$ is uniquely characterized by the following two properties:
1. $\otimes$ preserves limits and colimits.
1. the [[sphere spectrum]] is the [[monoidal unit]]/[[tensor unit]] wrt $\otimes$.
**definition** A [[symmetric monoidal (infinity,1)-category]] structure on an [[(infinity,1)-category]] $C$ is given by the following data:
1. another [[(infinity,1)-category]] $C^\otimes$ with an [[(infinity,1)-functor]] $C^\otimes \to N(\Gamma)$ that is a [[coCartesian fibration]]
where $\Gamma$ is [[Segal's category]] with objects finite [[pointed object|pointed]] [[set]]s and morphisms basepoint preserving [[function]]s between sets.
such that $C^\otimes_{\langle 1\rangle} \simeq C$
where $C^\otimes_{\langle 1\rangle}$ is the fiber
over $\langle 1\rangle = \{*,1\}$, i.e. the pullback
$$
\array{
C^\otimes_{\langle 1\rangle} &\to& C^\otimes
\\
\downarrow &pullback& \downarrow
\\
\{\langle 1\rangle\} &\to& N(\Gamma)
}
$$
>here should go some pictures that illustarte this. But see the first few pages of [[Noncommutative Algebra]] for the intuition and motivation.
so let $C$ now be a [[symmetric monoidal (infinity,1)-category]].
**definition** A [[commutative monoid in an (infinity,1)-category|commutative monoid in]] $C$ is a [[section]] $s$ of the structure map mentioned above $C^\otimes \to N(\Gamma) \stackrel{s}{\to} C^\otimes$.
The monoid object itself is the image of $\langle 1 \rangle$ under $s$, $A = s(\langle 1 \rangle)$. (Sort of. I think the whole point is that we don't ever say something like "this _particular_ $A$ is _the_ monoid object". Rather, the picture should roughly be that we have all of the standard diagrams describing a commutative monoid object, except that the various objects in the diagrams are _not necessarily the same object_. However, these _a priori_ different objects will be _a fortiori_ homotopy equivalent, so that in particular the usual picture will reappear in the homotopy category. Moreover, of course, these diagrams will not be strictly commutative, but commutative up to coherent homotopy, so that in particular the usual strict commutativity reappears after passage to the homotopy category.)
> There is one more condition on $s$, though.
**definition** an [[E-infinity ring]] spectrum is a [[commutative monoid in an (infinity,1)-category]] in the [[stable (infinity,1)-category of spectra]] $Sp$.
$E_\infty$-rings themselves form an [[(infinity,1)-category]]. And this has all [[limit]]s and [[colimit]]s (see DAG III 2.1, 2.7), so we can talk about sheaves of $E_\infty$ rings!
# part 3 - brave new schemes #
Now the theory of [[scheme]]s and [[derived scheme]]s, but not over [[simplicial commutative ring]]s, but over [[E-infinity ring]]s.
So we are trying to guess the content of the not-yet-existsting
* [[Jacob Lurie]], [[Spectral Schemes]].
Let $A$ be an [[E-infinity ring]].
Define its [[spectrum of an E-infinity ring]] $Spec A$ as the [[ringed space]] $(|Spec A|, \mathcal{O}_{Spec A})$ whose underlying [[topological space]] is the ordinary spectrum of the degree-0 ring
$$
|Spec A| := Spec \pi_0 A
$$
and where $\mathcal{O}_{Spec A}$ is given on Zariski-opens $D(f)$ for any $f \in \pi_0 A$ by
$$
\mathcal{O}_{Spec A}(D(f))
:=
A[f^{-1}]
\,.
$$
Here $A \to A[f^{-1}]$ is characterized by the following equivalent ways:
1. $\pi_\bullet A \to \pi_*(A[f^{-1}])$ identify $\pi_{\bullet}(A[f^{-1}])$ with $\pi_\bullet$
1. $\forall$ $E_\infty$-rings the induced map
$Hom(A[f^{-1}],B) \to Hom(A,B)$ is a [[homotopy equivalence]] of the left hand side with the subspace of the right hand side which takes $f \in \pi_0 A$ to an invertible element of $\pi_0 B$.
This geometry over [[E-infinity ring]]s is in [[spectral algebraic geometry]]/[[brave new algebra|brave new algebraic geometry]].
The analog for [[simplicial commutative ring]]s instead of is what is discussed at [[derived scheme]].
**theorem** ([[Jacob Lurie]])
If $X$ s a space and $\mathcal{O}$ a sheaf of [[E-infinity ring]]s then $(X,\pi_0 \mathcal{O}_X)$ is a classical [[scheme]] and $\pi_n \mathcal{O}_X$ is a quasicoherent $\pi_0 \mathcal{O}_X$-[[module]].
**theorem** there exists a [[derived Deligne-Mumford stack]] $(M_{1,1}, \mathcal{O}^{der}_{M_{1,1}})$ such that $(M_{1,1}, \pi_0 \mathcal{O}^{der}_{M_{1,1}})$ is the ordinary [[Deligne-Mumford stack|DM-]] [[moduli stack]] of [[elliptic curve]]s.
#References#
* [[Paul Goerss]], [[Topological Algebraic Geometry - A Workshop]] |
A Survey of Elliptic Cohomology - elliptic curves | https://ncatlab.org/nlab/source/A+Survey+of+Elliptic+Cohomology+-+elliptic+curves |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
+-- {: .standout}
This is a sub-entry of
* [[A Survey of Elliptic Cohomology]]
and
* [[Gromov-Witten invariants]]
see there for background and context.
This entry contains a basic introduction to [[elliptic curve]]s and their [[moduli space]]s.
=--
Previous:
* [[A Survey of Elliptic Cohomology - cohomology theories]]
* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]
* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]
Next:
* [[A Survey of Elliptic Cohomology - equivariant cohomology]]
>the following are rough unpolished notes taken more or less verbatim from some seminar talk -- needs attention, meaning: **somebody should go through this and polish**
***
#Contents#
* toc
{:toc}
#elliptic curves#
**Definition** An [[elliptic curve]] over $\mathbb{C}$ is equivalently
* a [[Riemann surface]] $X$ of [[genus]] 1 with a fixed point $P \in X$
* a quotient $\mathbb{C}/\Lambda$ where $\Lambda$ is a [[lattice]] in $\mathbb{C}$;
* a compact complex [[Lie group]] of dimension 1.
* a [[smooth scheme|smooth]] [[algebraic curve]] of degree 3 in $\mathcal{P}$.
**Remark** The third definition is the one that is easiest to generalize. For our simple purposes, though, the second one will be the most convenient.
From the second definition it follows that to study the
[[moduli space of elliptic curves]] it suffices to study the [[moduli space]] of [[lattice]]s in $\mathbb{C}$.
**Definition** A **[[framed elliptic curve]]** is an [[elliptic curve]] $(X,P)$ (in the sense of the first definition above) together with an ordered basis $(a,b)$ of $H_1(X, \mathbb{Z})$ with $(a \cdot b) = 1 $
A **framed lattice** in $\mathbb{C}$ is a lattice $\Lambda$ together with an ordered basis $(\lambda_1, \lambda_2)$ of $\Lambda$ such that $Im(\lambda_2/\lambda_1) \gt 0$.
(So turning $\lambda_1$ to $\lambda_2$ in the plane means going counterclockwise).
#moduli spaces of elliptic curves#
this implies that
the [[upper half plane]] $\mathfrak{h}$ is in bijection with framed lattices in $\mathbb{C}$ which in turn is in bijection with [[isomorphism]] classes of framed elliptic curves over $\mathbb{C}$
$$
\mathfrak{h}
\simeq
\{framed~lattices~in~\mathbb{C}\}
\simeq
\{framed~elliptic~curves~over~\mathbb{C}\}/_\sim
$$
and we have
$$
\{elliptic~curves~over~\mathbb{C}\}_\sim
\simeq
\mathfrak{h}/{SL_2(\mathbb{Z})}
$$
where $SL_2(\mathbb{Z}) = \left\{ \left(\array{a & b \\ c & d }\right)| a d - c d = 1\right\}$ acts by
$$
\tau \mapsto \frac{a \tau + b}{c \tau + d}
$$
**Claim** the quotient $\mathfrak{h}/_{SL_2(\mathbb{Z})}$ is [[biholomorphic map|biholomorphic]] to the disk and has a unique structure of a [[Riemann surface]] which makes the quotient map
$\mathfrak{h} \to \mathfrak{h}/SL_2(\mathbb{Z})$ a [[holomorphic map]]
>**warning** possibly something wrong here, audience doesn't believe the bit about the disk
**definition** write $M_{1,1} := \mathfrak{h}/SL_2(\mathbb{Z})$
**definition** a **homolorphic family of [[elliptic curve]]s** over a [[complex manifold]] $T$ is
* a holomorphic map $\pi : X \to T$
* together with a [[section]] $s : T \to X$ of $\pi$ such that for any $t \in T$ the pair $(X_t, s(t))$ is an [[elliptic curve]] (using the first definition above).
For every family
$$
\array{
X \\ \downarrow^\pi \\ T
}
$$
we would like to have $F \to M_{1,1}$
$$
\array{
X \simeq \phi^* F &\to& F
\\
\downarrow & & \downarrow
\\
T &\to& M_{1,1}
}
$$
where
$$
\phi: t \mapsto [X_t, s(t)]
$$
such that
* $\phi : T \to M_{1,1}$ is a [[holomorphic map]]
* every [[holomorphic map]] $T \to M_{1,1}$ corresponds to a family over $t$;
* there is a universal family over $M_{1,1}$
This is _impossible_ . One can construct explicit counterexamples. These counterexamples involved [[elliptic curve]]s with nontrivial [[automorphism]]s.
For instance
$$
\{
(x,y,z) \in \mathbb{P}^2 \times X :
y^2 = x(x-1)(x-\lambda)
\}
\to
X := \mathbb{P}^1 - \{0,1,\infty\}
$$
> but see the discussion at [[moduli space]] for a discussion of the statement "it's te automorphisms that prevent the [[moduli space]] from existing"
consider
$$
\mathbb{Z}^2 \hookrightarrow \mathbb{C} \times \mathfrak{h}
$$
given by
$$
(m,n) :
(z,\tau)
\mapsto
(z + m \tau + n, \tau)
$$
Then consider the family
$$
\array{
E := \mathbb{C}/_{\mathbb{Z}^2} \times \mathfrak{h}
\\ \downarrow \\ \mathfrak{h}
}
$$
is a family of [[elliptic curve]]s over $\mathfrak{h}$
and $E_\tau = \mathbb{C}/{\Lambda_\tau}$ with
$$
\Lambda_{\tau} := \mathbb{Z}\cdot 1 \oplus \mathbb{Z}\cdot \tau
$$
is a family of framed elliptic curves.
**fact** the space $\mathfrak{h}$ with the family $E \to \mathfrak{h}$ is a [[fine moduli space]] for [[framed elliptic curve]]s.
Consider any map $\phi : T \to \mathfrak{h}$
with pullback of the universal family
$$
\array{
X \stackrel{?}{\to} \phi^* E &\to & E
\\
\downarrow && \downarrow
\\
T &\stackrel{\phi}{\to}&
\mathfrak{h}
}
$$
**claim** for every point $t \in T$ there is an open neighbourhood $t_0 \in U \hookrightarrow T$ such that one can choose [[differential form|1-forms]] $\omega_t$ on $X_\tau$ which vary holomorphically with respect to $t$.
Notice that _locally_ every family of elliptic curves is framed (since we can locally extend a choice of basis for $H_1$). So
$$
\array{
&& \mathfrak{h}
\\
&& \downarrow^{SL_2(\mathbb{Z})}
\\
M_{1,1} &\stackrel{Id}{\to}&
M_{1,1}
}
$$
at $i$ and $\rho = e^{2\pi i/6}$ , $C = \{\pm I\}$
isn't locally liftable at $i$ and $\rho$ so it is not a univresal family of unframed curves.
#orbifolds#
**definition** A **basic pointed orbifold** (basic meaning global) is a triple $X//\Gamma := (X,\Gamma,\rho)$, where
* $X$ is a [[connected space|connected]] and [[simply connected space|simply connected]] [[topological space]] (or in other variants a [[complex manifold]] or whatever is under consideration)
* $\Gamma$ is a discrete [[group]]
* $\rho : \Gamma \to Aut(X)$ is a group homomorphism
(here "pointed" because we specified the action $\rho$ instead of its iso-class under the following morphisms)
A morphism from $(X,\gamma, \rho)$ to $(X', \Gamma', \rho')$ is a pair
$$
(f,\phi)
$$
where
* $f : X \to X'$ is a continuous map
* $\phi : \Gamma \to \Gamma'$ is a group homomorphism
such that for all $\gamma \in \Gamma$
$$
\array{
X &\stackrel{f}{\to}& X'
\\
\downarrow^{\gamma} && \downarrow^{\phi(\gamma)}
\\
X &\stackrel{f}{\to}& X'
}
$$
This really leads an enlargement of the plain category of spaces:
**remark** We have a faithful embedding of spaxces into orbifolds defined this way: for any connected semi-locally simply connected space $X$ with [[universal cover]] $\tilde X$
we have
$$
X \mapsto \tilde X //\pi_1(X)
$$
> **warning** notice all the simply-connectedness assumoptions above for making sense of this
**remark** let $X$ be a [[nice topological space]]. Let $G = \pi_1(X)$ be its first [[homotopy group]] and let a discrete group $\Gamma$ action on $X$.
then define
$$
\tilde \Gamma :=
\left\{
(\gamma,g)
\in
\Gamma \times Aut(\tilde X)
|
\array{
\tilde X &\stackrel{g}{\to}& \tilde X
\\
\downarrow^p && \downarrow^p
\\
X &\stackrel{\gamma}{\to}& X
}
\right\}
$$
then we have an exact sequence
$$
1 \to G \to \tilde \Gamma \to \Gamma \to 1
$$
where $G \to \tilde \Gamma$ is given by $(g \mapsto (1,g))$ and $\tilde \Gamma \to \Gamma$ by $(\gamma,g) \mapsto \gamma$.
**definition**
For an [[orbifold]] $(X,\Gamma,\rho)$ write $I \times (X,\Gamma,\rho) := (I \times X, \Gamma, \rho)$.
Then a **homotopy** from $(f,\phi)$ to $(f',\phi') : (X,\Gamma, \rho) \to (X',\Gamma', \rho')$
is a map
$$
(F,\Psi) : I \times (X, \Gamma, \rho) \to (X', \Gamma', \rho')
$$
such that
* $\Psi = \phi = \phi'$
* $f(-) = F(0,-)$, $f'(-) = F(1,-)$
now write
$$
S^1 := (\mathbb{R}, \mathbb{Z})
$$
(the circle regarded as a global orbifold)
**definition**
The first [[homotopy group]] for our definition of orbifold is:
$$
\pi_1(X//\gamma)
\simeq
\{
homotopy~classes~of~maps~S^1 \to X//\Gamma
\}
$$
**exercise** show that this is $\cdots \simeq \Gamma$
>(recall again the simply-connectness assumoption!!)
*definition** A morphism
$$
(f,\phi) : (X,\Gamma) \to (X',\Gamma')
$$
is a [[weak homotopy equivalence]] if $\phi$ is an ismorphism and $H_\bullet(f) : H_\bullet(X) \to X_\bullet(X')$.
**note** Let $E \Gamma$ be a contractible space on which $\Gamma$ acts properly, dic. and free, then
$$
(E \Gamma \times X, \Gamma)
\stackrel{(f,\phi)}{\to}
(X,\Gamma)
$$
with $\phi = Id_\Gamma$ and $f$ the projection is a [[weak homotopy equivalence]].
**definition** a [[local system]] $V$ on $(X,\Gamma)$ with fiber $V$ a group homomorphism $\Gamma \to Aut(V)$ with
**definition**
Introduce the following notation for [[homotopy group]]s, [[homology]] and [[integral cohomology]] of our [[orbifold]]s with coefficients in a local system:
* $\pi_n(X//\Gamma)$ := \pi_n(X)$ for $n \geq 2$
* $H_\bullet(X//\Gamma, V) := H_\bullet(E \Gamma \times_\Gama X, V)$
* $H^\bullet(X//\Gamma, V) := H^\bullet(E \Gamma \times_\Gama X, V)$
**example** ${*}//\Gamma$ has a [[weak homotopy equivalence]] to the [[classifying space]] $\mathcal{B}\Gamma$
it follows that for local system $V$ we have
$$
H^\bullet({*}//\Gamma, V) = H^\bullet_{gp}(\Gamma,V)
$$
where on the right we have [[group cohomology]]
***
We have all kinds of constructions on orbifolds by saying they are structures on $X$ with suitable extension of the action of $\Gamma$ to them
A **[[vector bundle]] on an orbifold** $V \to X//\Gamma$ is a vector bundle $V \to X$ with isomorphism action by $\Gamma$ specified, covering that on $X$.
for instance the [[tangent bundle]] of $X //\Gamma$ is given by $(T X)//\Gamma \to X//\Gamma $ in the obvious way.
**definition** say that $\Gamma$ acts virtually freely if $\exists$ a finite index subgroup $\Gamma'$ of $\Gamma$ which acts freely on $X$.
**note** $SL_2(\mathbb{Z})$ acts virtually freely on $\mathfrak{h}$
$$
SL_2(\mathbb{Z})[m]
\to
SL_2(\mathbb{Z})
\to SL_2(\mathbb{Z}/ m \mathbb{Z})
$$
Let $\Gamma' \lt \Gamma$ be a finite index subgroup which acts freely on $X$.
set
$$
X' := X// \Gamma';
$$
the map
$$
X/\Gamma' \to X//\Gamma
$$
**must be viewed as an unramified covering of degree $[\Gamma:\Gamma']$.**
> supposedly important statement
**definition**
the [[Euler characteristic]] of a global orbifold is
$$
\chi(X//\Gamma) :=
\frac{1}{[\Gamma: \Gamma']} \chi(X/\Gamma')
$$
> compare [[groupoid cardinality]]
#moduli stack/orbifold of elliptic curves#
**definition**
Define now the global orbifold
$$
\mathcal{M}_{1,1} := \mathfrak{h}//SL_2(\mathbb{Z})
$$
**proposition**
$$
H_1(\mathcal{M}_{1,1}, \mathbb{Z})
= \mathbb{Z}/12\mathbb{Z}
$$
$$
H^1(\mathcal{M}_{1,1}, \mathbb{Z}) = 0
$$
$$
H^2(\mathcal{M}_{1,1}, \mathbb{Z})
= \mathbb{Z}/12 \mathbb{Z}
$$
$$
H_\bullet(\mathcal{M}_{1,1}, \mathbb{Q})
\simeq
H_\bullet(M_{1,1}, \mathbb{Q})
$$
and similarly for [[integral cohomology]]
$$
\chi(\mathcal{M}_{1,1}) = -\frac{1}{12}
$$
$$
Pic(\mathcal{M}_{1,1}) \simeq \mathbb{Z}/12\mathbb{Z}
$$
# topological invariants of the moduli stack #
Since the upper half plane is contractible, the homotopy type of $\mathfrak{h}//\mathbb{Z}_2$ are the same as that of $* // \mathbb{Z}_2$ and similarly for the (group)cohomology
$$
H^\bullet(\mathcal{M}_{1,1}, \mathbb{Z})
= H\bullet(SL_2(\mathbb{Z}), \mathbb{Z})
$$
and similalry for homology.
In particular
$$
H_1(\mathcal{M}_{1,1}, \mathbb{Z})
\simeq
SL_2(\mathbb{Z})^{ab} \simeq \mathbb{Z}/12\mathbb{Z}
$$
for all $m \in \mathbb{N}$ then
$$
1 \to SL_2(\mathbb{Z})[m]
\to
SL_2(\mathbb{Z})
\to
SL_2(\mathbb{Z}/m \mathbb{Z})
\to
1
$$
so that
$$
H^1(\mathcal{M}_{1,1}, \mathbb{Z})
=
0
$$
**fact** the group $SL_2(\mathbb{Z})[m]$ is free for $m \gt 2$.
so far all $\mathbb{Q}$-representations $V$ we have
$$
H^k(SL_2(\mathbb{Z}), V)
\simeq
H^l(SL_2(\mathbb{Z}), V)^{SL_2(\mathbb{Z}/m\mathbb{Z})}
$$
due to the freeness we have also that
$$
H^k(SL_2(\mathbb{Z}), V) = 0
$$
for $k \geq 2$
and hence
$$
H^2(SL_2(\mathbb{Z}), \mathbb{Z})
$$
is [[torsion]]
$$
\cdots \simeq
Hom(H_1(SL_2(\mathbb{Z}), \mathbb{Z}), \mathbb{Q}/\mathbb{Z})
\simeq
\mathbb{Z}/12 \mathbb{Z}
\,.
$$
**proposition**
as [[orbifold]]s, we have an isomorphism
$$
\mathcal{M}_{1,1} \simeq
X//(S_3 \times C_2)
$$
where
$$
X :=
\mathbb{P}^1 - \{0,1, \infty\}
$$
and $S_3$ acts on that by permuting $0,1, \infty$. (Think of $\mathbb{P}^1$ as the [[Riemann sphere]]: there is a unique holomorphic automorphism of that permuting these three points in a given fashion.) While $C_2$ acts trivially.
*proof**
$$
\array{
1 \to SL_2(\mathbb{Z})[2]
&\to&
SL_2(\mathbb{Z})
&\to&
S_3 \simeq SL_2(\mathbb{Z}/2\mathbb{Z})
&\to&
1
\\
&&& {}_{\phi}\searrow& \downarrow
\\
&&&& PSL_2(\mathbb{Z})
}
$$
now $PSL_2(\mathbb{Z})[2]$ is known to be torsion free. It acts in a standard way on the [[upper half plane]] $\mathfrak{h}$.
A little discussion shows that
$$
\mathfrak{h}/PSL_2(\mathb{Z})[2]
\simeq
X
$$
this implies that
$$
PSL_2(\mathbb{Z})[2]
\simeq
F_2
$$
the [[free group]] on two generators.
Then the second but last map
$$
1 \to C_2 \to SL_2(\mathbb{Z})[2]
\to
F_2
\to 1
$$
has a [[section]], from which we get that
$$
SL_2(\mathbb{Z})[2]
\simeq
F_2 \times C_2
$$
and so
$$
X//(C_2 \times S_3)
\simeq
(X//C_2) // S_3
\simeq
((\mathfrak{h}// SPL_2(\mathbb{Z})[2])//C_2)//S_3
\simeq
((\mathfrak{h}//PSL_2(\mathbb{Z}[2]))//S_3)
\simeq
\mathfrak{h}/ SL_2(\mathbb{Z})
$$
which is the **end of the proof**.
**corollary** The [[Euler characteristic]] of the [[moduli stack]] of [[elliptic curve]]s is
$$
\chi(\mathcal{M}_{1,1}) = \frac{-1}{12}
\,.
$$
now consider the line bundle
$$
\array{
\mathbb{C} \times \mathfrak{h}
\\
\downarrow
\\
\mathfrak{h}//SL_2(\mathbb{Z})
}
$$
with action on the total space for $k \in\mathbb{Z}$
$$
\left(
\array{
a & b \\ c & d
}
\right)
:
(z, \tau)
\mapsto
(c \tau + d)^k z, \frac{a \tau + b}{c \tau + d}
$$
call this [[line bundle]] on the [[moduli stack]] $\mathcal{L}_k \to \mathcal{M}_{1,1}$. We will see that all line bundles are isomorphic to one of these.
**remark**
$$
f : \mathfrak{h} \to \mathcal{C}
$$
is a [[section]] of $\mathcal{L}_k$ iff
$$
f\left(
\frac{a \tau + b}{c \tau + d}
\right)
=
(c \tau + d)^k f(\tau)
$$
hence precisely if it defines a [[modular function]] of weight $k$! This gives a geometric interpretation of modular functions.
$$
\array{
\mathbb{C} \times \mathfrak{h}
\\
\downarrow
\\
\mathfrak{h}
}
$$
and define an action of $G := SL_2(\mathbb{Z}) \letimes \mathbb{Z}^2$
where $\mathbb{Z}^2$ acts on $SL_2(\mathbb{Z})$ by
$$
\left(
\array{
a & b \\ c & d
}
\right)
: (m , n)
\mapsto
a m + b m, c m + d n
$$
and on $\mathbb{C} \times \mathfrak{h}$ by
$$
(m, n) : (z,\tau) \mapsto (z + m \tau + n, \tau)
$$
the resulting bundle
$$
\array{
\mathbb{C} \times \mathfrak{h}//G
\\
\downarrow
\\
\mathcal{M}_{1,1}
}
$$
we call
$$
\mathcal{E} \to \mathcal{M}_{1,1}
$$
**theorem** for any [[complex manifold]] $T$ there is a bijection between families of elliptic curves over $T$ and [[orbifold]] maps $T \to \mathcal{M}_{1,1}$ classify them.
Suppose we have an "isotrivial family" (meaning all fibers are isomorphic elliptic curves, i.e. a fiber bundle of elliptic curves)
$$
\array{
\\
\downarrow
\\
T &\stackrel{\phi}{\to}& \mathcal{M}_{1,1}
}
$$
recall that the group that defines $T$ as an [[orbifold]] is the first [[homotopy group]] $\pi_1(T)$.
The only condition that we get from the definition of orbifold maps is that
$$
\phi : \pi_1(T) \to SL_2(\mathbb{Z})
$$
factors through the [[stabilizer group]] $\simeq Aut(E_p)$ of our base point $p \in \mathcal{M}_{1,1}$
# compactified moduli stack #
one can see that over _compact_ $T$ with $\mathcal{M}_{1,1}$ we cannot have nontrivial famlies _without_ singular fibers.
To get around that we want a **compactification** $\bar \mathcal{M}_{1,1}$ of the [[moduli stack]].
also fur purposes of intersection theory, we need to further compactify.
recall the description of $\mathcal{M}_{1,1}$ as a weak quotient of $\mathbb{P}^1$. Then consider:
**definition**
Let
$$
\bar \mathcal{M}_{1,1} := \mathbb{P}^1//(C_2 \times S_3)
$$
otice that this is now an [[orbifold]] which is no longer _basic_ by the above definition. In fact, we can cover it by charts of basic orbifolds as follows: consider
$$
\array{
&& \mathfrak{h}//(C_2 \times \mathbb{Z})
\\
& \swarrow && \searrow
\\
\mathfrak{h}//SL_2(\mathbb{Z})
&&&&
\mathbb{D}//C_2
}
$$
>with the arrows being maps of orbifolds whose precise details I haven't typed.
then let $\mathbb{D}^*$ be the punctured disk and realize the diagram
$$
\array{
&& \mathbb{D}^*//C_2
\\
& \swarrow && \searrow
\\
\mathcal{M}_{1,1}
&&&&
\mathbb{D}//C_2
}
$$
where the right morphism is just the inclusion
now we build a chart of $\bar \mathcal{M}_{1,1}$ consisting of the two patches $\mathcal{M}_{1,1}$ and $\mathbb{D}/C_2/$
from this we get the alternative
**definition**
$$
\bar \mathcal{M}_{1,1}
:=
\mathcal{M}_{1,1} \coprod_{\mathbb{D}^*//C_2}
\mathbb{D}//C_2
$$
the [[colimit]] on the right manifestly glues in the "point at infinity" that is not hit by the map $\mathbb{D}^*//C_2 \to \mathcal{M}_{1,1}$.
# Gromov-Witten invariants #
**definition** A **stable curve** (over $\mathcal{C}$) of genus $g$ with $n$ marked points is a proper, connected curve with $n$ smooth marked points such that all singularities are nodes and such that the the [[automorphism]] group (of autos respecting the smooth marked points) is finite,
$$
|Aut(C)| \lt + \infty
$$
and such that the [[arithmetic genus]] is $g$.
Now $\bat \mathcal{M}_{g,n}$ is the [[fine moduli space]] for smooth curves of genus $g$.
There is a [[line bundle]]
$$
\mathcal{T}_i^* \to \bar \mathcal{M}_{g,n}
$$
built fiberwise from the cotangent spaces of the elliptic curves.
one of them is obtained from one of the $n$ sections $s_i$ of the universal family $\mathcal{F} \to \bar \mathcal{M}_{g,n}$. The fiber over a point is the cotangent space of the elliptic curve over that point at this section.
Write for the first [[Chern class]]
$$
c_1(\mathcal{T}_i^*) = \Psi_i
$$
$$
k_1, \cdots, k_n
\in \mathbb{Z}_{\geq 0}
$$
such that
$$
\sum_{i = 1}^n k_i = 3 g - 3 + n
$$
then we get numbers called the [[Gromov-Witten invariants]] ("of the point")
$$
\langle
\tau_{k_1}, \cdots, \tau_{k_n}
\rangle_g
:=
\int_{\bar \mathcal{M}_{1,1}}
\prod_{i = 1}^n
\Psi_i^{k_i}
$$
## example: $\langle \tau_1\rangle_1$ ##
Let $x, y$ by affine coordinates on $\mathbb{P}^2$
Let $f(x,y)$ and $g(x,y)$ be two generic cubics, in particular there are nine joint zeros
$$
|\langle (x,y)| f(x,y) = g(x,y) = 0\rangle| = 0
$$
called $p_1, \cdots, p_9$.
define then
$$
F :=
\left\lbrace
(x,y,t)
\in \mathbb{P}^2 \times \mathbb{P}^1
:
f(x,y) - t g(x,y) = 0
\right\rbrace
$$
and consider
$$
\array{
F
\\
\downarrow^{pr_2}
\\
\mathbb{P}^1
&\stackrel{\phi}{\to}&
\bar \mathcal{M}_{1,1}
\\
&\searrow & \downarrow^{q}
\\
&& \bar M_{1,1}
}
$$
That map $q$ has degree $\frac{1}{2}$ (!) since $\mathbb{P}^1 \to \bar \mathcal{M}_{1,1}$ has degree 12
we also find that the diaginal map $\mathbb{P}^1 \to \bar M_{1,1}$ has degree 12. It follows that $\phi$ has degree 24:
$$
deg(\phi) = 24
\,.
$$
Now let $\mathcal{T}_i^* \to \bar \mathcal{M}_{g,n}$ be one of these line bundles. Consider the [[pullback]] $\phi^*(\mathcal{T}_1)$
then by some argument not reproduced here we find
$$
\int_{\mathbb{P}^1} c_1(\phi^*(\mathcal{T}_1)^*)
\,.
$$
Then since the order of $\phi$ is 24 we find that the first [[Gromov-Witten invariant]] is
$$
\langle
\tau_1
\rangle_1
=
\frac{1}{24}
\,.
$$
#extending structures to the compactified moduli space#
recall that the [[moduli stack]] of [[elliptic curve]] is, as a global [[orbifold]]
$$
\mathcal{M}_{1,1} := \mathfrak{h}//SL_2(\mathbb{Z})
$$
and also
$$
\cdots \simeq (\mathbb{P}^1 - \{0,1,\infty\})//(C_2 \times S_3)
$$
and there is a [[line bundle]] on this given by
$$
\mathcal{L}_k :=
(\mathbb{C} \times \mathfrak{h})//SL_2(\mathbb{Z})
$$
where the action is given by
$$
\left(
\array{
a& b \\ c & d
}
\right)
: (z, \tau)
\mapsto
(c\tau + d)^{k} z, \frac{a \tau + b}{c \tau + d}
$$
since $SL_2(\mathbb{Z})^{ab} = \mathbb{Z}/12\mathbb{Z}$
one finds the [[Picard group]]
$$
Pic \mathcal{M}_{1,1} \simeq \mathbb{Z}/12\mathbb{Z}
$$
meromorphic (holomorphic) sections $f$ of $\mathcal{L}_k$ are [[modular function]]s of weight $k$, i.e. $f : \mathfrak{h} \to \mathbb{C}$ such that
$$
\forall \gamma = (\frac{a b}{c d}) \in SL_2(\mathbb{Z})
:
f(\gamma \tau) = (c \tau + d)^k f(\tau)
$$
the **universal elliptic curve** over $\mathcal{M}_{1,1}$ is
$$
\mathcal{E} := (\mathbb{C} \times \mathfrak{h})//(SL_2(\mathbb{Z}) \ltimes \mathbb{Z}^2)
$$
Then we ended last time with describing the compactified moduli space
$$
\bar \mathcal{M}_{1,1} := \mathbb{P}^1//(C_2 \times S_3)
$$
## extending the line bundles ##
**proposition** $\mathbb{L}_k$ has a universal extension $\bar \mathcal{L}_k$ to $\bar \mathcal{M}_{1,1}$
**proof**
take
$$
(\mathbb{C} \times \mathbb{D}//C_2) \to (\mathbb{D}//C_2)
$$
where $C_2$ acts by
$$
\pm 1 \;\; (z,\tau) \mapsto (\pm^k z , \tau)
$$
note that since $\left(\array{1 & n \\ 0 1}\right) \in S_2(\mathbb{Z})$ for all $n \in \mathbb{Z}$ ay [[modular function]] $f : \mathfrak{h} \to \mathbb{C}$
$$
f(\tau)
=
\sum{-\infty}^\infty
a_n q^ n
$$
where $q := e^{2 \pi i \tau}$
is called a holomorphic **[[modular form]]** of weight $k$ if $f : \mathfrak{h} \to \mathbb{C}$ is holomorphic and $a_n = 0$ for all $n \lt 0$
**remark** modular forms of weight $k$ are in bijection with sections of the line bundle $\bar \mathcal{L}_k$.
**example** for any [[lattice]] $\Lambda$ in $\mathbb{C}$ and for any $k \gt 2$ we have
$$
S_k(\Lambda) := \Sum_{0 \neq \lambda \in \Lambda}
\frac{1}{\lambda^k}
$$
obviously for all $u \in \mathbb{C}^*$ $S_k(u \Lambda) = u^{-k} S_k(\Lambda)$
and for all $\tau \in \mathfrak{h}$ with $C_k(\tau) := S_k(\Lambda_\tau)$ it follows that $G_k : \mathfrak{h} \to \mathbb{C}$ is holomorphic
since $\Lambda_{\gamma \tau} = (c \tau + d)^{-1} \Lambda_\tau$ it follows that $G_k$ is a modular function of weight $k$
**fact** $G_{2k} = 2 \zeta(2 k) ü 2 \frac{(2 \pi i)^{2k}}{(2k-1)!} \sum_{n=1}^\infty b_{2k-1}(n)q^n$
with $b_k(n) := \sum_{d|n} d^k$ and where $\zeta$ is the [[zeta-function]]
it follows that
$G_{2k}$ is a modular form of weight $2k$ (which is not a cusp form).
an important cusp form is
setting $g_2 = 60 G_4$ and $g_3 = 140 G_6$
the [[modular form]]
$$
\Delta := g_2(\tau)^3 - 27 g_3(\tau)^2
$$
is a cusp form of weight 12. $\Delta$ does not have any 0 in $\mathfrak{h}$ and it has a simple zero at $q = 0$.
we have an isomorphism
$$
\bar \mathcal{L}_{12}
\simeq
\mathcal{O}_{\bar \mathcal{M}_{1,1}(\infty)}
$$
where on the right is the sheaf with at most a pole at $\infty$. This isomorphism going from right to left is induced by multiplication with $\Delta$.
we have an exact sequence
$$
0 \to \mathbb{Z} \to Pic(\bar \mathcal{M}_{1,1})
\to \mathbb{Z}/12\mathbb{Z} \to 0
$$
where the first nontrivial map sends 1 to $\bar \mathcal{L}_{12}$ and the second one $\bar \mathcal{L}_1$ to the generator.
set for all $k$
$$
M_k := \{modular~forms~of~weight~k\}
$$
$$
M_k^\circ := \{cusp~forms~of~weight~k\}
$$
**proposition** $M_\bullet$ is an even graded algebra freely generated by $G_4$ and $G_6$ and the ideal $M_\bullet^\circ$ is generated by $\Delta$.
the dimensions are
$$
dim M_{2k} =
\left\{
floor k/6 & for k = 1 mod 6
\\
1+ floor k/6 & otherwise
\right.
$$
> ???????
## extending the universal family of elliptic curves ##
Recall the three definition of [[elliptic curve]]s from above.
Now a fourth definition:
**definition** an elliptic curve is a smooth curve of degree 3 in $\mathbb{C}\mathbb{P}^2$ together with a point in it.
* that this equation implies the first one above follows from the genus formula, which says that a degree $n$ curve as in the definition has genus $g = \frac{(n-1)(n-2)}{2}$
* that the first def implies this one |
A Survey of Elliptic Cohomology - equivariant cohomology | https://ncatlab.org/nlab/source/A+Survey+of+Elliptic+Cohomology+-+equivariant+cohomology | <div class="rightHandSide toc">
[[!include cohomology - contents]]
***
[[!include higher algebra - contents]]
</div>
+-- {: .standout}
This is a sub-entry of
* [[A Survey of Elliptic Cohomology]].
See there for background and context.
This entry considers [[equivariant cohomology]] from the perspective of algebraic geometry.
=--
Previous:
* [[A Survey of Elliptic Cohomology - cohomology theories]]
* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]
* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]
* [[A Survey of Elliptic Cohomology - elliptic curves]]
Next:
* [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]
* [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]
>the following are rough unpolished notes taken more or less verbatim from some seminar talk -- needs attention, meaning: **somebody should go through this and polish**
***
#Contents#
* toc
{:toc}
# Introduction #
The slogan is: for $A$ a [[cohomology theory]], finding [[equivariant cohomology]] $A$-theory corresponds to finding [[group scheme]] over $A({*})$.
The main example we'll be looking at here is complex [[K-theory]].
# Notions of equivariant cohomology #
## Borel equivariant cohomology ##
**Remark** if $A_G$ is an [[equivariant cohomology]] theory and if $Y \to X$ is a $G$-[[principal bundle]] then we want that $A_G(Y) \simeq A(X)$.
So whenever we have a $G$-space where the $G$-[[action]] is free enough.
Let $X$ be a $G$-space, form the [[Borel construction]] $\mathcal{E}G \times_G X$ with $\mathcal{E}G \to \mathcal{B}G$ the [[universal principal bundle]].
Then we can _define_
$$
A_G^{Borel}(X) := A(\mathcal{E}G \times_G X)
\,.
$$
> notice that $\mathcal{E}G \times_G X$ is the realization of the [[action groupoid]] $X//G$. This Borel equivariant cohomology theory is what is discussed currently at the entry [[equivariant cohomology]]. The following will actually define a refinement of the discussion currently at [[equivariant cohomology]].
**Problem** If the cohomology theory is given by a geometric model, such as [[topological K-theory]] in terms of [[vector bundle]]s or [[elliptic cohomology]] potentially in a [[geometric model for elliptic cohomology]], then the above notion of equivariant cohomology need not coincide with the cohomology theory given by the equivariant version of these geometric models. In particular, equivariant [[vector bundle]]s are geometric cocycles of equivariant [[K-theory]] $K_G$ and there is a morphism
$$
K_G(X) \to K_G^{Borel}(X)
$$
but it is not an [[isomorphism]]. Instead, $K_G$ is a _completion_ of $K_G^{Bor}$.
Here $K_G(X)$ is the [[Grothendieck group]] of [[equivariant vector bundle]]s over the [[G-space]] $X$ (say $G$ is a compact [[Lie group]]).
## Grothendieck ring of equivariant vector bundles ##
**Definition** An [[equivariant vector bundle]] over $X$ is
* a [[G-space]] $E$ and $G$-equivariant map $p : E \to X$ such that this is a (complex, here) [[vector bundle]] of finite rank
* for each $g \in G$ the map $g : E_x \to E_{g x}$ is linear.
Morephism are the obvious $G$-equivariant morphisms of [[vector bundle]]s.
*examples**
1. $X$ a trivial $G$-space,then a $G$-equivariant vector bundle is a family of complex representations;
1. for $E \to X$ a [[vector bundle]] the $k$th tensor power of $E$ is a $\Sigma_k$-equivariant vector bundle;
1. if $G$ acts smoothly on $X$ then the complexified [[tangent bundle]] $T X \otimes \mathbb{C} \to X$ is a $G$-equivariant vector bundle.
**remark** The [[category]] of $G$-[[equivariant vector bundle]], has
* [[direct sum]] $\oplus$
* [[tensor product]] $\otimes$
And we can pull back $Vect^G$ \to $Vecg^H$ along any group homomorphism $\phi : H \to G$
So we are entitled to say
**definition** the [[Grothendieck group]] of $Vect^G(X)$ is
$$
K_G(X) := Groth(E \to X, \oplus)
\,.
$$
With the remaining [[tensor product]] $\otimes$ this yields a commutative [[ring]].
**proposition**
1. if $X = pt$ then $K_G(X) \simeq Rep(G)$ is the [[representation ring]] of $G$.
1. in general, $K_G(X)$ is an algebra over $Rep(G)$.
1. if $G$ [[free action|acts freely]] on $X$, then $K_G(X) \simeq K(X/G)$.
So in particular
$$
K(\mathcal{E}G \times_G X)
\simeq
K_G(\mathcal{E}G \times X)
$$
so we get a map
$$
\alpha : K_G(X) \to K_G(\mathcal{E}G \times X)
\simeq
K(\mathcal{E}G \times_G X)
:= K_G^{Borel}(X)
$$
**theorem** (Atiyah-Segal) This $\alpha$ induces an [[isomorphism]]
$$
\hat K_G(X) \simeq K_G^{Borel}(X)
$$
where
$$
\hat K_G(X) := \lim_\leftarrow K_G(X)/I_G^n K_G(X)
$$
where $I_G = ker(Rep(G) \simeq K_G({*}) \to K_G(\mathcal{E}G) \simeq K(\mathcal{B}G)
\stackrel{\epsilon}{\to} \mathbb{Z})$
**consider** $X = {*}$, $G = S^1$, $\mathbb{C}P^\infty \simeq \mathcal{B}S^1$
$$
\alpha : Rep(G)
\to
K(\mathcal{B}G)
= K(\mathbb{C}P^\infty)
\simeq
\mathbb{Z}[ [ t ] ]
$$
1. since $S^1$ is an abelian group, every [[irreducible representation]] is 1-dimensional
$$
\phi : S^1 \to \mathbb{C}^\times
$$
1. $\chi : S^1 \hookrightarrow \mathbb{C}^times$
$$
Rep(G) \simeq \mathbb{Z}[\chi, \chi^{-1}]
\,.
$$
## algebraic interpretation ##
**goal now** find an algebraic interpretation of $\alpha$ such that
$$
Rep(S^1) = \mathcal{o}_{G_m}
$$
and
$$
Rep(\mathbb{C}P^\infty) = \mathcal{o}_{\hat G_m}
$$
adopt the [[functor of points]] perspective
$$
X : CRings \to Set
$$
a [[functor]].
For $A \in $ [[CRing]] get a spectrum
$$
Spec A : R \mapsto CRing(A,R)
$$
for $X$ a functor, it is an [[affine scheme]] if it is a [[representable functor]] in that there is $A$ with $X \simeq Spec A$.
**examples**
1. $\mathbf{A}^n(R) := R^n$
1. $\hat \mathbf{A}^n(R) := Nil(R)^n$
1. $G_m(R) := R^\times \hookrightarrow \mathbf{A}^1(R)$
1. $\mathbb{P}^n(R) := R^{n+1}/\sim$, where $\sim$ is multiplication by $R^\times$
**proposition**
$G_m$ is affine.
**proof** $A = \mathbb{Z}[x,x^{-1}]$, let $u \in Spec(\mathbf{A}(R))$ a map $u A \to R$, then define
$$
\phi : Spec A \to G_m
$$
by
$$
u \mapsto u(x)
$$
Conversely, given $v \in G_m(R) = R^\times$, define
$$
\Psi : G_m \to Spec A
$$
by
$$
v \mapsto \Psi(v)
$$
with
$$
\Psi(v)(\sum_k a_k x^k) := \sum_k a_k v^k
$$
**endofproof**
similarly, $\mathbf{A}^n \simeq Spec \mathbb{Z}[x_1, \cdots, x_n]$
# group schemes #
Given a functor $X : CRing \to Set$ define the ring of funtions $\mathcal{o}_X$ as
$$
\mathcal{o}_X := Hom_{Func(CRing,Set)}(X, \mathbf{A}^1)
$$
in the [[functor category]].
> notice notation: this is global sectins of the [[structure sheaf]], not the structure sheaf itself, properly speaking
we have
$$
\mathcal{o}_{Spec A}
\simeq
A
$$
so that in particular
$$
\mathcal{o}_{G_m} \simeq \mathbb{Z}[x,x^{-1}]
\,.
$$
**definition**
Let $X$ be an [[affine scheme]] and $Y : CRing \to Set$ a [[functor]] with a [[natural transformation]] $p : Y \to X$. A **system of formal coordinates** is a sequence of maps
$$
X_i : Y \to \hat \mathbf{A}^1
$$
such that
$$
a \mapsto (x_1(a), \cdots, x_n(a), p(a))
\in
\hat \mathbf{A}^n \times X
$$
is an [[isomorphism]]. A $Y$ that admits a system of formal coordinates is a **[[formal scheme]]** over $X$.
> **warning** very restrictive definition. See [[formal scheme]]
A **[[formal group]]** $G$ over a [[scheme]] $X$ is a one-dimensional [[formal scheme]] with specified abelian [[group]] structure on each [[fiber]] $p^{-1}\{x\}$.
This means that there is a natural map
$$
\sigma : G \times_X G \to G
$$
and a natural map $\zeta : X \to G$ which maps $x \mapsto 0 \in p^{-1}\{x\}$.
**definition (formal multiplicative group)**
define $\hat G_m$ on each $R \in CRing$ by
$$
\hat G_m(R)
=
\left\{
1+n | n \in Nil(R)
\right\}
$$
which is a group under multiplication.
there is an [[isomorphism]] of underlying [[formal scheme]]s
$$
\hat G_m \simeq \hat \mathbf{A}^1
$$
We compute $\mathcal{o}_{\hat \mathbf{A}^1} \simeq \mathcal{o}_{\hat G_m}$ in two ways:
1. Recall that $\hat \mathbf{A}^1$ can be defined as $Spf \; \mathbb{Z} [t]$, so the global sections of the structure sheaf (which is what we have been calling $\mathcal{o}$) is
$$\mathcal{o}_{\hat \mathbf{A}^1} = \lim_\rightarrow \mathbb{Z}[t]/ (t^n) = \mathbb{Z} [[t]] .$$
1. We can also see this in the functor of points perspective. Consider the functor $\mathrm{Spec} \; \mathbb{Z} [t]/ (t^n)$, then for any ring $R$ $$\hat \mathbf{A}^1 (R) = \lim_\rightarrow \mathrm{Spec} \; \mathbb{Z} [t]/ (t^n) (R).$$
By the universal property of colimits we have
$$\mathrm{Nat} (\hat \mathbf{A}^1 , \mathbf{A}^1 ) \simeq \lim_\leftarrow \mathrm{Nat} (\mathrm{Spec} \; \mathbb{Z}[t]/ (t^n) , \mathbf{A}^1) \simeq \mathbb{Z} [[t]].$$
$$
\mathcal{o}_{\hat G_m} \simeq \mathbb{Z}[ [ t] ]
\,.
$$
**recall** we have a morphism $\alpha : Rep(S^1) \to K(\mathbb{C}P^\infty)$
such that
$$
\mathcal{o}_{G_m}
\simeq
\mathbb{Z}[x,x^{-1}]
\simeq
Rep(S^1)
\to
K(\mathbb{C}P^\infty) \simeq
\mathbb{Z}[ [ t ] ]
\simeq
\mathcal{o}_{\hat G_m}
$$
is the canonical inclusion
$$
\mathcal{o}_{G_m} \to \mathcal{o}_{\hat G_m}
$$
**exercise**
$$
x \in \mathbb{Z}[x,x^{-1}] \simeq \mathcal{o}_{G_m}
$$
is the natural transformation $R^\times \to R$
and
$$
t \in \mathbb{Z}[ [t] ] \simeq
\mathcal{o}_{\hat G_m}
$$
is the natural transformation
$$
\{1 + Nil(R)\}
\to
Nil(R)
\to
R
$$
so that we get a map
$$
\mathcal{o}_{G_m} \to \mathcal{o}_{\hat G_m}
$$
by sending $x$ to $1 + t$, and this corresponds to taking the [[germ]] of functions at $1 \in G_m$
#lesson #
given $G$ an algebraic group such that the [[formal spectrum]] $Spf A(\mathbb{C}P^\infty)$ is the completion $\hat G$, define $A_{S^1}({*}) := \mathcal{o}_{G}$ then passing to germs gives a completion map
$$
A_{S^1}({*}) \to A(\mathbb{C}P^\infty) =
A^{Bor}_{S^1}({*})
$$
|
A Survey of Elliptic Cohomology - formal groups and cohomology | https://ncatlab.org/nlab/source/A+Survey+of+Elliptic+Cohomology+-+formal+groups+and+cohomology |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
+-- {: .standout}
This is a sub-entry of
* [[A Survey of Elliptic Cohomology]]
see there for background and context.
This entry disscusses basics of [[formal group law]]s arising from [[periodic cohomology theory|periodic]] [[multiplicative cohomology theory|multiplicative]] [[cohomology theory|cohomology theories]]
=--
Previous:
* [[A Survey of Elliptic Cohomology - cohomology theories]]
Next:
* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]
## rough notes from a talk##
>the following are rough unpolished notes taken more or less verbatim from some seminar talk -- needs **somebody to go through it and polish it**
**Formal groups and elliptic cohomology.**
In all of the following, all [[cohomology theory|cohomology theories]] are [[multiplicative cohomology theory|multiplicative]] and all [[formal group law]]s are one-dimensional (and commutative).
**[[A Survey of Elliptic Cohomology - cohomology theories|Last time]].** we saw that orienting a [[periodic cohomology theory|periodic]] [[even cohomology theory]] gives a [[formal group law]]
over the [[cohomology ring]] $A^0(\bullet)$. (Note: $A^0$ and not $A^\bullet$ because of the periodicity property.)
**Today** we discuss a generalization of the above statement:
orienting a [[weakly periodic cohomology theory|weakly periodic]] [[even cohomology theory]] $A$ gives a [[formal group]] over $A^0(\bullet)$.
In particular, [[elliptic cohomology]] theories give [[elliptic curve]]s over $A^0(\bullet)$.
##Formal group laws and Landweber's criterion##
[[formal group law|Formal group law]]s of dimension $1$ over $R$ are classified by [[morphism]]s from the [[Lazard ring]] to $R$.
We can define $A_f^n(X)=MP^n(X)\otimes_{MP(\bullet)}R$.
Here $MP$ denotes [[complex cobordism cohomology theory|complex cobordism]], in particular $MP(\bullet)$ is isomorphic to [[Lazard ring|Lazard's ring]].
**Definition.** A sequence $v_0,\ldots,v_n$ of elements of $R$ is regular
if [[endomorphism]]s of $R/(v_0,\ldots,v_{k-1})$ given by multiplication by $v_k$
are injective for all $0\le k\le n$.
**[[Landweber criterion]]** Let $f(x,y)$ be a formal group law and $p$ a prime,
$v_i$ the coefficient of $x^{p^i}$ in $[p]_f(x)=x+_f\cdots+_fx$.
If $v_0,\ldots,v_i$ form a regular sequence for all $p$ and $i$ then $f(x,y)$ gives a
[[cohomology theory]] via the formula with [[tensor product]] above.
**Example.** $g_a(x,y)=x+y$, $[p]_a(x)=px$, $v_0=p$, $v_i=0$ for all $i\ge1$;
regularity condtions imply that
the zero map $R/(p)\to R/(p)$ must be injective. The last statement implies that
$R$ contains the rational numbers as a subring.
Note that $HP^*(X,R)=\prod_k H^{n+2k}(X,R)$ is a [[cohomology theory]] over any [[ring]] $R$.
**Example.** $g_m(x,y)=xy$, $[p]_m(x)=(x+1)^p-1$, $v_0=p$, $v_1=1$, $v_i=0$ for all $i \gt 1$.
The regularity conditions are trivial.
Hence we know that $K^*(X)=MP^*(X)\otimes_{MP(\bullet)} \mathbb{Z}$ is a cohomology theory.
## Formal groups from formal group laws##
Given a commutative topological
$R$-algebra $A$ and a [[formal group law]] $f(x,y)$ if $f(a,b)$ converges for all $a,b\in A$ and the formula giving an inverse to $a$ converges for all $a\in A$,
we get an abelian [[group]] $(A,+_f)$, where $a+_f b=f(a,b)$.
**Example.** For any $A$ the pair $(N(A),+_f)$ is an abelian group, where
$N(A)$ denotes the set of nilpotent elements of $A$.
**Example.** Let $A$ be an oriented complex oriented cohomology theory.
Then computing [[Chern class]]es of [[line bundle]]s is the same as evaluating the
formal group law of $A$ on some algebra.
Recall that [[line bundle]]s on $X$ are [[classifying space|classified]] by maps from $X$ to $\mathbb{C}P^\infty$,
pairs of line bundles are classified by maps to $\mathbb{C}P^\infty \times \mathbb{C}P^\infty$,
and tensor product of line bundles gives a map $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$.
Now apply cohomology functor to the sequence $X\to \mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$.
We have a degree 0 element $t$ in the cohomology of $\mathbb{C}P^\infty$.
Its image in the cohomology of $\mathbb{C}P^\infty \times \mathbb{C}P^\infty$ is a formal group law.
The image of this formal group law in the cohomology of $X$ makes sense
if $X$ is a finite cell complex so that $A^0(X)$ is a nilpotent algebra.
**Question:** When do two formal group laws yield isomorphic groups?
**Definiton.** A homomorphism of formal group laws $f$ and $g$ over $A$
is a formal power series $\phi\in A[x]$ such that $\phi(f(x,y))=g(\phi(x),\phi(y))$.
(The constant term of~$\phi$ is zero.)
Hence formal group laws form a category.
**Example.** If $R$ contains rational numbers as a subring,
then we have two canonical homomorphisms. The first one is $\exp\colon g_a\to g_m$, where
$\exp(x)=\sum_{k \gt 0}x^k/k!$. Its inverse is $\log\colon g_m\to g_a$, where
$\log(x)=\sum_{k \gt 0}(-1)^{k+1}x^k/k$.
This shows up in cohomology as Chern character.
(Isomorphism from $K^n(X)\otimes_{\mathbb{Z}} \mathbb{Q}$ to $\prod_kH^{n+2k}(X,\mathbb{Q})$.
**Formal groups.** A [[formal group]] is a group in the [[category]] of [[formal scheme]]s.
A [[formal scheme]] $\hat{Y}$ is defined for any [[closed subscheme|closed immersion]] of [[scheme]]s $Y \hookrightarrow X$.
Intuitively the [[formal scheme]] $\hat Y$ is the $\infty$-jet bundle in the normal direction of $Y$ inside of $X$.
**Definition.** The locally [[ringed space]] $\hat Y$ is defined as the [[topological space]] $Y$
with structure sheaf $lim O_X/{\mathcal{I}}^n$, where $\mathcal{I}$ is the defining sheaf of ideals of the closed immersion $Y\hookrightarrow X$.
(Where $Y$ is a closed subscheme of $X$.)
**Examples.** $X=\hat Y$ when $Y=X$. $\mathrm{Spec} k[t]=X$, $Y=V(t)$, $\hat X=k[t,t^{-1}]$.
In fact not every locally noetherian formal scheme can be obtained as a completion of a single noetherian scheme in another scheme; such formal schemes are called *algebraizable*.
**Definition. (formal spectrum)** The [[formal spectrum]] $\mathrm{Spf} R$ of a commutative noetherian ring $R$ with a specified ideal $I \subset R$ whose powers define a local basis of a topology around $0$ which is Hausdorff, is the locally [[ringed space]] with the underlying [[topological space]] $\mathrm{Spec} R/I$ whose global sections of the [[structure sheaf]] are the [[limit]]
$$
O_{\mathrm{Spf} R}(\mathrm{Spf} R)=\lim_n (R/I^n)
\,.
$$
(This is incomplete description, one needs to talk sheaves of ideals instead)
### formal group laws from elliptic curve ###
Recall from the above that a given a [[formal group law]] $F(x,y) \in R[ [x,y] ]$ we get te structure of a [[formal group]] on the [[formal spectrum]] $Spf$ by taking the product to be given by
$$
\array{
Spf R[[x,y]] \simeq
Spf[[x]] \times Spf R[[y]]
&\to& Spf R[[z]]
\\
f(x,y)&\leftarrow |& z
}
$$
Isomorphic [[formal group law]]s give [isomorphism|isomorphic]] (of [[formal group]]s) if $G$ a [[formal group]] has $G \simeq Spf R[ [z] ]$; we must choose such an iso to get a [[formal group law]].
Now we get [[formal group]]s from [[elliptic curve]]s over $R$
**Definition** An **[[elliptic curve]]** over a commutative [[ring]] $R$ is a [[group object]] in the [[category]] of [[scheme]]s over $R$ that is a relative 1-dimensional, , [[smooth scheme|smooth curve]], [[proper scheme|proper]] curve over $R$.
This implies that it has [[genus]] 1. (by a direct argument of the Chern class of the tangent bundle.)
Given an [[elliptic curve]] over $R$, $E \to Spec R$, we get a [[formal group]] $\hat E$ by completing $D$ along its identity [[section]] $\sigma_0$
$$
E \to Spec(R) \stackrel{\sigma_0}{\to} E
$$
(the one dual to the map that maps everything to $0 \in R$), we get a [[ringed space]] $(\hat E, \hat O_{E,0})$
**example** if $R$ is a [[field]] $k$, then the [[structure sheaf]] $\hat O_{E,0} \simeq k[ [z] ]$
then
$$
\hat O_{E \times E, (0,0)} \simeq
\hat O_{E,0}
\hat \otimes_k
\hat O_{E,0}
\simeq
k[[x,y]]
$$
**example** **(Jacobi quartics)**
$$
y^2 = 1- 2 \delta x^2 + \epsilon x^4
$$
defines $E$ over $R = \mathbb{Z}[Y_Z,\epsilon, \delta]$.
The corresponding [[formal group law]] is **Euler's formal group law**
$$
f(x,y) = \frac{x\sqrt{1- 2 \delta y^2 + \epsilon y^4}
+ y \sqrt{1- 2 \delta x^2 + \epsilon x^4}}
{1- \epsilon x^2 y^2}
$$
if $\Delta := \epsilon(\delta^2 - \epsilon)^2 \neq 0$ then this is a non-trivial elliptic curve.
If $\Delta = 0$ then $f(x,y) \simeq G_m, G_a$ (additive or multiplicative formal group law corresponding to [[ordinary cohomology]] and [[topological K-theory]] [[KU]], respectively).
## weakly periodic cohomology theories and formal groups ##
A [[multiplicative cohomology theory|multiplicative]] [[cohomology theory]] $A$ is **[[weakly periodic cohomology theory|weakly periodic]]** if the natural map
$$
A^2({*}) \otimes_{A^0({*})} A^n({*}) \stackrel{\simeq}{\to}
A^{n+2}({*})
$$
is an [[isomorphism]] for all $n \in \mathbb{Z}$.
Compare with the notion of a [[periodic cohomology theory]].
# Relation to formal groups #
One reason why weakly periodic cohomology theories are of interest is that their [[cohomology ring]] over the space $\mathbb{C}P^\infty$ defines a [[formal group]].
To get a [[formal group]] from a [[weakly periodic cohomology theory|weakly periodic]], [[even cohomology theory|even]] [[multiplicative cohomology theory|multiplicative]] [[cohomology theory]] $A^\bullet$, we look at the induced map on $A^\bullet$ from a morphism
$$
i_0 : {*} \to \mathbb{C}P^\infty
$$
and take the kernel
$$
J := ker(i_0^* : A^0(\mathbb{C}P^\infty) \to A^0({*}))
$$
to be the [[ideal]] that we complete along to define the [[formal scheme]] $Spf A^0(\mathbb{C}P^\infty)$ (see there for details).
Notice that the map from the point is unique only up to [[homotopy]], so accordingly there are lots of chocies here, which however all lead to the same result.
The fact that $A$ is weakly periodic allows to reconstruct the [[cohomology theory]] essentially from this [[formal scheme]].
To get a [[formal group law]] from this we proceed as follows: if the [[Lie algebra]] $Lie(Spf A^0(\mathbb{C}P^\infty))$ of the [[formal group]]
$$
Lie(Spf A^0(\mathbb{C}P^\infty))
\simeq
ker(i_0^*)/ker(i_0^*)^2
$$
is a free $A^0({*})$-module, we can pick a generator $t$ and this gives an [[isomorphism]]
$$
Spf(A^0(\mathbb{C}P^\infty)) \simeq Spf(A^0({*})[[t]])
$$
if $A^0(\mathbb{C}P^\infty) A^0({*})[ [t] ]$ then $i_0^*$ "forgets the $t$-coordinate".
**Definition** An **elliptic cohomology theory** over $R$ is
* a commutative [[ring]] $R$
* an [[elliptic curve]] $E/R$
* a [[weakly periodic cohomology theory|weakly periodic]], [[multiplicative cohomology theory|multiplicative]], [[even cohomology theory|even]] [[cohomology theory]] $A^\bullet$
* [[isomorphism]]s $A^0({*}) \simeq R$ and $\hat E \simeq Spf(A^0(\mathbb{C}P^\infty))$.
So we have on one side
$$
\array{
\hat E &\stackrel{\simeq}{\to}& Spf A^0(\mathbb{C}P^\infty)
\\
\downarrow && \downarrow
\\
Spec R &\stackrel{\simeq}{\to}&
Spec A^0({*})
\\
\downarrow^{\sigma_0} && \downarrow
\\
\hat E &\stackrel{\simeq}{\to}& Spf A^0(\mathbb{C}P^\infty)
}
$$
We can check that the [[Landweber exactness criterion]] is satisfied for the [[formal group law]] of the [[Jacobi quartic]], i.e. for [[Euler's formal group law]] over $\mathbb{Z}[\Delta^{-1}, \epsilon, \delta, 1/2]$, so this provides an example of an [[elliptic cohomology]] theory.
$$
A^n_G(X) = M P (X) \otimes_{M P({*})} \mathbb{Z}[\Delta^{-1}, \epsilon, \delta, 1/2]
$$
|
A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves | https://ncatlab.org/nlab/source/A+Survey+of+Elliptic+Cohomology+-+the+derived+moduli+stack+of+derived+elliptic+curves |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
>**Abstract** We sketch some basic ideas ([[Jacob Lurie]]'s ideas, that is) about [[higher geometry]] motivated from the [[higher category theory|higher]] version of the [[moduli stack]] of [[elliptic curve]]s: the derived moduli stack of [[derived elliptic curve]]s. We survey aspects of the theory of [[generalized scheme]]s and then sketch how the derived moduli stack of derived elliptic curves is an example of a generalized scheme modeled on the formal dual of [[E-∞ ring]]s.
+-- {: .standout}
This is a sub-entry of
* [[A Survey of Elliptic Cohomology]]
see there for background and context.
=--
For fully appreciating the details of the main theorem here the material discussed in the previous sessions (and a little bit more) is necessary, but our exposition of [[generalized scheme]]s is meant to be relatively self-contained (albeit necessarily superficial).
This are the entries on the previous sessions:
* [[A Survey of Elliptic Cohomology - cohomology theories]]
* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]
* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]
* [[A Survey of Elliptic Cohomology - elliptic curves]]
* [[A Survey of Elliptic Cohomology - equivariant cohomology]]
* [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]
* [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]
***
# the derived moduli stack of derived elliptic curves #
* automatic table of contents goes here
{:toc}
## Motivation and Statement
In the context of [[elliptic cohomology]] one assigns to every [[elliptic curve]] $\phi$ over a [[ring]] $R$ a [[cohomology theory]] [[Brown representability theorem|represented]] by an [[E-∞ ring]] [[spectrum]] $E_\phi$.
Since, by definition, we may identify the [[elliptic curve]] $\phi$ over $R$ with a patch $\phi : Spec R \to \mathcal{M}_{1,1}$ of the [[moduli stack]] $\mathcal{M}_{1,1}$ of elliptic curves, this assignment
$$
\mathcal{O}^{Der} : \phi \mapsto E_\phi
$$
looks like an [[E-∞ ring]] valued [[structure sheaf]] on $\mathcal{M}_{1,1}$.
There is a very general theory of [[higher geometry]] for [[nLab:generalized scheme|generalized spaces]] with generalized [[nLab:structure sheaf|structure sheaves]]. Using this one may regard the pair
$$
(\mathcal{M}_{1,1}, \mathcal{O}^{Der})
$$
as a [[structured (infinity,1)-topos|structured space]] that is a "derived" [[Deligne-Mumford stack]].
The central theorem about [[elliptic cohomology]] of [[Jacob Lurie]], refining the Goerss-Hopkins-Miller theorem says that
+-- {: .standout}
**the central theorem, first version**
The [[moduli stack]] $\mathcal{M}_{1,1}$ of [[elliptic curve]]s equipped with the [[E-∞ ring]]-valued [[structure sheaf]] $\mathcal{O}^{Der}$ may be regarded as the _derived moduli stack_ of [[derived elliptic curve]]s in that for any [[E-∞ ring]] $R$ the space of derived stack morphisms
$$
(Spec R, R) \to (\mathcal{M}_{1,1}, \mathcal{O}^{Der})
$$
is equivalent to the space of [[derived elliptic curve]]s over $R$.
=--
After we have looked at some concepts in [[higher geometry]] a bit more closely below, we will restate this in slightly nicer fashion.
## References
A sketch of what this theorem means and how it is proven is part of the content of
* [[Jacob Lurie]], [[A Survey of Elliptic Cohomology]].
and goes back to Jacob Lurie's PhD thesis (listed [[Jacob Lurie|here]]).
The general theory for the context of [[higher geometry]] invoked here has later been spelled out in
* [[Jacob Lurie]], [[Structured Spaces]]
The special case of the general theory that is needed here, where the coefficient objects of structure sheaves are [[E-∞ ring]]s, is described in
* [[Jacob Lurie]], [[Spectral Schemes]],
while the general theory of [[E-∞ ring]]s themselves, in the [[(∞,1)-category]] theory context needed here, is developed in
* [[Jacob Lurie]], [[higher algebra|Commutative geometry]].
## Notions of Space {#NotionsOfSpace}
The statement that we are after really lives in the context of [[higher geometry]] (often called "derived geometry"). Here is an outline of the central aspects.
The **central ingredient** which we choose at the beginning to get a theory of [[higher geometry]] going is an [[(∞,1)-category]] $\mathcal{G}$ whose objects we think of as **model spaces** : the simplest objects exhibiting the geometric structures that we mean to consider.
+-- {: .standout}
**Examples for categories of model spaces**
* with smooth structure
* $\mathcal{G} = $ [[Diff]], the category of smooth [[manifold]]s;
* $\mathcal{G} = \mathbb{L}$, the category of [[smooth locus|smooth loci]];
* without smooth structure
* $\mathcal{G} = (C Ring^{fin})^{op}$, the formal dual of [[CRing]]: the category of (finitely generated) algebraic [[affine scheme]]s;
* $\mathcal{G} = (sC Ring^{fin})^{op}$, the formal dual of [[simplicial object]]s in [[CRing]];
* $\mathcal{G} = (E_\infty Ring^{fin})^{op}$, the formal dual of [[E-∞ ring]]s: the category of (finitely generated) algebraic derived [[affine scheme]]s.
=--
These [[(∞,1)-category|(∞,1)-categories]] $\mathcal{G}$ are naturally equipped with the structure of a [[site]] (and a bit more, which we won't make explicit for the present purpose). Following [[Jacob Lurie]] we call such a $\mathcal{G}$ a **[[geometry (for structured (infinity,1)-toposes)|geometry]]** .
We want to be talking about generalized spaces _modeled on_ the objects of $\mathcal{G}$. There is a hierarchy of notions of what that may mean:
+-- {: .standout}
**Hierarchy of generalized spaces modeled on $\mathcal{G}$**
$$
\array{
\mathcal{G}
&\stackrel{Spec^{\mathcal{G}}}{\hookrightarrow}&
Sch(\mathcal{G})
&\hookrightarrow&
\mathcal{L}Top(\mathcal{G})^{op}
&\hookrightarrow&
Sh_{(\infty,1)}(Pro(\mathcal{G}))
\\
\\
model spaces
&&
spaces locally like model spaces
&&
concrete spaces coprobeable by model spaces
&&
spaces probeable by model spaces
\\
\\
affine\;\mathcal{G}-schemes
&&
\mathcal{G}-schemes
&&
\mathcal{G}-structured\;(\infty,1)-toposes
&&
\infty-stacks\;on\;\mathcal{G}
\\
\stackrel{tame\;but\;restrictive}{\leftarrow} &
&&&&
&
\stackrel{versatile\;but\;possibly\;wild}{\to}
}
$$
=--
We explain what this means from right to left.
### spaces probeable by model spaces: $\infty$-stacks
An object $X$ _probeable_ by objects of $\mathcal{G}$ should come with an assignment
$$
X : (U \in \mathcal{G}) \mapsto (X(U) \in \infty Grpd)
$$
of an [[∞-groupoid]] of possible ways to probe $X$ by $U$, for each possible $U$, natural in $U$. More precisely, this should define an object in the [[(∞,1)-category of (∞,1)-presheaves]] on $\mathcal{G}$
$$
X \in PSh(\mathcal{G})
=
Funct(\mathcal{G}^{op}, \infty Grpd)
$$
But for $X$ to be _consistently_ probeable it must be true that probes by $U$ can be reconstructed from overlapping probes of pieces of $U$, as seen by the [[coverage|topology]] of $\mathcal{G}$. More precisely, this should mean that the [[(∞,1)-presheaf]] $X$ is actually an object in an [[(∞,1)-category of (∞,1)-sheaves]] on $\mathcal{G}$
$$
X \in
Sh(\mathcal{G}) \stackrel{}{\hookrightarrow}
PSh(\mathcal{G})
\,.
$$
Such objects are called [[∞-stack]]s on $\mathcal{G}$. The [[(∞,1)-category]] $Sh(\mathcal{G})$ is called an [[∞-stack]] [[(∞,1)-topos]].
A supposedly pedagogical discussion of the general philosophy of [[∞-stacks]] as probebable spaces is at [[motivation for sheaves, cohomology and higher stacks]].
The [[∞-stack]]s on $\mathcal{G}$ that are used in the following are those that satisfy [[descent]] on [[?ech cover]]s. But we will see [[(∞,1)-topos]]es of [[∞-stack]]s that may satisfy different descent conditions, in particular with respect to [[hypercover]]s. Every [[∞-stack]] [[(∞,1)-topos]] has a [[hypercompletion]] to one of this form.
For concretely working with [[hypercomplete (∞,1)-topos]]es it is often useful to use [[models for ∞-stack (∞,1)-toposes]] in terms of the [[model structure on simplicial presheaves]].
+-- {: .standout}
$$
\array{
Sh^{hc}_{(\infty,1)}(C)
&\stackrel{\stackrel{\;\;\;\;\;lex\;\;\;\;\;\;}{\leftarrow}}
{\hookrightarrow}&
PSh_{(\infty,1)}(C)
&& \text{abstract nonsense def of (∞,1)-topos}
\\
\uparrow^{\simeq} &&
\uparrow^{\simeq}
&& \text{Lurie's theorem}
\\
([C^{op}, SSet]_{loc})^\circ
&\stackrel{\stackrel{Bousfield\;loc.}{\leftarrow}}{\to}&
([C^{op}, SSet]_{glob})^\circ
&&
\text{model category of simplicial presheaves}
}
$$
=--
+-- {: .un_remark}
###### Warning
This discussion here is glossing over all set-theoretic size issues. See [[Structured Spaces|StSp, warning 2.4.5]].
=--
### concrete spaces co-probeable by model spaces: structured $(\infty,1)$-toposes
Spaces probeable by $\mathcal{G}$ in the above sense can be very general. They need not even have a _concrete underlying space_ , even for general definitions of what _that_ might mean.
**(Counter-)Example** For $\mathcal{G} = $ [[Diff]], for every $n \in \mathbb{N}$ we have the [[∞-stack]] $\Omega_{cl}^n(-)$ (which happens to be an ordinary [[sheaf]]) that assigns to each manifold $U$ the set of closed [[differential form|n-form]]s on $U$. This is important as a generalized space: it is something like the rational version of the [[Eilenberg-MacLane space]] $K(\mathbb{Z}, n)$. But at the same time this is a "wild" space that has exotic properties: for instance for $n=3$ this space has just a single point, just a single curve in it, just a single surface in it, but has many nontrivial probes by 3-dimensional manifolds.
In the classical theory for instance of [[ringed space]]s or [[diffeological space]]s a _concrete underlying space_ is taken to be a [[topological space]]. But this in turn is a bit _too_ restrictive for general purposes: a topological space is the same as a [[localic topos]]: a [[category of sheaves]] on a [[category of open subsets]] of a [[topological space]]. The obvious generalization of this to [[higher geometry]] is: an [[n-localic (∞,1)-topos]] $\mathcal{X}$.
This makes us want to say and make precise the statement that
+-- {: .standout}
An **concrete [[∞-stack]]** $X$ is one which has an _underlying_ [[(∞,1)-topos]] $\mathcal{X}$:
the collection of $U$-probes of $X$ is a [[subobject]] of the collection of [[(∞,1)-topos]]-morphisms from $U$ to $\mathcal{X}$:
$$
X(U) \subset \mathcal{L}Top(\mathcal{G})^{op}(Sh_{\infty}(U),\mathcal{X})
$$
=--
We think of $\mathcal{X}$ as the [[(∞,1)-topos]] of [[∞-stack]]s on a category of open subsets of a would-be space $X$, only that this would be space $X$ might not have an independent existence as a space apart from $\mathcal{X}$. The available entity closest to it is the [[terminal object]] ${*}_{\mathcal{X}} \in \mathcal{X}$.
To say that $\mathcal{X}$ is _modeled on $\mathcal{G}$_ means that among all the [[∞-stack]]s on the would-be space a [[structure sheaf]] of functions with values in objects of $\mathcal{G}$ is singled out: for each object $V \in \mathcal{G}$ there is a [[structure sheaf]] $\mathcal{O}(-,V) \in \mathcal{X}$, naturally in $V$.
This yields an [[(∞,1)-functor]]
$$
\mathcal{O} : \mathcal{G} \to \mathcal{X}
\,.
$$
We think of $X$ as being a concrete space _co-probebale_ by $\mathcal{G}$ (we can map from the concrete $X$ into objects of $\mathcal{G}$).
Such an $\mathcal{O}$ is a _consistent_ collection of coprobes if coprobes with values in $V$ may be reconstructed from co-probes with values in pieces of $V$.
More precisely:
+-- {: .un_def}
###### Definition
**($\mathcal{G}$-structure, [[Structured Spaces|StSp, def. 1.2.8]])**
An [[(∞,1)-functor]] $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ is a **$\mathcal{G}$-valued structure sheaf** on the [[(∞,1)-topos]] if
* it preserves finite [[limit]]s
* and sends covering coproducts $(\coprod_i U_i) \to U$ to [[effective epimorphism]]s.
A pair $(\mathcal{X}, \mathcal{O})$ of an [[(∞,1)-topos]] $\mathcal{X}$ equipped with $\mathcal{G}$-valued [[structure sheaf]] $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ we call a [[structured (∞,1)-topos]].
=--
In summary:
+-- {: .standout}
A **concrete [[∞-stack]] $X$ modeled on $\mathcal{G}$** is
* an [[(∞,1)-topos]] $\mathcal{X}$ ("of $\infty$-stacks on $X$")
* equipped with a $\mathcal{G}$-valued structure sheaf $\mathcal{O}$ in the form of a finite limits and cover preserving functor $\mathcal{O} : \mathcal{G} \to \mathcal{X}$.
=--
The fundamental **example** for [[structured (∞,1)-topos]]es
are provided by the objects of $\mathcal{G}$ themselves, which are canonically equipped with a $\mathcal{G}$-structure as follows.
+-- {: .un_theorem}
###### Theorem
**([[Structured Spaces|StSp, thm. 2.1.1]])**
Let $f : \mathcal{G} \to \mathcal{G}'$ be a morphism of [[geometry (for structured (infinity,1)-toposes)|geometries]],
then the obvious [[(∞,1)-functor]] $f^* : \mathcal{L}Top(\mathcal{G}) \to \mathcal{L}Top(\mathcal{G}')$ admits a [[left adjoint]]
$$
f^*
:
\mathcal{L}Top(\mathcal{G}')
\stackrel{\leftarrow}{\to}
\mathcal{L}Top(\mathcal{G})
:
Spec_{\mathcal{G}}^{\mathcal{G}'}
$$
called the **relative spectrum functor**.
=--
For $\mathcal{G}$ any [[geometry (for structured (infinity,1)-toposes)|geometry]], write $\mathcal{G}_{disc}$ for the [[geometry (for structured (infinity,1)-toposes)|geometry]] obtained from this by forgetting its [[coverage|Grothendieck topology]] and instead using the discrete topology where only equivalences cover.
Notice
that we may identify $\mathcal{G}_{disc}$-structures on
the archetypical [[(∞,1)-topos]] [[∞Grpd]], being finite [[limit]]-preserving functors $\mathcal{G}_{disc}^{op} \to \infty Grpd$
with [[ind-object]]s in $\mathcal{G}^{op}$, hence with the
opposite of [[pro-object]]s in $\mathcal{G}$. This gives a canonical inclusion
$$
Pro(\mathcal{G})
\hookrightarrow
\mathcal{L}Top(\mathcal{G})^{op}
\,.
$$
+-- {: .un_def}
###### Definition
**([[Structured Spaces|StSp, def. 2.1.2]])**
The composite [[(∞,1)-functor]]
$$
Spec^{\mathcal{G}}
:
Pro(\mathcal{G})^{op}
\hookrightarrow
\mathcal{L}Top(\mathcal{G}_{disc})
\stackrel{Spec_{\mathcal{G}}^{\mathcal{G}_{disc}}}{\to}
\mathcal{L}Top(\mathcal{G})
$$
we call the **absolute spectrum functor**
=--
This [[category theory|abstract nonsense]] is reassuring, but we want a more concrete definition of what such $Spec^{\mathcal{G}} U$ is like:
+-- {: .un_def}
###### Definition
**([[Structured Spaces|StSp, def. 2.2.9]])**
For every $U \in \mathcal{G}$ there is naturally induced a [[coverage|topology]] on the [[over category]] $Pro(\mathcal{G})/U$. Define the [[(∞,1)-topos]]
$$
Spec U := Sh_{(\infty,1)}(Pro(\mathcal{G})/U)
\,,
$$
naturally to be thought of as the [[(∞,1)-topos]] of [[∞-stack]]s _on $U$_ .
This is canonically equipped with a [[(∞,1)-functor]]
$$
\mathcal{O}_{Spec X}
:
\mathcal{G} \to Spec X
\,.
$$
=--
And this is indeed the concrete underlying space produced by the absolute spectrum functor:
+-- {: .un_theorem}
###### Theorem
**[[Structured Spaces|StSp, prop. 2.2.11, thm. 2.2.12]])**
For every $U \in \mathcal{G}$ the pair $(Spec U, \mathcal{O}_{Spec U})$ is indeed a [[structured (∞,1)-topos]] and is indeed equivalent to the $Spec^{\mathcal{G}} U$ defined more abstractly above.
=--
**Example** For $\mathcal{G} = (C Ring^{fin})^{op}$ with the standard [[coverage|topology]] we have that 0-localic $\mathcal{G}$-structured spaces are _[[locally ringed space]]s_ , while $\mathcal{G}_{disc}$-structured 0-localic spaces are just arbitrary [[ringed space]]s. Applying the above machinery to this situaton gives a spectrum functor that takes a [[ring]] $R$ first to the [[ringed space]] $({*,R})$ and this then to the [[locally ringed space]] $(Spec R, R)$.
### Spaces locally like model spaces: generalized schemes
We have seen that $\mathcal{G}$-[[structured (∞,1)-topos]]es are those general spaces modeled on $\mathcal{G}$ that are well-behaved in that at least they do have an "underlying topological structure" in the form of an underlying [[(∞,1)-topos]]. But such concrete spaces may still be very different from the model objects in $\mathcal{G}$.
In parts this is desireable: many objects that one would naturally build out of the objects in $\mathcal{G}$, such as mapping spaces $[\Sigma,X]$, are much more general than objects in $\mathcal{G}$ but do live happily in $\mathcal{L}Top(\mathcal{G})^{op}$.
But in many situations one would like to regard $\mathcal{G}$-[[structured (∞,1)-topos]]es that are not globally but _locally_ equivalent to objects in $\mathcal{G}$. This is supposed to be captured by the following definition.
+-- {: .un_def}
###### Definition
**[[Structured Spaces|StSp, def. 2.3.9]]
A [[structured (∞,1)-topos]] $(\mathcal{X}, \mathcal{O})$ is a **$\mathcal{G}$-generalized scheme** if
* there exists a collection $\{V_i \in \mathcal{X}\}$
* such that
* this covers $\mathcal{X}$ in that the canonical morphism
$$
(\coprod_i V_i) \to {*}_{\mathcal{X}}
$$
to the [[terminal object]] in $\mathcal{X}$
is an [[effective epimorphism]]
* the [[structured (∞,1)-topos]]es
$(\mathcal{X}/V_i, \mathcal{O}|_{V_i})$ induced by
the $V_i$ are model spaces in that there exists
$\{U_i \in \mathcal{G}\}$ and equivalences
$$
(\mathcal{X}/V_i, \mathcal{O}|_{V_i})
\simeq
Spec^{\mathcal{G}} U_i
$$
=--
**Examples**
> **warning** the following statements really pertain to pregeometries, not geometries. for the moment this here is glossing over the difference between the two. See [[geometry (for structured (∞,1)-toposes)]] for the details.
* ordinary smooth [[manifold]]s are [[n-localic (infinity,1)-topos|0-localic]] [[Diff]]-[[generalized scheme]]s ([Structured Spaces|StSp, ex. 4.5.2]])
* ordinary [[schemes]] are those $(CRing^{fin})^{op}$-[[generalized scheme]]s whose underlying [[(∞,1)-topos]] is
[[n-localic (infinity,1)-topos|0-localic]] and whose [[structure sheaf]] is [[n-truncated object of an (infinity,1)-category|0-truncated]] ([Structured Spaces|StSp, prop. 4.2.9]])
* [[Deligne-Mumford stack]]s are 1-localic $(CRing^{fin})_{et}^{op}$-[[generalized scheme]]s ([Structured Spaces|StSp, prop. 4.2.9]])
* This last statement is then the basis for calling a general $(CRing^{fin})_{et}^{op}$-[[generalized scheme]] a **derived Deligne-Mumford stack
* Finally, to make contact with the application to the derived moduli stack of derived elliptic curves, it seems that in [[Spectral Schemes]] a derived Deligne-Mumford stack (with derived in the sense of having replaced ordinary commutative rings by [[E-∞ ring]]s) is gonna be a 1-localic $(E_\infty Ring^{fin})^{op}$-[[generalized scheme]].
## The derived moduli space of elliptic curves
With the above machinery for [[higher geometry]] in hand, we now set out to describe the particular application that we are interested in: the study of the derived [[moduli stack]] of [[derived elliptic curve]]s.
### Derived elliptic curves
* [[derived elliptic curve]]
$$
A \mapsto E(A)
$$
### The derived moduli stack
Lurie's discussion of the derived moduli stack $(\mathcal{M}_{1,1}, \mathcal{O}^{Der})$ is more than a re-interpretation of the Goerss-Miller-Hopkins theorem. It is in particular a re-derivation of this result, from the following perspective
+-- {: .standout}
**the central statement, conceptually**
**Input** We have the $(E_\infty Ring)^{op}$-probeable space
$$
(E : R \mapsto \{derived\;elliptic\;curves\;over\;R\})
\in Sh(E_\infty Ring^{op})
\,.
$$
**Question**: Does this happen to even be a $E_\infty Ring^{op}$-[[generalized scheme]]?
**Answer** Yes. It is actually a [[derived Deligne-Mumford stack]].
=--
Let $\mathcal{M}_{1,1}$ be the ordinary [[moduli stack]] of [[elliptic curve]]s.
Using constructions in [[elliptic cohomology]] we may associate to each [[elliptic curve]] over $R$, i.e. each morphism $\phi : Spec R \to \mathcal{X}$, an [[E-infinity ring]] $E_\phi$ -- the multiplicative spectrum that represents the elliptic cohomology theory given by $T$.
This gives an $E_\infty$-ring valued structure sheaf
$$
\mathcal{O}^{Der} : (\phi : Spec R \to \mathcal{X}) \mapsto
E_\phi
\,.
$$
**Question** What, if anything, is this derived stack a derived [[moduli stack]] of?
### the classification of derived elliptic curves ###
The big theorem is that the derived space $(\mathcal{X}, \mathcal{O}^{Der})$ classifies derived elliptic curves over $E_\infty$-rings
This is the theorem that we said above we wanted to consider, stated now a little bit more precisely.
+-- {: .un_theorem }
###### Theorem
**(J. Lurie)**
For
* $A$ any [[E-∞ ring]]
* and $E(A)$ is the space of [[derived elliptic curve]]s over $A$ (the realization of the topological category of elliptic curves over $A$).
we have an equivalence
$$
Hom(Spec A, (\mathcal{X}, \mathcal{O}^{Der}))
\simeq
E(A)
$$
naturally in $A$.
=--
+-- {: .proof}
###### Proof
Jacob Lurie writes that the proof proceeds alonmg these steps. Details will be discussed in the next session.
1. consider the presheaf of preoriented ellitptic curves $E'(A)$ first
1. observe that this restricted to ordinary rings produces the ordinary moduli stack
1. notice that every oo-stack with good deformation theory that restricts this way is a derived Deligne-Mumford stack $(\mathcal{X}, \mathcal{O}')$ that assigns connective $E_\infty$-rings over affines
1. let $\omega$ be the line bundle on $\mathcal{M}_{1,1}$ regarded as a coheren sheaf. There is then from the preorientation of the universal curve over $(\mathcal{M}, \mathcal{O}')$ a morphism
$$
\beta : \omega \to \pi_2 \mathcal{O}'
$$
1. let $\mathcal{O}$ be the sheaf obtained from $\mathcal{O}'$ by inverting $\beta$
1. show that
1. for $n = 2 k $ we have an isomorphism $\omega^k \to \pi_{2 k}\mathcal{O}$
1. for $n = 2 k + 1$ we have an isomorphism $0 \to \pi_{2k +1}\mathcal{O}$
strategy: reduce to neighbourhood of a point
1. notice that this implies the desired statement
=--
[[!redirects derived moduli stack of derived elliptic curves]]
|
A Survey of Elliptic Cohomology - towards a proof | https://ncatlab.org/nlab/source/A+Survey+of+Elliptic+Cohomology+-+towards+a+proof |
<div class="rightHandSide toc">
[[!include higher algebra - contents]]
</div>
>**Abstract** This entry attempts to give an outline of a proof of Lurie's main theorem.
+-- {: .standout}
This is a sub-entry of
* [[A Survey of Elliptic Cohomology]]
see there for background and context.
=--
Here are the entries on the previous sessions:
* [[A Survey of Elliptic Cohomology - cohomology theories]]
* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]
* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]
* [[A Survey of Elliptic Cohomology - elliptic curves]]
* [[A Survey of Elliptic Cohomology - equivariant cohomology]]
* [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]
* [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]
* [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves]]
***
# Towards a proof #
* automatic table of contents goes here
{:toc}
Recall the main theorem.
+-- {: .un_theorem }
###### Theorem
**(J. Lurie)**
For
* $A$ any [[E-∞ ring]]
* and $E(A)$ is the space of oriented [[derived elliptic curve]]s over $A$ (the realization of the topological category of elliptic curves over $A$).
There exists a derived Deligne-Mumford stack $M^{Der} = (M, O^{Der})$ such that we have an equivalence
$$
Hom(Spec A, M^{Der})
\simeq
E(A)
$$
natural in $A$.
And $O^{Der}$ provides the lift of Goerss-Hopkins-Miller.
=--
##Preliminaries##
Recall that for $A$ an $E_\infty$-ring a [[derived elliptic curve]] $F$ is a commutative [[derived group scheme]] over $A$ such that $F_0$ over $\pi_0 A$ is an elliptic curve.
Denote by $E' (A)$ the space of preoriented (derived) elliptic curves (so equipped with a map $\mathbb{C} P^\infty \to F ( \mathrm{Spec} A )$. And $E(A)$ the space of oriented elliptic curves.
Note that a map $\mathrm{Spec} A \to M^{Der}$ is a map
$\mathrm{Spec} \pi_0 A \to M_{1,1}$ and a map of rings
$O (\mathrm{Spec} \pi_0 A) \to A$.
##Representability##
In his thesis, Lurie proves the following.
**Proposition.** Let $F$ be a functor from connective $E_\infty$-ring spectra to spaces s.t.
1. The restriction of $F$ to discrete rings is represented by a (classical) DM-stack $X$, i.e. $F(R) \simeq \mathrm{Nerv} X(R)$;
1. $F$ is a sheaf with respect to the etale topology;
1. $F$ has a good deformation theory.
Then there exists a derived DM-stack $(X , \hat O )$ representing $F$ s.t. $\hat O (U)$ is connective for $U$ affine.
**Examples.**
1. The functor $E'$: observe that every classical elliptic curve over a discrete $R$ has a unique preorientation. Hence $E'$ is represented by DM-stack $(M, O' )$.
1. The functor $E$: the theorem doesn't apply as a discrete ring cannot be weakly periodic.
**Claim.** $(M, O')$ represents $E'$ for all $E_\infty$-rings, so we dropped connectivity.
_Proof._ Recall the map $A \to \tau_{\ge 0} A$ to the connected cover.
1. Let $A$ be an $E_\infty$-ring, then we have an equivalence
$$\mathrm{Hom} (O' (\mathrm{Spec} \pi_0 \tau_{\ge 0} A) , \tau_{\ge 0} A ) \to \mathrm{Hom} (O' (\mathrm{Spec} \pi_0 A) , A).$$
1. $E' ( \tau_{\ge 0} A \simeq E' (A)$ since every elliptic curve is by definition flat.
We need the following to prove the claim.
**Proposition.** The functor $M \mapsto M \otimes_{\tau_{\ge 0} A} A,$ from flat modules over $\tau_{\ge 0} A$ to flat modules over $A$ is an equivalence.
_Proof of proposition (sketch)._
Let $M,N$ be $A$-modules then there is a spectral sequence
$$\mathrm{Tor}_{p}^{\pi_* A} (\pi_* M, \pi_* N )_q \Rightarrow \pi_{p+q} (M \otimes_A N ) .$$
Suppose $M$ is flat, then
$$\mathrm{Tor}_{p}^{\pi_* A} (\pi_* M , \pi_* N) = \pi_* N \otimes_{\pi_0 A} \pi_0 M$$
if $p=0$ and 0 otherwise. Thus,
$$\pi_* (M \otimes_A N ) \simeq \pi_* N \otimes_{\pi_0 A} \pi_0 M .$$
So we have
$$ F: \mathrm{Mod}^{flat} (\tau_{\ge 0} A ) \leftrightarrow \mathrm{Mod}^{flat} (A) : G .$$
This is an equivalence and $F$ respects the monoidal structure, hence the equivalence extends to the categories of algebras and the proposition (and hence claim) is proved.
We need to know that $\pi_i O'$ are coherent sheaves over $M_{1,1}$, where a [[coherent sheaf]] is an assignment $\mathrm{Spec} R \to M_{1,1}$ which behaves well under finite limits. Let $\omega$ be the line bundle of invariant differentials on $M_{1,1}$ so that is
$$ \omega = e^* \Omega_{F |_{\mathrm{Spec} R}}. $$
Recall that a preorientation determines a map $\beta : \omega \to \pi_2 (O' )$. So define a sheaf of $E_\infty$-rings $O$ as $O' [ \beta^{-1} ]$ which is characterized (maybe) by
$$ \pi_n O = \lim ( \pi_{n+2k} O' \otimes_{O_{1,1}} \omega^{-k} ) .$$
**Remark.**
* This formula comes from a much simpler situation...Let $R$ be (an honest to God) ring, $x \in R$ then
$$ R[x^{-1}] \simeq \lim (R \stackrel{x}{\to} R \stackrel{x}{\to} \dots )$$
* Suppose $U \to M_{1,1}$ with $U$ affine, $\omega$ restricted to $U$ is trivialized then $O' (U) =A$ and $\beta \in \pi_2 A$. Hence, $O' [\beta^{-1} ] (U) = A [ \beta^{-1} ]$ and we get the formula above $\pi_i (A [ \beta^{-1} ]) = (\pi_i A) [\beta^{-1} ]$.
More generally, for $U \to M_{1,1}$ we have that $O(U)$ is weakly periodic, so
$$ \pi_2 O (U) \otimes_{\pi_0 O(U)} \pi_n O(U) \to \pi_{n+2} O (U) $$
is an equivalence.
* $(M, O)$ classifies oriented elliptic curve. Let $F$ over $U = \mathrm{Spec} B$ be a preoriented elliptic curve over $B$, so we have the classifying map $f: O' (U) = A \to B$ and this is an orientation iff
$$ \pi_n B \otimes_{\pi_0 B} \omega \to \pi_{n+2} B $$
is an isomorphism. This map can be identified with $\times \beta$, so the preorientation is an orientation iff there is a unique factorization through $O(U) = A [ \beta^{-1} ]$.
###Reduction###
**Claim.** To prove the theorem it is enough to show
1. $O_{1,1} = \pi_0 O' \to \pi_0 O$ is an isomorphism;
1. For $n$ odd, the sheaf $\pi_n O =0$.
_Proof._ Suppose (1) and (2) hold. Let $f: \mathrm{Spec} R \to M_{1,1}$ be etale for $R$ discrete. We must show that $O (\mathrm{Spec} R)$ is an elliptic cohomology theory associated to $f$. Condition (1) ensures $\pi_0 A \simeq R$, (2) guarantees evenness and from above we have weakly periodic. We must show that $\mathrm{Spf} A^{0} (\mathbb{C}P^\infty ) \simeq \hat E_f$ which follows from having an orientation.
###Reducing to a Local Calculation###
We wish to show that $\pi_n O = 0$ for $n$ odd. From above, it suffices to show that
$$ f_k : \pi_{n+2k} O' \otimes \omega^{-k} \to \pi_n O $$
is zero for all $k$. Note that $\mathrm{im} (f_k )$ is a quotient of $\pi_{n+2k} O' \to \mathrm{im} (f_k )$ which is coherent. Suppose $p$ is an etale cover of $M_{1,1}$ then
$\mathrm{im} (f_k ) = 0$ iff $p^{i} \mathrm{im} (f_k )=0$. We can find an etale cover by a disjoint union of level 3 and level 4 modular forms denoted $\mathrm{Spec} R$.
That $M:= p^{i} \mathrm{im} (f_k ) =0$ is equivalent to $M \otimes_R R/m$ for all $m \in R$. It is not difficult to show that all residue fields $R/m$ are finite in this case.
Now it is enough to show condition (2) formally locally as $\hat R_m /m \simeq R_m / m$.
##Key Ingredients##
In the previous section we had a moduli stack preoriented elliptic curves $(M,O')$. The structure sheaf of $O'$ took values in _connected_ $E_\infty$-rings. We had a refinement $(M, O)$ which was a moduli stack for oriented elliptic curves. From the orientation condition we deduced that the structure sheaf took values in _weakly periodic_ $E_\infty$-rings.
Further, we showed how to reduce the main theorem to a (formal) local computation. That is, we only need to consider $\hat R_m$, the completion of a ring localized at a maximal ideal.
We delay the completion of the proof until later, but now we introduce the key technical tool: $p$-divisible groups.
###$p$-divisible Groups###
Let $R$ be a complete, local ring (e.g. the $p$-adic integers $\mathbb{Z}_p$) and $E_0$ an [[elliptic curve]] over $R_0 := R/M = \mathbf{F}_q$. What do we need in order to lift $E_0$ to an elliptic curve over $R$?
Let $p$ be a prime (say the characteristic of $\mathbf{F}_q$) and $E$ an [[elliptic curve]] over $R$. Using the _multiplication by_ $p$ map $p^n \colon E \to E$ we can define a sheaf of Abelian groups
$$ E[p^\infty] := \mathrm{colim} \; E[p^n],$$
where $E[p^n]$ is the kernel of the map $p^n$. That is, $E[p^n]$ corresponds to the $p$-torsion points of $E$.
**Definition.** A $p$-divisible group $\mathfrak{I}$ over $R$ is a sheaf of Abelian groups on the flat site of schemes over $R$ such that
1. $p^n : \mathfrak{I} \to \mathfrak{I}$ is surjective;
1. $\mathfrak{I} = \mathrm{colim} \; \mathfrak{I} [p^n]$ where $\mathfrak{I} = \mathrm{ker} \; (p^n \colon \mathfrak{I} \to \mathfrak{I})$.
1. $\mathfrak{I}[p^n]$ is a finite, flat, commutative $R$-group scheme. Note that finite means that $\mathfrak{I}$ is affine and whose global sections is a finite $R$-module.
For instance, the constant sheaf $\underline{\mathbb{Z}_p}$ is a $p$-divisible group.
###Serre-Tate Theory###
Now let $R$, in addition to above, be Noetherian with residue field $\mathbf{F}_q$ for $q=p^n$.
**Theorem (Serre-Tate).**
Let $\overline{E}$ be an elliptic curve over $\mathbf{F}_q$, then there is an equivalence of categories between elliptic curves over $R$ that restrict to $\overline{E}$ and the category of $p$-divisible groups $\mathfrak{I}$ over $R$ such that the restriction of $\mathfrak{I}$ to $\mathbf{F}_q$ is $\overline{E} [p^\infty ]$.
The theorem is somewhat surprising as, _a priori_, the latter category sees only torsion phenomena of the elliptic curves.
###Derived Serre-Tate Theory###
**Definition.** Let $A$ be an $E_\infty$-ring. A functor $\mathfrak{I}$ from commutative $A$-algebras to topological Abelian groups is a $p$-divisible group if
1. $B \mapsto \mathfrak{I} (B)$ is a sheaf;
1. $p^n : \mathfrak{I} \to \mathfrak{I}$ is surjective;
1. $(\mathrm{ho})\mathrm{colim} \; \mathfrak{I} [p^n] \simeq \mathfrak{I}$, where as above $\mathfrak{I} [p^n] := (\mathrm{ho}) \mathrm{ker} \; p^n$;
1. $\mathfrak{I}[p^n]$ is a derived commutative group scheme over $A$ which is finite and flat.
If $\mathfrak{I}$ is a $p$-divisible group over $\mathbf{F}_q$, then $\mathfrak{I} [p]$ is a finite $\mathbf{F}_q$-module of dimension $r$ called the _rank_ of $\mathfrak{I}$.
**Proposition.** If $\mathfrak{I} = E [p^\infty]$, for $E$ an elliptic curve, then $\mathfrak{I}$ has rank 2.
One can verify the proposition over $\mathbb{C}$ pretty easily; it is more subtle over a finite field. We have a derived version of the Serre-Tate theorem.
**Theorem (Serre-Tate).** Let $A$ be an $E_\infty$-ring such that $\pi_0 A$ is a complete, local, Noetherian ring and $\pi_i A$ are finitely generated $\pi_0 A$-modules. Let $E_0$ be a (derived) elliptic curve over $\pi_0 A / M$, then there is an equivalence of $\infty$-categories: elliptic curves over $A$ that restrict to $E_0$ and $p$-divisible groups over $A$ that restrict to $E_0 [p^\infty]$.
###Proof of the Classical Theorem###
We give a proof of the classical result, however the proof is quite formal and should carry over to the derived setting.
Let $R$ be a ring as in the theorem such that $N \cdot R =0$ for some $N \in \mathbb{N}$ and let $I \subset R$ be a nilpotent ideal, so $I^{r+1} =0$, and set $R_0 = R/I$. Let $\mathfrak{I} \colon R-\mathrm{alg} \to \mathrm{Ab}$ and
$$ \mathfrak{I}_I (A) := \mathrm{ker} \; ( \mathfrak{I} (A) \to \mathfrak{I} (A / I \cdot A ) .$$
Further, define
$$ \overline{\mathfrak{I}} (A) := \mathrm{ker} \; (\mathfrak{I}(A) \to \mathfrak{I} (A / \mathrm{nil}R \cdot A)) \subset \mathfrak{I}_I (A) .$$
**Lemma.** If $\mathfrak{I}$ is a formal group over $R$, then $\mathfrak{I}_I$ is annihilated by $N^r$.
**Proposition.** Let $\mathfrak{I}$, $\mathfrak{H}$ be elliptic curves or $p$-divisible groups over $R$ and let $N=p^n$. Denote by $\mathfrak{I}_0$ and $\mathfrak{H}_0$ the restriction of $\mathfrak{I}$ and $\mathfrak{H}$ to $R_0$-algebras. Then
1. $\mathrm{Hom}_{R-grp} (\mathfrak{I}, \mathfrak{H})$ and $\mathrm{Hom}_{R_0-grp} (\mathfrak{I}_0 , \mathfrak{H}_0 )$ have no $N$-torsion;
1. $\mathrm{Hom} (\mathfrak{I} , \mathfrak{H}) \to \mathrm{Hom} (\mathfrak{I}_0 , \mathfrak{H}_0 )$ is injective;
1. For $f_0 \colon \mathfrak{I}_0 \to \mathfrak{H}_0$ there is a unique homomorphism $N^r f \colon \mathfrak{I} \to \mathfrak{H}$ which lifts $N^r \cdot f_0$;
1. $f_0 \colon \mathfrak{I}_0 \to \mathfrak{H}_0$ lifts to $f \colon \mathfrak{I} \to \mathfrak{H}$ if and only if $N^r f$ annihilates $\mathfrak{I}[N^r] \subset \mathfrak{I}$.
Note that if $E$ is an elliptic curve over $R$ then the $E_0$ above is given by
$$E_0 = E \times_{\mathrm{Spec} \; R} \mathrm{Spec} \; R/I .$$
Using the previous results one can prove the following alternate version of the Serre-Tate theorem.
**Theorem (alternative Serre-Tate).**
Let $R$ be a ring with $p$ nilpotent and $I \subset R$ a nilpotent ideal. Let $R_0 = R/I$. We have a categorical equivalence: elliptic curves over $R$ and the category of triples $\{ (E, E_0 [p^\infty] , \epsilon )\}$; where $E$ is an elliptic curve over $R_0$, $E_0 [p^\infty]$ is a $p$-divisble group over $R$, and $\epsilon \colon E_0 [p^\infty] \to E[p^\infty]_0$ is a natural isomorphism.
###Adding in Completions###
We really want to consider elliptic curves over $R$ completed by an ideal $m$, this is the $\hat{R}_m$ from far above. We can reduce this problem to that of elliptic curves over $R$ and a system of $p$-divisible groups over $R/m^n$ by combining the Serre-Tate theorem and the following theorem of Grothendieck.
**Theorem (Formal GAGA).** Let $X, Y$ be exceedingly nice schemes over $R$ and $\hat X$, $\hat Y$ be their formal completions, then there is a bijection
$$\mathrm{Hom}_{R} (X,Y) \leftrightarrow \mathrm{Hom}_{\hat R} (\hat X, \hat Y).$$
##Deformation Theory##
Fix a morphism $\mathrm{Spec} \; \mathbf{F}_q \to M_{1,1}$, that is an [[elliptic curve]] $E_0 /k$. Let $\mathbf{O}_k$ be the sheaf over $M_{1,1}$ that classifies deformations of $E_0$ to oriented elliptic curves over $A$ where $A$ is an $E_\infty$-ring with $\pi_0 A$ a complete, local ring.
Let us assume for the moment that $M_{1,1} = \mathrm{Spec} \; R$ (more generally we pass to an affine cover). One can show that $\mathbf{O}_k$ moreover classifies oriented $p$-divisible groups which deform $E_0 [p^\infty]$.
###Deformations of FGLs and Lubin-Tate Spectra###
Recall that for $F,G$ [[formal group law]]s over $R$ a morphism is $f \in R [ [x] ]$ with $f(0) =0$ such that $f(X +_F Y) = f(X) +_G f(Y)$. Now, let $k$ be a field of characteristic $p$ and $F,G$ formal group laws over $k$. Let $f: F \to G$ be a morphism, then
$$f = x^{p^n} + \dots$$
where $n$ is the [[height]] of $f$. For $F$ a formal group law the height of $F$, $\mathrm{ht} \; F$, is the height of $[p]F$, that is the multiplication by $p$ map.
Let $\Gamma$ be a formal group law over $k$, then a deformation of $F$ consists of a complete local ring $B$ ($B/M = k$) and $F$ a formal group law over $B$ such that $p_* F = \Gamma$, where $p: B \to B/M$ is the canonical surjection. To such deformations $F_1 , F_2$ are isomorphic if there is an isomorphism $f: F_1 \to F_2$ which induces the identity on $p_* F_1 = p_* F_2 = \Gamma$.
**Theorem (Lubin-Tate).** Let $k, \Gamma$ as above with $\mathrm{ht} \; \Gamma \lt \infty$ then there exists a complete local ring $E(k, \Gamma )$ with residue field $k$ and a formal group law over $E(k, \Gamma)$, $F^{univ}$ which reduces to $\Gamma$ such that there is a bijection of sets
$$\mathrm{Hom} (E(k, \Gamma) , R) \to \mathrm{Deform} (\Gamma, R) .$$
If $k = \mathbf{F}_p$, then $E(k, \Gamma) \simeq \mathbb{Z}_p [ [ u_1 , \ldots , u_{n-1} ] ]$ where $\mathrm{ht} \; \Gamma =n$. More generally, if $k= \mathbf{F}_q$, then $E(k, \Gamma) \simeq W\mathbf{F}_q [ [ u_1 , \ldots , u_{n-1} ] ]$, that is, the Witt ring.
Let $E(k, \Gamma), F^{univ}$ as in the theorem, then we define $\overline{F}^{univ}$ over $E(k, \Gamma) [u^\pm]$, degree $u=2$, by
$ \overline{F}^{univ} (X,Y) = u^{-1} F^{univ} (uX , uY). $ We then define a homology theory as
$$( E_{k,\Gamma})_* (X) = E(K,\Gamma)[ [u^\pm] ] \otimes_{MU_*} MU_* (X).$$
Define a category (really a stack and let $p$ and $n$ vary) $\mathbf{FG}$ of pairs $(k,\Gamma)$ where $\mathrm{char} \; k =p$ and $\mathrm{ht} \; \Gamma =n$. By associating to any such pair its Lubin-Tate theory we get a functor from $\mathbf{FG}$ to multiplicative cohomology theories.
**Theorem (Hopkins-Miller I).** This functor lifts to $E_\infty$-rings.
**The philosophy is that there should be a sheaf of $E_\infty$-rings on the stack of formal groups $\mathbf{FG}$ with global sections the sphere spectrum. Then tmf and taf are low height approximations.**
Let $E_{\infty}^{LT}$ be the subcategory of $E_\infty$-rings such that the associated cohomology theory is isomorphic to some $E_{k,\Gamma}$.
**Theorem (Hopkins-Miller II).** $\pi : E_{\infty}^{LT} \to \mathbf{FG}$ is a weak equivalence of topological categories. That is, the lift above is pretty unique.
This implies Hopkins-Miller I by taking a Kan extension $\mathbf{FG} \to E_\infty$ along the inclusion $E_{\infty}^{LT} \hookrightarrow E_\infty$.
##The Final Sprint##
Let us sketch how to proceed...Let $\mathfrak{I}$ be a $p$-divisible group over $A$ with $\pi_0 A$ complete and local. Then $\mathfrak{I}$ fits in an exact sequence
$$ 0 \to \hat \mathfrak{I} \to \mathfrak{I} \to \mathfrak{I}_{et} \to 0 .$$
There are two cases: either the underlying elliptic curve $E_0$ is super singular ($\mathrm{ht} \; E_0 =2$), else $E_0$ is ordinary.
###The Supersingular Case###
In this case $\widehat{E_0} [p^\infty] = E_0 [p^\infty]$, so one can show $\widehat{\mathfrak{I}} = \mathfrak{I}$ for every deformation. Now an orientation means
$$\widehat{\mathfrak{I}} \simeq \mathrm{Spf} \; A^{\mathbb{C}P^\infty}[p^\infty],$$
so $E_{k, \widehat{E_0}}$ classifies deformations to oriented $p$-divisible groups, hence $\mathbf{O}_k \simeq E_{k, \widehat{E_0}}$. By construction $E_{k,\widehat{E_0}}$ is even.
###The Ordinary Case###
This is more subtle (see DAG IV 3.4.1 for some hints).
First we can analyze the formal part of the exact sequence and see that deformations to formal $p$-divisible groups of height 1 are classified by $E_{k, \widehat{E_0}} [ p^\infty]$. By analyzing extensions etale $k$-algebras we see that $\mathbf{O}_k \simeq E_{k,\widehat{E_0}}^{\mathbb{C}P^\infty}$. The homotopy groups are then $E_{k , \widehat{E_0}} [ [t] ]$ and are odd as the degree of $t$ is 0.
|
A-brane | https://ncatlab.org/nlab/source/A-brane |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[branes]] of the [[A-model]] [[topological string]], hence a [[TQFT]]-shadow of the [[D-branes]] of the [[superstring]]. Form the [[Fukaya category]] of [[target space|target]] [[spacetime]].
## Related concepts
* [[2d TQFT]], [[TCFT]]
* [[Kapustin-Witten TQFT]]
* [[B-brane]], [[homological mirror symmetry]]
[[!include table of branes]]
## References
* [[Anton Kapustin]], [[Dmitri Orlov]], _Remarks on A-branes, Mirror Symmetry, and the Fukaya category_, J. Geom. Phys. 48 (2003), no. 1, 84--99 ([arXiv:hep-th/0109098](http://arxiv.org/abs/hep-th/0109098))
[[!redirects A-branes]] |
A-hat genus | https://ncatlab.org/nlab/source/A-hat+genus |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
### In terms of an operator index
For $X$ a [[smooth manifold]] of even [[dimension]] and with [[spin structure]], write $\mathcal{S}(X)$ for the [[spin bundle]] and
$$
\mathcal{S}(X) \simeq \mathcal{S}^+(X) \oplus \mathcal{S}^-(X)
$$
for its decomposition into [[chiral spinor]] bundles. For $(X,g)$ the [[Riemannian manifold]] structure and $\nabla$ the corresponding [[Levi-Civita connection|Levi-Civita]] [[spin connection]] consider the map
$$
c \circ \nabla
\;\colon\;
\Gamma(\mathcal{S}^+(X))
\to
\Gamma(\mathcal{S}^-(X))
$$
given by [[composition|composing]] the action of the [[covariant derivative]] on [[sections]] with the [[symbol map]]. This is an [[elliptic operator]]. The [[index of a Dirac operator|index]] of this operator is called the **$\hat A$-[[genus]]**.
### In terms of the universal $Spin$-orientation of $KO$
More abstractly, there is the universal [[orientation in generalized cohomology]] of [[KO]] over [[spin structure]], known as the [[Atiyah-Bott-Shapiro orientation]], which is a [[homomorphism]] of [[E-∞ rings]] of the form
$$
M Spin \longrightarrow KO
$$
from the universal [[spin structure]] [[Thom spectrum]]. The $\hat A$-genus
$$
\Omega_\bullet^{SO}\longrightarrow \pi_\bullet(KO)\otimes \mathbb{Q}
$$
is the corresponding homomorphism in [[homotopy groups]].
## Properties
### Characteristic series
The [characteristic series](genus#LogarithmAndCharacteristicSeries) of the $\hat A$-genus is
$$
\begin{aligned}
K_{\hat A}(e)
& =
\frac{z}{e^{z/2} - e^{-z/2}}
\\
&=
\exp\left(
- \sum_{k \geq 2} \frac{B_k}{k} \frac{z^k}{k!}
\right)
\end{aligned}
\,,
$$
where $B_k$ is the $k$th [[Bernoulli number]] ([Ando-Hopkins-Rezk 10, prop. 10.2](#AndoHopkinsRezk10)).
### Relation to the Todd genus
{#RelationToTheToddGenus}
On an [[almost complex manifold]] $M_{\mathrm{U}}$, the [[Todd class]] coincides with the [[A-hat genus|A-hat class]] up to the [[exponential]] of half the [[first Chern class]]:
$$
Td(M_{\mathrm{U}})
\;=\;
\big(e^{c_1/2} \hat A\big)(M_{\mathrm{U}})
\,.
$$
(e.g. [Freed 87 (1.1.14)](#Freed87)).
In particular, on manifolds $M_{S\mathrm{U}}$ with [[special unitary group|SU]]-[[G-structure|structure]], where $c_1 = 0$, the Todd class is actually equal to the [[A-hat genus|A-hat class]]:
$$
Td(M_{S\mathrm{U}})
\;=\;
\hat A(M_{S\mathrm{U}})
\,.
$$
\linebreak
Given the [[complexification]] of a [[real vector bundle]] $\mathcal{X}$ to a [[complex vector bundle]] $\mathcal{E} \otimes \mathbb{C}$, the $\hat A$-class of $\mathcal{E}$ is the [[square root]] of the [[Todd class]] of $\mathcal{E} \otimes \mathbb{C}$ (e.g. [de Lima 03, Prop. 7.2.3](#deLima03)).
### As a Rozansky-Witten invariant
+-- {: .num_prop #RozanskyWittenWilsonLoopOfUnknotIsSquareRootOfAHat}
###### Proposition
**([[Rozansky-Witten Wilson loop of unknot is A-hat genus|Rozansky-Witten Wilson loop of unknot is square root of A-hat genus]])**
For $\mathcal{M}^{4n}$ a [[hyperkähler manifold]] (or just a [[holomorphic symplectic manifold]]) the [[Rozansky-Witten invariant]] [[Wilson loop observable]] associated with the [[unknot]] in the [[3-sphere]] is the [[square root]] $\sqrt{{\widehat A}(\mathcal{M}^{4n})}$ of the [[A-hat genus]] of $\mathcal{M}^{4n}$.
=--
This is [Roberts-Willerton 10, Lemma 8.6](Rozansky-Witten+Wilson+loop+of+unknot+is+A-hat+genus#RobertsWillerton10), using the [[Wheels theorem]] and the [[Hitchin-Sawon theorem]].
## Related entries
* [[Todd genus]]
* [[index of an elliptic complex]]
[[!include genera and partition functions - table]]
## References
Review:
* {#Freed87} [[Daniel Freed]], Sections 1.1, 1.2 of: _Geometry of Dirac operators_, 1987 ([pdf](http://www.ma.utexas.edu/users/dafr/DiracNotes.pdf), [[FreedGeometryOfDiracOperators.pdf:file]])
The $\hat A$-genus as the index of the spin complex is discussed for instance in:
* {#Gilkey} [[Peter Gilkey]], Section 3 of: _The Atiyah-Singer Index Theorem -- Chapter 5_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/gilkey3.pdf))
* {#deLima03} Levi Lopes de Lima, _The Index Formula for Dirac operators: an Introduction_, 2003 ([pdf](https://impa.br/wp-content/uploads/2017/04/PM_10.pdf))
The relation of the characteristic series to the [[Bernoulli numbers]] is made explicit for instance in prop. 10.2 of
* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))
A construction via a [[1-dimensional Chern-Simons theory]] is in
* [[Owen Gwilliam]], [[Ryan Grady]], _One-dimensional Chern-Simons theory and the $\hat A$-genus_ ([arXiv:1110.3533](http://arxiv.org/abs/1110.3533))
[[!redirects A-hat genera]] |
A-infinity algebra > history | https://ncatlab.org/nlab/source/A-infinity+algebra+%3E+history | < [[A-infinity algebra]]
[[!redirects A-infinity algebra -- history]] |
A-infinity category > history | https://ncatlab.org/nlab/source/A-infinity+category+%3E+history | < [[A-infinity category]]
[[!redirects A-infinity category -- history]] |
A-infinity operad | https://ncatlab.org/nlab/source/A-infinity+operad | # The $A_\infty$ operad
* table of contents
{: toc}
##Idea
An __$A_\infty$ operad__ is an [[operad]] over some [[enriched category theory|enriching category]] $C$ which is a (free) [[homological resolution|resolution]] of the standard [[associative operad]] enriched over $C$ (that is, the operad whose algebras are [[monoids]]).
Important examples, to be discussed below, include:
* The topological operad of Stasheff associahedra.
* The little $1$-cubes operad.
* The standard dg-$A_\infty$ operad.
* The standard categorical $A_\infty$ operad.
An $A_\infty$ operad, like the standard associative operad, can be defined to be either a _symmetric_ or a _non-symmetric_ operad. On this page we assume the non-symmetric version. When regarded as a symmetric operad, an $A_\infty$ operad may also be called an $E_1$ operad.
An [[algebra over an operad]] over an $A_\infty$ operad is called an __$A_\infty$-object__ or [[A-∞ algebra]], where -object is often replaced with an appropriate noun; thus we have the notions of **$A_\infty$-[[A-infinity-space|space]]**, **$A_\infty$-[[A-infinity-algebra|algebra]]**, and so on. In general, $A_\infty$-objects can be regarded as 'monoids up to coherent homotopy.' Likewise, a [[category over an operad|category]] over an $A_\infty$ operad is called an $A_\infty$-[[A-infinity-category|category]].
Some authors use the term '$A_\infty$ operad' only for a _particular_ chosen $A_\infty$ operad in their chosen ambient category, and thus use '$A_\infty$-object' and '$A_\infty$-category' for algebras and categories over this particular operad. The $A_\infty$ operads discussed below are common choices for this 'standard' $A_\infty$ operad.
## The topological Stasheff associahedra operad
### Definition
Let $\{K(n)\}$ be the sequence of [[associahedron|Stasheff associahedra]]. This is naturally equipped with the structure of a (non-symmetric) operad $K$ enriched over [[Top]] called the **topological Stasheff associahedra operad** or simply the **Stasheff operad**. Since each $K(n)$ is [[contractible space|contractible]], $K$ is an $A_\infty$ operad.
The original article that defines associahedra, and in which the operad $K$ is used implicitly to define $A_\infty$-[[topological space]]s, is ([Stasheff](#Stasheff)).
A textbook discussion (slightly modified) is in [MarklShniderStasheff, section 1.6](#MarklShniderStasheff)
### Properties
Stasheff's $A_\infty$-operad is the relative [[Boardman-Vogt resolution]] $W([0,1], I_* \to Assoc)$ where $I_*$ is the operad for [[pointed object]]s [BergerMoerdijk](#BergerMoerdijk).
## The little $1$-cubes operad
Let $\mathcal{C}_1(n)$ denote the configuration space of $n$ disjoint intervals linearly embedded in $[0,1]$. Substitution gives the sequence $\{\mathcal{C}_1(n)\}$ an operad structure, called the **little 1-cubes operad**; it is again an $A_\infty$ operad. This is a special case of the [[little n-cubes operad]] $\mathcal{C}_n$, which is in general an $E_n$ [[E-n operad|operad]].
The little $n$-cubes operads (in their symmetric version) were among the first operads to be explicitly defined, in the book that first explicitly defined operads: [The geometry of iterated loop spaces](#May).
##The standard dg-$A_\infty$ operad
The **standard dg-$A_\infty$ operad** is the dg-operad (that is, operad enriched in [[cochain complex]]es $Ch^\bullet(Vect)$)
* freely generated from one $n$-ary operation $f_n$ for each $n \geq 1$, taken to be in degree
$2 - n$;
* with the differential of the $n$th generator given by
$$ -
\sum_{j+p+q = n}^{1 \lt p \lt n} (-1)^{j p + q} a_{p,j,n} ,$$
where $a_{p,j,n}$ is $f_p$ attached to the $(j+1)$st input of $f_{n-p+1}$.
This can be shown to be a standard free resolution of the linear associative operad in the context of dg-operads; see [Markl 94](http://arxiv.org/abs/hep-th/9411208), [proposition 3.3](http://arxiv.org/PS_cache/hep-th/pdf/9411/9411208v1.pdf#page=13); therefore it is an $A_\infty$ operad.
It can also be shown to be _isomorphic_ to the operad of top-dimensional (cellular) chains on the topological Stsheff associahedra operad. This is discussed on [pages 26-27](http://arxiv.org/PS_cache/hep-th/pdf/9411/9411208v1.pdf#page=26) of [Markl 94](http://arxiv.org/abs/hep-th/9411208)
In the dg-context it is especially common to say '$A_\infty$-algebra' and '$A_\infty$-category' to mean specifically algebras and categories over _this_ operad. The explicit description of this operad given above means that such $A_\infty$-algebras and categories can be given a fairly direct description without explicit reference to operads.
In addition to
* Martin Markl, _Models for operads_ ([arXiv](http://arxiv.org/abs/hep-th/9411208))
another reference is section 1.18 of
* Yu. Bespalov, V. Lyubashenko, O. Manzyuk, _Pretriangulated $A_\infty$-categories_, Proceedings of the Institute of Mathematics of NAS of Ukraine, vol. 76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008, 598 ([ps.gz](http://www.math.ksu.edu/~lub/cmcMonad.gz))
A relation of the linear dg-$A_\infty$ operad to the Stasheff associahedra is in the proof of proposition 1.19 in Bespalov et al.
## The standard categorical $A_\infty$ operad
Let $O$ be the operad in [[Set]] freely generated by a single binary operation and a single nullary operation. Thus, the elements of $O(n)$ are ways to associate, and add units to, a product of $n$ things. Let $B(n)$ be the [[indiscrete category]] on the set $O(n)$; then $B$ is an $A_\infty$ operad in [[Cat]]. $B$-algebras are precisely (non-strict, biased) [[monoidal category|monoidal categories]], and $B$-categories are precisely (biased) [[bicategory|bicategories]].
If instead of $O$ we use the $Set$-operad freely generated by a single $n$-ary operation for every $n$, we obtain a $Cat$-operad whose algebras and categories are _unbiased_ monoidal categories and bicategories.
## Related concepts
* **$A_\infty$-operad**
* [[E-∞ operad]]
* [[L-∞ operad]]
## References
* [[Jim Stasheff]], _Homotopy associative H-spaces I_, _II_, Trans. Amer. Math. Soc. 108 (1963), 275-312
* [[Peter May]], _The Geometry of Iterated Loop Spaces_, Springer 1972 ([doi:10.1007/BFb0067491](https://doi.org/10.1007/BFb0067491), [pdf](http://www.math.uchicago.edu/~may/BOOKS/gils.pdf))
* [[Martin Markl]], Steve Shnider, [[Jim Stasheff]], _Operads in Algebra, Topology and Physics_ ([web](http://books.google.de/books?id=fMhZjT9lQo0C&pg=PA56&lpg=PA56&dq=Stasheff+associahedra&source=bl&ots=ZuGXjT4zbp&sig=V-taGG2LHS0msHK-PTxmUXXCvEY&hl=de#PPP1,M1))
* [[Clemens Berger]], [[Ieke Moerdijk]], _Resolution of coloured operads and rectification of homotopy algebras_ ([arXiv:math/0512576](http://arxiv.org/abs/math/0512576))
{#BergerMoerdijk}
[[!redirects A-infinity-operad]]
[[!redirects A-∞ operad]]
[[!redirects A-∞-operad]]
[[!redirects A-∞ operads]]
[[!redirects A-∞-operads]]
[[!redirects E1-operad]]
[[!redirects E1-operads]] |
A-infinity operad > history | https://ncatlab.org/nlab/source/A-infinity+operad+%3E+history | < [[A-infinity operad]]
[[!redirects A-infinity operad -- history]] |
A-infinity-algebra | https://ncatlab.org/nlab/source/A-infinity-algebra |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An _$A_\infty$-algebra_ is a [[monoid]] internal to a [[homotopical category]] such that the [[associativity]] law holds not as an equation, but only up to higher [[coherent]] [[homotopy]].
## Definition
+-- {: .num_defn}
###### Definition
An **$A_\infty$-algebra** is an [[algebra over an operad]] over an [[A-∞ operad]].
=--
## Realizations
### In chain complexes
Let here $\mathcal{E}$ be the [[category of chain complexes]]
$\mathcal{Ch}_\bullet$. Notice that often in the literature this choice of $\mathcal{E}$ is regarded as default and silently assumed.
An $A_\infty$-algebra in chain complexes is concretely the following data.
A chain $A_\infty$-algebra is the structure of a degree 1 [[coderivation]]
$$
D : T^c V \to T^c V
$$
on the reduced tensor coalgebra $T^c V = \oplus_{n\geq 1} V^{\otimes n}$ (with the standard noncocommutative coproduct, see [[differential graded Hopf algebra]]) over a [[graded vector space]] $V$ such that
$$
D^2 = 0
\,.
$$
Coderivations on free coalgebras are entirely determined by their "value on cogenerators", which allows one to decompose $D$ as a sum:
$$
D = D_1 + D_2 + D_3 + \cdots
$$
with each $D_k$ specified entirely by its action
$$
D_k : V^{\otimes k} \to V
\,.
$$
which is a map of degree $2-k$ (or can be alternatively understood as a map $D_k : (V[1])^{\otimes k}\to V[1]$ of degree $1$).
Then:
* $D_1 : V\to V$ is the _differential_ with $D_1^2 = 0$;
* $D_2 : V^{\otimes 2} \to V$ is the _product_ in the algebra;
* $D_3 : V^{\otimes 3} \to V$ is the _associator_ which measures the failure of $D_2$ to be associative;
* $D_4 : V^{\otimes 4} \to V$ is the _pentagonator_ (or so) which measures the failure of $D_3$ to satisfy the pentagon identity;
* and so on.
One can also allow $D_0$, in which case one talks about weak $A_\infty$-algebras, which are less understood.
There is a resolution of the operad $\mathrm{Ass}$ of associative algebras (as operad on the category of chain complexes) which is called the $A_\infty$-operad; the algebras over the $A_\infty$-[[A-infinity operad|operad]] are precisely the $A_\infty$-algebras.
A **morphism of $A_\infty$-algebras** $f : A\to B$ is a collection $\lbrace f_n\rbrace_{n\geq 1}$ of maps
$$
f_n : (A[1])^{\otimes n}\to B[1]
$$
of degree $0$ satisfying
$$
\sum_{0\leq i+j\leq n} f_{i+j+1}\circ(1^{\otimes i}\otimes D_{n-i-j}\otimes 1^{\otimes j})
= \sum_{i_1+\ldots+i_r=n} D_r\circ (f_{i_1}\otimes\ldots \otimes f_{i_r}).
$$
For example, $f_1\circ D_1 = D_1\circ f_1$.
#### Rectification
+-- {: .num_theorem }
###### Theorem
**(Kadeishvili (1980), Merkulov (1999))**
If $A$ is a [[dg-algebra]], regarded as a strictly associative $A_\infty$-algebra, its [[chain homology and cohomology|chain cohomology]] $H^\bullet(A)$, regarded as a [[chain complex]] with trivial differentials, naturally carries the structure of an $A_\infty$-algebra, unique up to isomorphism, and is weakly equivalent to $A$ as an $A_\infty$-algebra.
=--
More details are at _[[Kadeishvili's theorem]]_.
+-- {: .num_remark }
###### Remark
This theorem provides a large supply of examples of $A_\infty$-algebras: there is a natural $A_\infty$-algebra structure on all cohomologies such as
* [[de Rham cohomology]]
* [[Hochschild cohomology]]
etc.
=--
### In Topological space
An $A_\infty$-algebra in [[Top]] is also called an _[[A-∞ space]]_ .
#### Examples
Every [[loop space]] is canonically an [[A-∞ space]]. (See there for details.)
#### Rectification
+-- {: .num_theorem }
###### Theorem
Every $A_\infty$-space is [[weak homotopy equivalence|weakly homotopy equivalent]] to a topological [[monoid]].
=--
This is a classical result by ([Stasheff 1963](#Stasheff63), [BoardmanVogt](#BoardmanVogt)). It follows also as a special case of the more general result on rectification in a [[model structure on algebras over an operad]] (see there).
### In spectra
See [[ring spectrum]] and [[algebra spectrum]].
## Related concepts
* **$A_\infty$-algebra**, [[A-∞-category]]
* [[augmented A-∞ algebra]]
* [[curved A-∞ algebra]]
* [[An-space]]
* [[A-n algebra]]
* [[E-∞ algebra]]
* [[L-∞ algebra]], .
[[!include k-monoidal table]]
[[!include deformation quantization - table]]
## References
A survey of $A_\infty$-algebras in chain complexes is in
* [[Bernhard Keller]], _A brief introduction to $A_\infty$-algebras_ ([pdf](http://people.math.jussieu.fr/~keller/publ/IntroAinfEdinb.pdf))
Classical articles on $A_\infty$-algebra in topological spaces are
* {#Stasheff63a} [[Jim Stasheff]], *Homotopy associativity of H-spaces I*, Trans. Amer. Math. Soc. **108** 2 (1963) 275-292 [[doi:10.2307/1993608](https://doi.org/10.2307/1993608)]
* {#Stasheff63b} [[Jim Stasheff]], *Homotopy associativity of H-spaces II* **108** 2 (1963) 293-312 [[doi:10.2307/1993609](https://doi.org/10.2307/1993609), [doi:10.1090/S0002-9947-1963-0158400-5](https://doi.org/10.1090/S0002-9947-1963-0158400-5)]
* [[Michael Boardman]] and [[Rainer Vogt]], _Homotopy invariant algebraic structures on topological spaces_ , Lect. Notes Math. 347 (1973).
{#BoardmanVogt}
A brief survey is in section 1.8 of
* [[Martin Markl]], Steve Shnider, [[Jim Stasheff|James D. Stasheff]], _Operads in algebra, topology and physics_, Math. Surveys and Monographs __96__, Amer. Math. Soc. 2002.
The 1986 thesis of [[Alain Prouté]] explores the possibility of obtaining analogues of [[minimal model]]s for $A_\infty$ algebras. It was published in TAC much later.
* [[Alain Prouté]], _Algèbres différentielles fortement homotopiquement associatives ($A_\infty$-algèbres)_, thesis, available as [Reprints in Theory and Applications of Categories, No. 21, 2011, pp. 1–99](http://www.logique.jussieu.fr/~alp/these_A_Proute-TAC.pdf)
[[!redirects A-infinity-algebras]]
[[!redirects A-infinity algebra]]
[[!redirects A-infinity algebras]]
[[!redirects A-∞ algebra]]
[[!redirects A-∞ algebras]]
[[!redirects A? algebra]]
[[!redirects A? algebras]]
[[!redirects A-∞-algebra]]
[[!redirects A-∞-algebras]]
|
A-infinity-category | https://ncatlab.org/nlab/source/A-infinity-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An _$A_\infty$-category_ is a kind of [[category]] in which the associativity condition on the [[composition]] of [[morphism]]s is relaxed "up to higher coherent homotopy".
The "A" is for Associative and the "${}_\infty$" indicates that associativity is relaxed up to higher homotopies without bound on the degree of the homotopies.
In the most widespread use of the word $A_\infty$-categories are _linear_ categories in that they have [[hom-object]]s that are [[chain complex]]es. These are really models/presentations for [[stable (∞,1)-category|stable (∞,1)-categories]].
If higher coherences in a linear $A_\infty$-category happen to be equal to identity morphisms
(which is encoded by the vanishing of the maps $m_n$ for $n\ge3$, defined below), it becomes the same as a [[dg-category]]. In fact, every linear $A_\infty$-category is $A_\infty$-equivalent to a [[dg-category]]. In this way, we have that $A_\infty$-categories related to [[dg-category|dg-categories]] as models for [[stable (∞,1)-category|stable (∞,1)-categories]] in roughly the same way as [[quasi-category|quasi-categories]] relate to [[simplicially enriched category|simplicially enriched categories]] as models for [[(∞,1)-category|(∞,1)-categories]]: the former is the general incarnation, while the latter is a [[semi-strict infinity-category|semi-strictified]] version.
### Ordinary linear $A_\infty$-categories
In what is strictly speaking a restrictive sense -- which is however widely and conventionally understood in [[homological algebra]] as the standard notion of $A_\infty$-category (see references below) -- the [[hom-space]]s of an $A_\infty$-category are taken to be linear spaces, i.e. [[module]]s over some [[ring]] or [[field]], and in fact [[chain complex]]es of such modules.
Therefore an $A_\infty$-category in this standard sense of [[homological algebra]] is a category which is in some way [[homotopical enrichment|homotopically enriched]] over a [[category of chain complexes]] $Ch$. Since a category which is [[enriched category|enriched]] in the ordinary sense of [[enriched category theory]] is a [[dg-category]], there is a close relation between $A_\infty$-categories and [[dg-category|dg-categories]].
$A_\infty$-categories in this linear sense are a [[horizontal categorification]] of the notion of [[A-infinity-algebra]]. As such they are to [[A-infinity-algebra]]s as [[Lie infinity-algebroid]]s are to [[L-infinity-algebra]]s. For this point of view see [Konsevich-Soibelman 08](#KonsevichSoibelman08).
+-- {: .num_defn}
###### Definition
A category $C$ such that
1. for all $X,Y$ in $Ob(C)$ the [[hom-set|Hom-set]]s $Hom_C(X,Y)$ are finite dimensional [[chain complex]]es of $\mathbf{Z}$-graded modules
2. for all objects $X_1,...,X_n$ in $Ob(C)$ there is a family of linear composition maps (the higher compositions)
$m_n : Hom_C(X_0,X_1) \otimes Hom_C(X_1,X_2) \otimes \cdots \otimes Hom_C(X_{n-1},X_n) \to Hom_C(X_0,X_n)$ of degree $n-2$ (homological grading convention is used) for $n\geq1$
3. $m_1$ is the differential on the chain complex $Hom_C(X,Y)$
4. $m_n$ satisfy the quadratic $A_\infty$-associativity equation for all $n\geq0$.
=--
$m_1$ and $m_2$ will be [[chain complex|chain map]]s but the compositions $m_i$ of higher order are not chain maps, nevertheless they are [[Massey product]]s.
The framework of dg-categories and dg-functors is too narrow for many problems, and it is preferable to consider the wider class of $A_\infty$-categories and $A_\infty$-functors. Many features of $A_\infty$-categories and $A_\infty$-functors come from the fact that they form a symmetric closed [[multicategory]], which is revealed in the language of [[comonad|comonads]].
From a higher dimensional perspective $A_\infty$-categories are weak $\omega$-categories with all morphisms invertible. $A_\infty$-categories can also be viewed as noncommutative formal dg-manifolds with a closed marked subscheme of objects.
### Examples (and remarks) ##
* Every [[dg-category]] may be regarded as a special case when there are no higher maps (trivial homotopies) of an $A_\infty$-category.
* Every $A_\infty$-category is $A_\infty$-equivalent to a [[dg-category]].
* This is a corollary of the $A_\infty$-categorical [[Yoneda lemma]].
* beware that this statement does not imply that the notion of
$A_\infty$-categories is obsolete (see section 1.8 in Bespalov et al.):
in practice it is often easier to work with a given naturally
arising $A_\infty$-category than constructing its equivalent [[dg-category]]
* for instance when dealing with a [[Fukaya A-infinity-category|Fukaya]] $A_\infty$-category;
* or when dealing with various constructions on dg-categories, for instance certain quotients,that naturally yield directly $A_\infty$-categories instead of dg-categories.
* The [[path space ]] of a topological space $X$
* The [[Fukaya category]] $Fuk(X)$ of a topological space $X$ -- a [[Calabi-Yau A-∞ category]]
* $A_\infty$-[[A-infinity-algebra|algebras]] are the $A_\infty$-categories with one object.
* For example, the delooping $\mathbf{B}\Omega{X}$ of [[loop space]] $\Omega{X}$ of a topological space $X$
### More general $A_\infty$-categories ##
In the widest sense, $A_\infty$-category may be used as a term for a category in which the [[composition]] operation constitutes an algebra over an [[operad]] which resolves in some sense the associative operad $Ass$.
One should be aware, though, that this use of the term is not understood by default in the large body of literature concerned with the above linear notion.
A less general but non-linear definition is fairly straight forward in any category in which there is a notion of _homotopy_ with the usual properties.
+-- {: .un_defn}
###### Definition
An $A_\infty$-category is a [[category over an operad|category over]]
the $A_\infty$-[[A-infinity operad|operad]]: e.g. the free resolution
in the context of dg-operads of the linear associative operad.
=--
+-- {: .num_example}
###### Examples
* [[Trimble n-category|Trimble n-categories]];
* also the classical notion of [[bicategory]] can be interpreted as an $A_\infty$-category in [[Cat]] for a suitable Cat-operad.
=--
## Related concepts
* [[A-∞ space]], [[associahedron]]
* [[A-∞ algebra]]
* [[stable (∞,1)-category]]
## References #
### For $A_\infty$-categories in the sense of homological algebra
For a short and precise introduction see
* B. Keller, _Introduction to $A_\infty$-algebras and modules_ (<a href="http://people.math.jussieu.fr/~keller/publ/ioan.dvi">dvi</a>, <a href="http://people.math.jussieu.fr/~keller/publ/ioan.ps">ps</a>) and Addendum (<a href="http://people.math.jussieu.fr/~keller/publ/addendum.ps">ps</a>), Homology, Homotopy and Applications 3 (2001), 1-35;
* B. Keller, _$A_\infty$ algebras, modules and functor categories_, (<a href="http://people.math.jussieu.fr/~keller/publ/ainffun.dvi">pdf</a>, <a href="http://people.math.jussieu.fr/~keller/publ/ainffun.ps">ps</a>).
and for a [[Fukaya category]]-oriented introduction see chapter 1 in
* P. Seidel, _Fukaya category and Picard-Lefschetz theory_
A very detailed treatment of $A_\infty$-categories is a recent book
* Yu. Bespalov, [[Volodymyr Lyubashenko]], O. Manzyuk, _Pretriangulated $A_\infty$-categories_, Proceedings of the Institute of Mathematics of NAS of Ukraine, vol. 76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008, 598 ([ps.gz](http://www.math.ksu.edu/~lub/cmcMonad.gz))
* notice: the ps.gz file has different page numbers than the printed version, but the numbering of sections and formulae is final. Errata to published version are [here](http://www.math.ksu.edu/~lub/cmcMoCor.pdf).
* Oleksandr Manzyk, _A-infinity-bimodules and Serre A-infinity-functors_, dissertation [pdf](https://kluedo.ub.uni-kl.de/files/1910/dissertation.pdf), [djvu](https://kluedo.ub.uni-kl.de/files/1910/dissertation.djvu); _Serre $A_\infty$ functors_, talk at Categories in geometry and math. physics, Split 2007, slides, [pdf](http://www.irb.hr/korisnici/zskoda/manzyukslides.pdf), work with [[Volodymyr Lyubashenko]]
The relation of $A_\infty$-categories to [[differential graded algebra]]s is emphasized in the introduction of
* {#KonsevichSoibelman08} [[Maxim Kontsevich]], [[Yan Soibelman]], _Notes on $A_\infty$-algebras, $A_\infty$-categories and non-commutative geometry. I_, in [[Anton Kapustin]], [[Maximilian Kreuzer]], [[Karl-Georg Schlesinger]] (eds.) _Homological mirror symmetry -- New developments and perspectives_, Springer 2008 ([arXiv:math/0606241](http://arxiv.org/abs/math/0606241), [doi:10.1007/978-3-540-68030-7](https://www.springer.com/de/book/9783540680291))
which mostly discusses just [[A-infinity-algebra]]s, but points out a generalizations to $A_\infty$-categories, see the overview on [p. 3](http://arxiv.org/PS_cache/math/pdf/0606/0606241v2.pdf#page=3)
Essentially the authors say that an $A_\infty$-category should be a non(-graded-)commutative [[dg-manifold]]/[[L-infinity-algebroid]].
More [[category theory]] and [[homotopy theory]] of $A_\infty$-categories is discussed in
* {#LefevreHasegawa03} Kenji Lefèvre-Hasegawa, _Sur les A-infini catégories_ ([arXiv:math/0310337](http://arxiv.org/abs/math/0310337))
* [[Bruno Valette]], _Homotopy theory of homotopy algebras_, Annales de l'Institut Fourier __70__:2 (2020) 683--738 [doi](https://10.5802/aif.3322) [Zbl:1335.18001](https://zbmath.org/?q=an:1335.18001) [arXiv:1411.5533](https://arxiv.org/abs/1411.5533)
See also
* [[Yong-Geun Oh]], [[Hiro Lee Tanaka]], _$A_\infty$-categories, their $\infty$-categories, and their localizations_ ([arXiv:2003.05806](https://arxiv.org/abs/2003.05806))
### For $A_\infty$-categories in the wider sense ##
If one understands $A_\infty$-category as "operadically defined higher category", then relevant references would include:
* Eugenia Cheng, _Comparing operadic definitions of $n$-category_ ([arXiv](http://arxiv.org/abs/0809.2070))
With operads modeled by [[dendroidal sets]], [[n-categories]] for low $n$ viewed as objects with an $A-\infty$-composition operation are discussed in section 5 of
* Andor Lucacs, _Cyclic Operads, Dendroidal Structures, Higher Categories_ ([pdf](http://igitur-archive.library.uu.nl/dissertations/2011-0211-200314/lukacs.pdf))
and
* Andor Lucacs, _Dendroidal weak 2-categories_ ([arXiv:1304.4278](http://de.arxiv.org/abs/1304.4278))
See also the references at _[[model structure on algebras over an operad]]_.
[[!redirects A-infinity category]]
[[!redirects A-infinity categories]]
[[!redirects A-∞ category]]
[[!redirects A-∞ categories]]
[[!redirects A? category]]
[[!redirects A? categories]]
[[!redirects A∞-category]]
[[!redirects A∞-categories]]
[[!redirects A-∞-category]]
[[!redirects A-∞-categories]] |
A-infinity-cocategory | https://ncatlab.org/nlab/source/A-infinity-cocategory |
The concept _$A_\infty$-cocategory_ is the result of combining [[A-infinity-category]] with [[cocategory]]. |
A-infinity-module | https://ncatlab.org/nlab/source/A-infinity-module | ## Idea
__$A_\infty$-module__ $M$ over an $A_\infty$-algebra $A$ is an object with an $\infty$-[[infinity-representation|representation]] of $A$, modelled in the same formalism as the $A_\infty$-algebras are (usually in $k$-linear context, most often in (∞,1)-categories of chain complexes). In particular, one considers $A_\infty$-modules over an associative algebra or a dg-algebra viewed as a special case of an $A_\infty$-algebra.
## Definition
...
## Literature
* [[Bernhard Keller]], _Introduction to A-infinity algebras and modules_, Homology Homotopy Appl. 3(1) (2001) 1--35 arXiv:[math.RA/9910179](https://arxiv.org/abs/math/9910179) addendum.[dvi](https://webusers.imj-prg.fr/~bernhard.keller/publ/addendum.dvi)
* [[Bernhard Keller]], _Bimodule complexes via strong homotopy actions_, Algebr. Represent. Theory 3(4) (2000) 357--376
* [[Yuri Berest]], Oleg Chalykh, _$A_\infty$-modules and Calogero-Moser spaces_, J. Reine Angew. Math. __607__ (2007) 69--112 [MR2009f:16019](http://www.ams.org/mathscinet-getitem?mr=2338121) [doi](https://doi.org/10.1515/CRELLE.2007.046)
category: algebra
|
A-infinity-ring | https://ncatlab.org/nlab/source/A-infinity-ring |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Higher linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Stable homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An $A_\infty$-ring is a [[monoid in an (∞,1)-category]] in an additive (that is, [[stable (infinity,1)-category|stable]]) [[(∞,1)-category]]. Alternatively one can take a model which is a (non-homotopic) additive monoidal category, but the monoid is replaced by an algebra over a resolution of the associative operad.
For example there is a variant for the [[stable (∞,1)-category of spectra]]. Sometimes this is called an [[associative ring spectrum]].
This may be modeled equivalently as an ordinary [[monoid]] with respect to the [[symmetric monoidal smash product of spectra]].
Notice the difference to an ordinary [[ring spectrum]] which which is not necessarily coherently homotopy-associative.
$A_\infty$-rings play the role of [[ring]]s in [[higher algebra]].
The higher analog of a commutative ring is an [[E-∞ ring]].
## Related concepts
* [[ring]], [[ring groupoid]]
* [[ring spectrum]], [[functor with smash products]]
* [[symmetric ring spectrum]]
* [[cohomology theory]]
* [[periodic ring spectrum]]
## References
See the references at *[[ring spectrum]]*.
Another version of the $A_\infty$-ring is simply what is usually called the $A_\infty$-[[A-infinity-algebra|algebra]] in the case when the [[ground ring]] is the ring of integers. See
* Gerald Dunn, _Lax operad actions and coherence for monoidal $n$-Categories, $A_{\infty}$ rings and modules_, Theory Appl. Cat. 1997, n.4 ([TAC](http://www.emis.de/journals/TAC/volumes/1997/n4/3-04abs.html))
[[!redirects A-infinity-rings]]
[[!redirects A-infinity ring]]
[[!redirects A-infinity rings]]
[[!redirects A∞-ring]]
[[!redirects A∞-rings]]
[[!redirects A-∞-ring]]
[[!redirects A-∞ ring]]
[[!redirects A-∞ rings]]
[[!redirects ∞-ring]]
[[!redirects ∞-rings]]
[[!redirects infinity-ring]]
[[!redirects infinity-rings]]
[[!redirects A-∞ ring spectrum]]
[[!redirects A-infinity ring spectrum]]
[[!redirects A-∞ ring spectra]]
[[!redirects A-infinity ring spectra]]
|
A-infinity-space | https://ncatlab.org/nlab/source/A-infinity-space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
An **$A_\infty$-space** is a [[homotopy type]] $X$ that is equipped with the structure of a [[monoid]] up to [[coherence|coherent]] higher [[homotopy]]:
that means it is equipped with
1. a binary product operation $\cdot \colon X \times X \to X$
1. a choice of [[associativity]] [[homotopy]]; $\eta_{x,y,z} : (x\cdot y) \cdot z \to x \cdot (y \cdot z)$;
1. a choice of [[pentagon law]] homotopy between five such $\eta$s;
1. and so ever on, as controlled by the [[associahedron|associahedra]].
In short one may say: an **$A_\infty$-space** is an [[A-∞ algebra]]/[[monoid in an (∞,1)-category]] in the [[(∞,1)-category]] [[∞Grpd]]/[[Top]]. See there for more details.
## Properties
### Relation to H-monoids
If in the definition of an $A_\infty$-space one discards all the higher homotopies and retains only the _existence_ of an [[associativity]]-[[homotopy]], then one has the notion of _[[H-monoid]]_. Put another way, An $A_\infty$-space in the [[(∞,1)-category]] [[∞Grpd]]/[[Top]] becomes an [[H-monoid]] in the [[homotopy category of an (∞,1)-category|homotopy]] [[Ho(Top)]]. And lifting an [[H-monoid]] structure to an $A_\infty$-space structure means lifting a monoid structure through the projection from the [[(∞,1)-category]] [[∞Grpd]]/[[Top]] to [[Ho(Top)]].
### Relation to $A_\infty$-categories
The [[delooping]] of an $A_\infty$-space is an [[A-∞ category]]/[[(∞,1)-category]] with a single object. (Beware that in standard literature "$A_\infty$-category" is often, but not necessarily, reserved for a _[[stable (∞,1)-category]]_).
There is an [[equivalence of (∞,1)-categories]] between pointed connected [[A-∞ categories]]/[[(∞,1)-categories]] and $A_\infty$-spaces.
## Related concepts
[[!include k-monoidal table]]
* [[model structure for dendroidal left fibrations]]
* [[H-space]], [[A-n space]]
* [[monoid]], [[monoidal groupoid]]
* [[k-tuply associative n-groupoid]]
* [[A-infinity category]]
## References
$A_\infty$-spaces were introduced by [[Jim Stasheff]] as a refinement of an [[H-group]] taking into account higher [[coherence|coherences]].
* [[Jim Stasheff]], _Homotopy associative H-spaces I_, _II_, Trans. Amer. Math. Soc. __108__ (1963), 275--312 [MR158400](http://www.ams.org/mathscinet-getitem?mr=158400)
[[!redirects A-∞-space]]
[[!redirects A∞-space]]
[[!redirects A-infinity-spaces]]
[[!redirects A-∞-spaces]]
[[!redirects A∞-spaces]]
[[!redirects A-infinity space]]
[[!redirects A-∞ space]]
[[!redirects A? space]]
[[!redirects A-infinity spaces]]
[[!redirects A-∞ spaces]]
[[!redirects A? spaces]] |
A-model | https://ncatlab.org/nlab/source/A-model |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
What is called the _A-model_ [[topological string]] is the 2-dimensional [[topological conformal field theory]] corresponding to the [[Calabi–Yau category]] called the [[Fukaya category]] of a [[symplectic manifold]] $(X,\omega)$. This is the [[Poisson sigma-model]] of the underlying [[Poisson manifold]] after appropriate [[gauge fixing]] ([AKSZ 97, p 19](#AKSZ)). The A-model on $X$ is effectively the [[Gromov–Witten theory]] of $X$.
The A-model arose in formal [[physics]] from considerations of [[string theory|superstring]]-propagation on [[Calabi-Yau spaces]]: it may be motivated by considering the [[vertex operator algebra]] of the 2d[[CFT|SCFT]] given by the [[supersymmetric sigma-model]] with [[target space]] $X$ and then deforming it such that one of the super-[[Virasoro algebra|Virasoro]] generators squares to $0$. The resulting "topologically twisted" algebra may then be read as being the [[BRST complex]] of a [[TCFT]].
One can also define an A-model for [[Landau-Ginzburg model|Landau–Ginzburg models]]. The category of [[D-brane|D-branes]] for the corresponding [[open string]] theory is given by the [[Fukaya–Seidel category]].
By [[homological mirror symmetry]], the A-model is dual to the [[B-model]].
## Properties
### Lagrangian
The [[action functional]] of the A-model is that associated by [[AKSZ theory]] to a Lagrangian [[submanifold]] in a [[target space|target]] [[symplectic Lie n-algebroid]] which is the [[Poisson Lie algebroid]] of a [[symplectic manifold]].
See the _[references on Lagrangian formulation](#LagrangianLit)_.
### Boundary theory / holography
On [[coisotropic submanifold|coisotropic]] [[branes]] in symplectic target manifolds that arise by complexification of [[phase spaces]], the boundary [[path integral]] of the A-model computes the [[quantization]] of that phase space. For details see
* [[quantization via the A-model]].
and
* [[holographic principle]].
### Second quantization / effective background field theory
{#SecondQuantization}
The [[second quantization]] [[effective field theory|effective]] background field theory defined by the [[perturbation series]] of the A-model string has been argued to be [[Chern-Simons theory]]. ([Witten 92](#Witten92), [Costello 06](#Costello06))
For more on this see at _[TCFT -- Worldsheet and effective background theories](http://ncatlab.org/nlab/show/TCFT#ActionFunctionals)_. A related mechanism is that of _[[world sheets for world sheets]]_.
## Related concepts
* [[schreiber:∞-Chern-Simons theory]]
* [[sigma-model]]
* [[AKSZ sigma-model]]
* [[Poisson sigma-model]]
* **A-model**, [[B-model]]
* [[half-twisted model]]
* [[Courant sigma-model]]
* [[Chern-Simons theory]]
* [[topological membrane]]
* [[topologically twisted D=4 super Yang-Mills theory]]
* [[Landau-Ginzburg model]]
[[!include gauge theory from AdS-CFT -- table]]
## References
### General
The A-model was first conceived in
* [[Edward Witten]], _Topological sigma models_, Commun. Math. Phys. __118__ (1988) 411--449, [euclid](http://projecteuclid.org/euclid.cmp/1104162092), [MR90b:81080](http://www.ams.org/mathscinet-getitem?mr=0958805)
An early review is in
* [[Edward Witten]]. _Mirror manifolds and topological field theory_, in: Essays on mirror manifolds, pp. 120–-158. Int. Press, Hong Kong, 1992. ([arXiv:hep-th/9112056](http://arxiv.org/abs/hep-th/9112056)).
The motivation from the point of view of [[string theory]] is reviewed for instance in
* [[Paul Aspinwall]], _D-Branes on Calabi-Yau Manifolds_ ([arXiv:hep-th/0403166](http://arxiv.org/abs/hep-th/0403166))
A summary of these two reviews is in
* H. Lee, _Review of topological field theory and homological mirror symmetry_ ([pdf](http://people.maths.ox.ac.uk/leeh/files/CYMSmini.pdf))
### Action functional {#LagrangianLit}
That the A-model [[Lagrangian]] arises in [[AKSZ theory]] by [[gauge fixing]] the [[Poisson sigma-model]] was observed in
* {#AKSZ} M. Alexandrov, [[Maxim Kontsevich|M. Kontsevich]], [[Albert Schwarz|A. Schwarz]], O. Zaboronsky, around page 19 in _The geometry of the master equation and topological quantum field theory_, Int. J. Modern Phys. A 12(7):1405--1429, 1997
with more details in
* {#BonechiCattaneoIraso16} Francesco Bonechi, [[Alberto Cattaneo]], Riccardo Iraso, _Comparing Poisson Sigma Model with A-model_ ([arXiv:1607.03411](http://arxiv.org/abs/1607.03411))
Review and further discussion includes
* Francesco Bonechi, [[Maxim Zabzine]], section 5.3 of _Poisson sigma model on the sphere_ ([arXiv:0706.3164](http://arxiv.org/abs/0706.3164))
Also
* [[Noriaki Ikeda]], _Deformation of graded (Batalin-Volkvisky) Structures_, in Dito, Lu, Maeda, [[Alan Weinstein]] (eds.) _Poisson geometry in mathematics and physics_ Contemp. Math. 450, AMS (2008)
Discussion of how the [[second quantization]] [[effective field theory]] given by the A-model [[perturbation series]] is [[Chern-Simons theory]] is in
* [[Edward Witten]], _Chern-Simons Gauge Theory As A String Theory_, Prog.Math. 133 (1995) 637-678 ([arXiv:hep-th/9207094](http://arxiv.org/abs/hep-th/9207094))
{#Witten92}
* [[Kevin Costello]], _Topological conformal field theories and gauge theories_ ([arXiv:math/0605647](http://arxiv.org/abs/math/0605647))
{#Costello06}
formalizing at least aspects of the observations in
[[!redirects A-model]]
[[!redirects A-models]] |
A-n space | https://ncatlab.org/nlab/source/A-n+space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
An **$A_n$-space** or **$A_n$-algebra in spaces** is a space (in the sense of an [[homotopy type]]/[[infinity-groupoid|$\infty$-groupoid]], usually presented by a [[topological space]] or a [[simplicial set]]) with a multiplication that is associative up to higher homotopies involving up to $n$ variables.
* An $A_0$-space is a pointed space.
* Same with an $A_1$-space.
* An $A_2$-space is an [[H-space]].
* An [[A_3-space|$A_3$-space]] is a homotopy associative H-space (but no coherence is required of the associator).
* An $A_4$-space has an [[associativity]] homotopy that satisfies the [[pentagon identity]] up to [[homotopy]], but no further [[coherence law|coherence]].
* ...
* An [[A-infinity space|$A_\infty$-space]] has all [[coherence|coherent]] higher associativity homotopies.
## Related concepts
* [[A-infinity space|$A_\infty$-space]]
* [[E-n algebra]]
* [[A2-groupoid]]
* [[A3-groupoid]]
* [[k-tuply associative n-groupoid]]
## References
The notion is originally due to:
* {#Stasheff63a} [[Jim Stasheff]], *Homotopy associativity of H-spaces I*, Trans. Amer. Math. Soc. **108** 2 (1963) 275-292 [[doi:10.2307/1993608](https://doi.org/10.2307/1993608)]
* {#Stasheff63b} [[Jim Stasheff]], *Homotopy associativity of H-spaces II* **108** 2 (1963) 293-312 [[doi:10.2307/1993609](https://doi.org/10.2307/1993609), [doi:10.1090/S0002-9947-1963-0158400-5](https://doi.org/10.1090/S0002-9947-1963-0158400-5)]
Early review:
* {#Adams78} [[John Adams]], §2.2 of: _Infinite loop spaces_, Hermann Weyl lectures at IAS 1974, Annals of Mathematics Studies **90**, Princeton University Press (1978) [[ISBN:9780691082066](https://press.princeton.edu/books/paperback/9780691082066/infinite-loop-spaces-am-90-volume-90), [doi:10.1515/9781400821259](https://doi.org/10.1515/9781400821259)]
[[!redirects A-n-space]]
[[!redirects An-space]]
[[!redirects A_n-space]]
[[!redirects A-n-spaces]]
[[!redirects An-spaces]]
[[!redirects A_n-spaces]]
[[!redirects A-n space]]
[[!redirects An space]]
[[!redirects A_n space]]
[[!redirects A-n spaces]]
[[!redirects An spaces]]
[[!redirects A_n spaces]]
[[!redirects A-n algebra]]
[[!redirects A-n algebras]]
[[!redirects A-1 algebra]]
[[!redirects A-1 algebras]]
[[!redirects A1 space]]
[[!redirects A1 spaces]]
[[!redirects A2 space]]
[[!redirects A2 spaces]]
[[!redirects A4 space]]
[[!redirects A4 spaces]]
[[!redirects A5 space]]
[[!redirects A5 spaces]]
[[!redirects A1-space]]
[[!redirects A1-spaces]]
[[!redirects A2-space]]
[[!redirects A2-spaces]]
[[!redirects A4-space]]
[[!redirects A4-spaces]]
[[!redirects A5-space]]
[[!redirects A5-spaces]] |
A-theory | https://ncatlab.org/nlab/source/A-theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Waldhausen's _A-theory_ ([Waldhausen 85](#Waldhausen85)) of a connected [[homotopy type]] $X$ is the [[algebraic K-theory]] of the [[suspension spectrum]] $\Sigma^\infty_+ (\Omega X)$ of the [[loop space]] $\Omega X$, hence of the [[∞-group ∞-rings]] $\mathbb{S}[\Omega X]$ of the [[looping]] [[∞-group]] $\Omega X$, hence the K-theory of the [[parametrized spectra]] over $X$ ([Hess-Shipley 14](#HessShipley14)).
## Related concepts
* [[iterated algebraic K-theory]]
* [[coring spectrum]]
## References
The definition is originally due to
* {#Waldhausen85} [[Friedhelm Waldhausen]], _Algebraic K-theory of spaces_ Algebraic and geometric topology (Ne Brunswick, N. J., 1983), 318–419, Lecture Notes in Math., 1126, Springer, Berlin, 1985
The interpretation in terms of certain [[module spectra]] over the [[Spanier-Whitehead dual]] of $X$ is due to
* [[Andrew Blumberg]], [[Michael Mandell]], _Derived Koszul Duality and Involutions in the Algebraic K-Theory of Spaces_, J. Topology 4 (2011), no. 2, 327-342 ([arXiv:0912.1670](https://arxiv.org/abs/0912.1670))
and the interpretation in terms of $\mathbb{S}[\Omega X]$-[[module spectra]] and [[Koszul duality|Koszul dually]] in terms of $\mathbb{S}[X]$-[[comodule spectra]] (using the canonical [[coring spectrum]]-[[structure]] of [[suspension spectra]]) is due to:
* {#HessShipley14} [[Kathryn Hess]], [[Brooke Shipley]], _Waldhausen K-theory of spaces via comodules_, Advances in Mathematics **290** (2016) 1079-1137 [[arXiv:1402.4719](https://arxiv.org/abs/1402.4719), [doi:10.1016/j.aim.2015.12.019](https://doi.org/10.1016/j.aim.2015.12.019)]
[[!redirects Waldhausen a-theory]]
|
A. Ch. Ganchev | https://ncatlab.org/nlab/source/A.+Ch.+Ganchev |
* [arXiv page](https://arxiv.org/search/hep-th?searchtype=author&query=Ganchev%2C+A)
## Selected writings
On [[WZW models]] at [[rational number|fractional]] [[level (Chern-Simons theory)|level]]:
* P. Furlan, [[A. Ch. Ganchev]], R. Paunov, [[Valentina B. Petkova]], *Solutions of the Knizhnik-Zamolodchikov Equation with Rational Isospins and the Reduction to the Minimal Models*, Nucl. Phys. B394 (1993) 665-706 ([arXiv:hep-th/9201080](https://arxiv.org/abs/hep-th/9201080), <a href="https://doi.org/10.1016/0550-3213(93)90227-G">doi:10.1016/0550-3213(93)90227-G</a>)
* [[A. Ch. Ganchev]], [[Valentina B. Petkova]], G. M. T. Watts, *A note on decoupling conditions for generic level $\widehat{s l}(3)_k$ and fusion rules*, Nucl. Phys. B **571** (2000) 457-478 $[$<a href="https://doi.org/10.1016/S0550-3213(99)00745-2">doi:10.1016/S0550-3213(99)00745-2</a>$]$
category: people |
A. G. Walker | https://ncatlab.org/nlab/source/A.+G.+Walker | Arthur Geoffrey Walker was a professor at the University of Liverpool,
working in [[differential geometry]], [[general relativity]], and [[cosmology]].
In particular, he is responsible for the [[Friedmann–Lemaître–Robertson–Walker metric]] in [[cosmology]].
## Selected writings
On [[vector fields as derivations]]:
* [[W. F. Newns]], [[A. G. Walker]], _Tangent Planes To a Differentiable Manifold_. Journal of the London Mathematical Society s1-31:4 (1956), 400–407 ([doi:10.1112/jlms/s1-31.4.400](https://doi.org/10.1112/jlms/s1-31.4.400))
category: people
[[!redirects Arthur Geoffrey Walker]]
[[!redirects Arthur G. Walker]]
[[!redirects Arthur Walker]]
|
A. J. Tolland > history | https://ncatlab.org/nlab/source/A.+J.+Tolland+%3E+history | < [[A. J. Tolland]]
|
A. P. Balachandran | https://ncatlab.org/nlab/source/A.+P.+Balachandran | __A. P. Balachandran__ is an indian-american theoretical physicist, retired from the Syracuse University.
* [Wikipedia entry](https://en.wikipedia.org/wiki/A._P._Balachandran)
## Selected writings
On the notion of [[entropy]] in [[quantum probability theory]]:
* [[A. P. Balachandran]], T. R. Govindarajan, Amilcar R. de Queiroz, A. F. Reyes-Lega, _Algebraic approach to entanglement and entropy_, Phys. Rev. A 88, 022301 (2013) ([arXiv:1301.1300](http://arxiv.org/abs/1301.1300))
* [[A. P. Balachandran]], A. R. de Queiroz, S. Vaidya, *Entropy of Quantum States: Ambiguities*, Eur. Phys. J. Plus 128, 112 (2013) ([arXiv:1212.1239](https://arxiv.org/abs/1212.1239), [doi:10.1140/epjp/i2013-13112-3](https://doi.org/10.1140/epjp/i2013-13112-3))
On [[fiber bundles in physics]]:
* [[A. P. Balachandran]], G. Marmo, B.-S. Skagerstam, A. Stern, _Gauge Symmetries and Fibre Bundles_, Lect. Notes in Physics 188, Springer-Verlag, Berlin, 1983 ([arXiv:1702.08910](https://arxiv.org/abs/1702.08910))
## Related $n$Lab pages
* [[fiber bundles in physics]]
* [[entanglement]],
* [[entropy]]
* [[Hopf fibration]]
category: people |
A. Yung | https://ncatlab.org/nlab/source/A.+Yung |
## Selected writings
On [[vortex strings]] in $\mathcal{N}=2$ super QCD identified as [[superstrings]] in [[type II string theory]] [[KK-compactification|compactified]] on a [[conifold]]:
* [[Mikhail Shifman]], [[A. Yung]], *Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories*, Rev. Mod. Phys. **79** 1139 (2007) [[doi:10.1103/RevModPhys.79.1139](https://doi.org/10.1103/RevModPhys.79.1139), [arXiv:hep-th/0703267](https://arxiv.org/abs/hep-th/0703267)
* [[Mikhail Shifman]], [[A. Yung]], *Supersymmetric Solitons*, Cambridge University Press (2009) [[doi:10.1017/CBO9780511575693](https://doi.org/10.1017/CBO9780511575693)]
* [[Mikhail Shifman]], [[A. Yung]], *Critical String from Non-Abelian Vortex in Four Dimensions*,
Physics Letters B **750** (2015) 416-419
[[arXiv:1502.00683](https://arxiv.org/abs/1502.00683), [doi:10.1016/j.physletb.2015.09.045](https://doi.org/10.1016/j.physletb.2015.09.045)
]
* P. Koroteev, [[Mikhail Shifman]], [[A. Yung]], *Studying Critical String Emerging from Non-Abelian Vortex in Four Dimensions*, Phys.Lett. B **759** (2016) 154-158
[[arXiv:1605.01472](https://arxiv.org/abs/1605.01472), [doi:10.1016/j.physletb.2016.05.075](https://doi.org/10.1016/j.physletb.2016.05.075)
]
and interpreted in [[AdS/QCD]], in the presence of the [[Kalb-Ramond field]]:
* [[A. Yung]], *NS Three-form Flux Deformation for the Critical Non-Abelian Vortex String* [[arXiv:2209.08118](https://arxiv.org/abs/2209.08118)]
category: people |
A.A. Sharapov | https://ncatlab.org/nlab/source/A.A.+Sharapov |
* [publications](http://arxiv.org/find/hep-th/1/au:+Sharapov_A/0/1/0/all/0/1)
## Selected writings
On [[non-Lagrangian field theory]] such as [[self-dual higher gauge theory]]:
* S. L. Lyakhovich, [[A. A. Sharapov]], _Quantizing non-Lagrangian gauge theories: an augmentation method_, JHEP 0701 047 (2007) [[arXiv:hep-th/0612086](http://arxiv.org/abs/hep-th/0612086)]
category: people
[[!redirects A. A. Sharapov]]
[[!redirects A. Sharapov]] |
A1 | https://ncatlab.org/nlab/source/A1 |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In the [[ADE-classification]], the items labeled $A1$ include the following:
1. as [[finite subgroups of SO(3)]]:
the [[cyclic group of order 2]]
$\mathbb{Z}/2$
1. as [[finite subgroups of SU(2)]]:
the [[cyclic group of order 2]]
$\mathbb{Z}/2$
1. as [[simple Lie groups]]: the [[special unitary group]] in 2 complex dimensions
[[SU(2)]]
1. as a [[Dynkin diagram]]/[[Dynkin quiver]]:
\begin{tikzpicture}
\node (center) at (0,0) {};
\draw[fill=black] (center) circle (.1);
\end{tikzpicture}
\linebreak
## Related concepts
[[!include ADE -- table]]
|
A1-cohesive homotopy type theory | https://ncatlab.org/nlab/source/A1-cohesive+homotopy+type+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
\tableofcontents
## Idea
**$\mathbb{A}^1$-cohesive homotopy type theory**, **affine-cohesive homotopy type theory**, or **algebraic-cohesive homotopy type theory** is a version of [[cohesive homotopy type theory]] which has an [[affine line]] [[type]] $\mathbb{A}^1$ and the [[axiom of A1-cohesion|axiom of $\mathbb{A}^1$-cohesion]]. $\mathbb{A}^1$-cohesive homotopy type theory provides a [[synthetic mathematics|synthetic]] [[foundation]] for [[A1-homotopy theory|$\mathbb{A}^1$-homotopy theory]], where the affine line is the standard [[affine line]] in the [[Nisnevich site]] of [[smooth schemes]] of [[finite type]] over a [[Noetherian scheme]].
### Motivic homotopy type theory
{#MotivicHomotopyTypeTheory}
[[Mitchell Riley]]'s [[bunched logic|bunched]] [[linear homotopy type theory]] [[Riley (2022)](dependent+linear+type+theory#Riley22)] is a [[synthetic mathematics|synthetic]] foundations for [[parameterized stable homotopy theory|parameterized]] [[stable homotopy theory]]. This means that there should be an $\mathbb{A}^1$-cohesive [[bunched logic|bunched]] [[linear homotopy type theory]] which behaves as a synthetic foundations for [[motivic homotopy theory]].
## Presentation
Similarly to [[real-cohesive homotopy type theory]], we assume a [[spatial type theory]] presented with crisp term judgments $a::A$. In addition, we also assume the spatial type theory has an [[affine line]] [[type]] $\mathbb{A}^1$, and $\mathbb{A}^1$-[[localization of a type at a family of functions|localizations]] $\mathcal{L}_{\mathbb{A}^1}(-)$.
Given a type $A$, let us define $\mathrm{const}_{A, \mathbb{A}^1}:A \to (\mathbb{A}^1 \to A)$ to be the type of all constant functions in the [[affine line]] $\mathbb{A}^1$:
$$\delta_{\mathrm{const}_{A, \mathbb{A}^1}}(a, r):\mathrm{const}_{A, \mathbb{A}^1}(a)(r) =_A a$$
There is an equivalence $\mathrm{const}_{A, 1}:A \simeq (1 \to A)$ between the type $A$ and the type of functions from the [[unit type]] $1$ to $A$.
Given types $B$ and $C$ and a function $F:(B \to A) \to (C \to A)$, type $A$ is **$F$-[[localization of a type at a family of functions|local]]** if the function $F:(B \to A) \to (C \to A)$ is an [[equivalence of types]].
A crisp type $\Xi \vert () \vdash A$ is **discrete** if the function $(-)_\flat:\flat A \to A$ is an [[equivalence of types]].
The **axiom of $\mathbb{A}^1$-cohesion** states that for the crisp affine line $\Xi \vert () \vdash \mathbb{A}^1 \; \mathrm{type}$, given any crisp type $\Xi \vert () \vdash A \; \mathrm{type}$, $A$ is discrete if and only if $A$ is $(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, \mathbb{A}^1})$-local.
This allows us to define discreteness for non-crisp types: a type $A$ is **discrete** if $A$ is $(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, \mathbb{A}^1})$-local.
The [[shape modality]] in $\mathbb{A}^1$-cohesive homotopy type theory is then defined as the $\mathbb{A}^1$-localization $\esh(A) \coloneqq \mathcal{L}_{\mathbb{A}^1}(A)$, which ensures that the shape of $\mathbb{A}^1$ itself is a [[contractible type]].
## See also
* [[cohesive homotopy type theory]]
* [[A1-homotopy theory|$\mathbb{A}^1$-homotopy theory]]
* [[axiom of A1-cohesion|axiom of $\mathbb{A}^1$-cohesion]]
## References
For the presentation of the underlying [[spatial type theory]] used for $\mathbb{A}^1$-cohesive homotopy type theory, see:
* [[Mike Shulman]], *Brouwer's fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))
[[!redirects A1-cohesive homotopy type theory]]
[[!redirects A1 cohesive homotopy type theory]]
[[!redirects affine-cohesive homotopy type theory]]
[[!redirects affine cohesive homotopy type theory]]
[[!redirects algebraic-cohesive homotopy type theory]]
[[!redirects algebraic cohesive homotopy type theory]] |
a1-meson | https://ncatlab.org/nlab/source/a1-meson |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The $a_1$-meson is the [[light meson]] which (in the [[Wigner classification]]) is the [[Lorentz group|Lorentz]] [[spin group]] [[pseudo-vector representation]] as well as the [[isospin]] [[vector representation]].
This is the [[chiral partner]] of the [[rho-meson]].
\begin{imagefromfile}
"file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
"width": 600,
"unit": "px",
"margin": {
"top": 0,
"right": 10,
"bottom": 0,
"left": 20
}
\end{imagefromfile}
## Related concepts
* [[b1-meson]]
## References
* [[Stefan Leupold]], Markus Wagner, _Chiral Partners in a Chirally Broken World_, International Journal of Modern Physics A Vol. 24, No. 02n03, pp. 229-236 (2009) ([arXiv:0807.2389](https://arxiv.org/abs/0807.2389))
* A. E. Dorokhova, N. I. Kochelevba. A. P. Martynenkoc. F. A. Martynenkoc. A. E. Radzhabovb, _The contribution of axial-vector mesons to hyperfine structure of muonic hydrogen_, Physics Letters B Volume 776, 10 January 2018, Pages 105-110 Physics Letters B ([doi:10.1016/j.physletb.2017.11.027](https://doi.org/10.1016/j.physletb.2017.11.027))
[[!redirects a1-mesons]]
|
A2 | https://ncatlab.org/nlab/source/A2 |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In the [[ADE-classification]], the items labeled $A2$ include the following:
1. as [[finite subgroups of SO(3)]]:
the [[cyclic group of order 3]]
$\mathbb{Z}/3$
1. as [[finite subgroups of SU(2)]]:
the [[cyclic group of order 3]]
$\mathbb{Z}/3$
1. as [[simple Lie groups]]: the [[special unitary group]] in 3 complex dimensions
[[SU(3)]]
1. as a [[Dynkin diagram]]/[[Dynkin quiver]]:
\begin{tikzpicture}
\node (left) at (-.5,0) {};
\node (right) at (.5,0) {};
\draw[fill=black] (left) circle (.1);
\draw[fill=black] (right) circle (.1);
\draw (left) to (right);
\end{tikzpicture}
\linebreak
## Related concepts
[[!include ADE -- table]]
|
A3 | https://ncatlab.org/nlab/source/A3 |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In the [[ADE-classification]], the items labeled $A3$ include the following:
1. as [[finite subgroups of SO(3)]]:
the [[cyclic group of order 4]]
$\mathbb{Z}/4$
1. as [[finite subgroups of SU(2)]]:
the [[dicyclic group]]
$Dic_1 \simeq \mathbb{Z}/4$
1. as [[simple Lie groups]]: the [[special unitary group]] in 4 complex dimensions
[[SU(4)]]
1. as a [[Dynkin diagram]]/[[Dynkin quiver]]:
\begin{tikzpicture}
\node (center) at (0,0) {};
\node (left) at (-1,0) {};
\node (right) at (1,0) {};
\draw[fill=black] (center) circle (.1);
\draw[fill=black] (left) circle (.1);
\draw[fill=black] (right) circle (.1);
\draw (center) to (left);
\draw (center) to (right);
\end{tikzpicture}
\linebreak
## Properties
### Exceptional isomorphism
{#ExceptionalIsomorphism}
The A3 [[Dynkin diagram]] coincides with the [[D3]] diagram, hence with the result of removing one node from the [[D4]] diagram. In the [[classification of simple Lie groups]] this coincidence reflects the exceptional [[isomorphism]] between [[SU(4)]] and [[Spin(6)]]:
\begin{tikzpicture}
\node (Spin6) at (0,1.4) {$\mathrm{Spin}(6)$};
\node (SU4) at (3.4,1.4) {$\mathrm{SU}(4)$};
\node at (1.7,1.4) {$\simeq$};
\node (center) at (0,0) {};
\node (topright) at (30:1) {};
\node (left) at (180-30:1) {};
\node (botright) at (0,-1) {};
\draw[fill=black] (center) circle (.1);
\draw[fill=black] (topright) circle (.1);
\draw[draw=lightgray, fill=lightgray] (botright) circle (.1);
\draw[fill=black] (left) circle (.1);
\draw (center) to (topright);
\draw[lightgray] (center) to (botright);
\draw (center) to (left);
\begin{scope}[shift={(3.4,0)}]
\node (center) at (0,0) {};
\node (left) at (-1,0) {};
\node (right) at (+1,0) {};
\draw[fill=black] (center) circle (.1);
\draw[fill=black] (left) circle (.1);
\draw[fill=black] (right) circle (.1);
\draw (center) to (left);
\draw (center) to (right);
\end{scope}
\end{tikzpicture}
## Related concepts
[[!include ADE -- table]]
[[!redirects D3]]
|
A4-spatial groupoid > history | https://ncatlab.org/nlab/source/A4-spatial+groupoid+%3E+history | [[!redirects A4-spatial groupoid]]
see discussion [here](https://nforum.ncatlab.org/discussion/14220/ktuply-associative-ngroupoid/?Focus=100308#Comment_100308) |
Aarne Ranta | https://ncatlab.org/nlab/source/Aarne+Ranta | [webpage](http://www.cse.chalmers.se/~aarne/)
## Selected writings
On [[dependent type theory]] and natural language (parsing and [[syntactic sugar|sugaring]]):
* {#Ranta93} [[Aarne Ranta]], _Type theory and the informal language of mathematics_, TYPES '93: Proceedings of the international workshop on Types for proofs and programs, pp. 352–365, ([pdf](/nlab/files/InfLang.pdf))
* {#Ranta94} [[Aarne Ranta]], §1.6, §1.7, §9 in: *Type-theoretical grammar*, Oxford University Press (1994) [[ISBN:9780198538578](https://global.oup.com/academic/product/type-theoretical-grammar-9780198538578)]
## Related pages
* [[dependent type theoretic methods in natural language semantics]]
category: people |
Aaron Bergman | https://ncatlab.org/nlab/source/Aaron+Bergman |
* [arXiv entry](http://arxiv.org/find/hep-th/1/au:+Bergman_A/0/1/0/all/0/1)
## Selected writings
On [[Bridgeland stability conditions]] for [[B-branes]] of the [[B-model]] [[topological string]]:
* [[Aaron Bergman]], _Stability Conditions and Branes at Singularities_, Journal of High Energy Physics 2008.10 (2008): 07 ([arXiv:hep-th/0702092](http://arxiv.org/abs/hep-th/0702092))
Application of [Block 2005](Lie+infinity-algebroid+representation#Block05) to [[D-branes]] in [[topological string theory]]:
* {#Bergman08} [[Aaron Bergman]], _Topological D-branes from Descent_ ([arXiv:0808.0168](http://arxiv.org/abs/0808.0168))
On [[non-geometric string vacua]] ([[T-folds]]):
* [[Aaron Bergman]], Daniel Robbins, _Ramond-Ramond Fields, Cohomology and Non-Geometric Fluxes_ ([arXiv:0710.5158](http://arxiv.org/abs/0710.5158))
On [[T-duality]]:
* [[Katrin Becker]], [[Aaron Bergman]], _Geometric Aspects of D-branes and T-duality_ ([arXiv:0908.2249](http://arxiv.org/abs/0908.2249))
category: people
|
Aaron F | https://ncatlab.org/nlab/source/Aaron+F |
### September 2012 ###
Wow, time flies! I'm now a Ph.D. student at UT Austin, and you can find some of my writing and stuff on my [department home page](http://www.ma.utexas.edu/users/afenyes). Things I'm thinking about these days include...
* Quadratic differentials, [spectral networks](http://arxiv.org/abs/1204.4824), and $SL_2\mathbb{C}$ connections on Riemann surfaces (for a reading course).
* Classical and quantum $\mathbb{Z}/2\mathbb{Z}$ gauge theories on compact surfaces (for a [math club](http://www.ma.utexas.edu/mathclub/) talk).
* What the $\clubsuit\ast\mho\heartsuit$ is going on in [this paper](http://arxiv.org/abs/1105.4170) (for a graduate geometry seminar).
* Markov algorithms (maybe for a middle school summer program somewhere, someday).
### October 2009 ###
I'm a master's student interested in mathematical physics. Or is it physical mathematics? Here are some things I've been wondering about lately...
* The [general boundary formulation](http://arxiv.org/abs/hep-th/0509122) of quantum field theory.
* Whether or not it would be pedagogically useful to introduce topology as a special case of [[pretopological space|pretopology]].
+-- {: .query}
Welcome, Aaron! It might be good if you used your full name, although I don\'t think anybody will try to force you.
It also helps if you log your changes in the [latest changes](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5) category of the Forum. In particular, people will be much more likely to look at your work on [[induced representation]] if you do that.
I have also thought that pretopological spaces are pedagogically simpler than topological spaces. I also like to motivate [[uniform spaces]] as a uniform version of pretopological rather than topological spaces (although we do have the theorem that any pretopological uniform space is topological). On the other hand, now that I\'m getting to [[locale]] theory, I\'m not as interested in pretopological spaces ....
---[[Toby Bartels]]
=--
category: people |
Aaron Greicius | https://ncatlab.org/nlab/source/Aaron+Greicius |
* [webpage](http://greicius.wordpress.com/about-3/)
category: people
|
Aaron Lauda | https://ncatlab.org/nlab/source/Aaron+Lauda |
* [website](https://sites.google.com/view/lauda-home/home)
## Selected writings
On [[Frobenius monads]] and [[ambidextrous adjunctions]] in [[2-categories]] more general than [[Cat]], with motivation from [[2d TQFT]]:
* {#Lauda05} [[Aaron Lauda]], *Frobenius algebras and ambidextrous adjunctions*, Theory and Applications of Categories **16** 4 (2006) 84-122 [[arXiv:math/0502550](http://arxiv.org/abs/math/0502550), [tac:16-04](http://www.tac.mta.ca/tac/volumes/16/4/16-04abs.html)]
## Videos
On [[knot theory]]:
* [[Aaron Lauda]], _Knot theory explained to newbies in 1 min 24 sec lightning idea_, USC Dornsife College of Letters, Arts and Sciences ([video](https://www.youtube.com/embed/kxaWqKM5JyQ)).
category: people
[[!redirects A. Lauda]] |
Aaron M. Smith | https://ncatlab.org/nlab/source/Aaron+M.+Smith |
* [GoogleScholar page](https://scholar.google.com/citations?user=e0yBkScAAAAJ)
## Selected writings
On [[infinity-local systems|$\infty$-local systems]] in the sense of [[(infinity,1)-vector bundles|$(\infty,1)$-vector bundles]] with [[flat infinity-connections|flat $\infty$-connections]]:
* [[Jonathan Block]], [[Aaron M. Smith]], *The higher Riemann--Hilbert correspondence*, Advances in Mathematics
**252** (2014) 382-405 [[arXiv:0908.2843](https://arxiv.org/abs/0908.2843), [doi:10.1016/j.aim.2013.11.001](https://doi.org/10.1016/j.aim.2013.11.001)]
category: people
[[!redirects Aaron Smith]]
|
Aaron Mazel-Gee | https://ncatlab.org/nlab/source/Aaron+Mazel-Gee | * [webpage](http://etale.site/)
## Selected writing
On [[tmf]]:
* _You could've invented $tmf$_, April 2013 ([pdf slides](http://math.berkeley.edu/~aaron/writing/ustars-tmf-beamer.pdf))
Introducing [[model (infinity,1)-category|model $\infty$-category]]-theory:
* [[Aaron Mazel-Gee]], *Model ∞-categories I: some pleasant properties of the ∞-category of simplicial spaces* ([arXiv:1412.8411](https://arxiv.org/abs/1412.8411))
* [[Aaron Mazel-Gee]], *Model ∞-categories II: Quillen adjunctions*, New York Journal of Mathematics **27** (2021) 508-550. ([arXiv:1510.04392](https://arxiv.org/abs/1510.04392), [nyjm:27-21](http://nyjm.albany.edu/j/2021/27-21.html))
* [[Aaron Mazel-Gee]], *Model ∞-categories III: the fundamental theorem*, New York Journal of Mathematics **27** (2021) 551-599 ([arXiv:1510.04777](https://arxiv.org/abs/1510.04777), [nyjm:27-22](http://nyjm.albany.edu/j/2021/27-22.html))
## Related entries
* [[tmf]]
* [[homotopy Kan fibration]]
category: people
[[!redirects amg]]
|
Aaron Stump | https://ncatlab.org/nlab/source/Aaron+Stump |
* [personal page](https://homepage.divms.uiowa.edu/~astump/)
## Selected writings
On [[Agda]] as a [[certified programming|verified]] [[functional programming language]]:
* [[Aaron Stump]], *Verified Functional Programming in Agda*, Association for Computing Machinery and Morgan & Claypool (2016) [[doi:10.1145/2841316](https://doi.org/10.1145/2841316), ISBN:978-1-970001-27-3]
category: people |
Ab | https://ncatlab.org/nlab/source/Ab |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
_$Ab$_ denotes the [[category]] of [[abelian group|abelian groups]]: it has abelian groups as [[object|objects]] and [[group homomorphisms]] between these as [[morphism|morphisms]].
The archetypical example of an abelian group is the group $\mathbb{Z}$ of integers, and for many purposes it is useful to think of $Ab$ equivalently as the [[category of modules]] over $\mathbb{Z}$
$$
Ab \simeq \mathbb{Z} Mod
\,.
$$
The category $Ab$ serves as the basic [[enriched category theory|enriching]] category in [[homological algebra]]. There [[Ab-enriched categories]] play much the same role as [[Set]]-enriched categories ([[locally small categories]]) play in general.
In this vein, the analog of $Ab$ in [[homotopy theory]] -- or rather in _[[stable homotopy theory]]_ -- is the category of [[spectra]], either regarded as the [[stable homotopy category]] or rather refined to the [[stable (infinity,1)-category of spectra]]. A spectrum is much like an abelian group up to [[coherence law|coherent]] [[homotopy]] and the role of the archetypical abelian group $\mathbb{Z}$ is the played by the [[sphere spectrum]] $\mathbb{S}$.
## Properties
### Free abelian groups
+-- {: .num_remark #FreeForgetful}
###### Remark
The category $Ab$ is a [[concrete category]], the [[forgetful functor]]
$$
U : Ab \to Set
$$
to [[Set]] sends a group, regarded as a [[set]] $A$ equipped with the [[structure]] $(+,0)$ of a chosen element $0 \in A$ and a binary, associative and 0-unital operation $+$ to its underlying set
$$
(A, +, 0) \mapsto A
\,.
$$
This functor has a [[left adjoint]] $F : Set \to Ab$ which sends a set $S$ to the [[free abelian group]] $\mathbb{Z}[S]$ on this set: the group of [[formal linear combinations]] of elements in $S$ with [[coefficients]] in $\mathbb{Z}$.
=--
### Direct sum, direct product and tensor product
{#DirectSumEtc}
We discuss basic properties of binary operations on the category of abelian groups: [[direct product]], [[direct sum]] and [[tensor product]]. Below in _[Monoidal and bimonoidal structure](#MonoidalAndBipermutativeStructure)_ we put these structures into a more abstract context.
+-- {: .num_prop }
###### Proposition
For $A, B \in Ab$ two [[abelian groups]], their
[[direct product]] $A \times B$ is the abelian group whose elements are pairs $(a, b)$ with $a \in A$ and $b \in B$, whose 0-element is $(0,0)$ and whose addition operation is the componentwise addition
$$
(a_1, b_1) + (a_2, b_2) = (a_1 + a_2, b_1 + b_2)
\,.
$$
This is at the same time [[generalized the|the]] [[direct sum]] $A \oplus B$.
Similarly for $I \in $[[FinSet]]$\hookrightarrow$ [[Set]] a [[finite set]], we have
$$
\oplus_{i \in I} A_i \simeq \prod_i A_{i}
\,.
$$
But for $I \in Set$ a set which is not finite, there is a difference: the [[direct sum]] $\oplus_{i \in I} A_i$ of an $I$-indexed family ${A_i}_{i \in I}$ of abelian groups is the sub-group of the direct product on those elements for which only finitely many components are non-0
$$
\oplus_{i \in I} A_i \hookrightarrow \prod_i A_i
\,.
$$
=--
+-- {: .num_example }
###### Example
The [[trivial group]] $0 \in Ab$ (the group with a single element) is a [[unit]] for the direct sum: for every abelian group we have
$$
A \oplus 0 \simeq 0 \oplus A \simeq A
\,.
$$
=--
+-- {: .num_example }
###### Example
In view of remark \ref{FreeForgetful} this means that the direct sum of ${\vert I \vert}$ copies of the additive group of [[integers]] with themselves is equivalently the [[free abelian group]] on $I$:
$$
\oplus_{i \in I} \mathbb{Z} \simeq \mathbb{Z}[I]
\,.
$$
=--
+-- {: .num_defn }
###### Definition
For $A$ and $B$ two [[abelian groups]], their [[tensor product of abelian groups]] is the group $A \otimes B$ with the property that a group homomorphism $A \otimes B \to C$ is equivalently a [[bilinear map]] out of the _set_ $A \times B$.
=--
See at _[[tensor product of abelian groups]]_ for details.
+-- {: .num_example }
###### Example
The [[unit]] for the tensor product of abelian groups is the additive group of [[integers]]:
$$
A \otimes \mathbb{Z} \simeq \mathbb{Z} \otimes A \simeq A
\,.
$$
=--
+-- {: .num_prop }
###### Proposition
The [[tensor product of abelian groups]] [[distributivity law|distributes]] over arbitrary [[direct sums]]:
$$
A \otimes (\oplus_{i \in I} B_i) \simeq \oplus_{i \in I} A \otimes B_i
\,.
$$
=--
+-- {: .num_example }
###### Example
For $I \in Set$ and $A \in Ab$, the [[direct sum]] of ${\vert I\vert}$ copies of $A$ with itself is equivalently the [[tensor product of abelian groups]] of the [[free abelian group]] on $I$ with $A$:
$$
\oplus_{i \in I} A \simeq (\oplus_{i \in I} \mathbb{Z}) \otimes A
\simeq (\mathbb{Z}[I]) \otimes A
\,.
$$
=--
### Symmetric monoidal and bimonoidal structure
{#MonoidalAndBipermutativeStructure}
With the definitions and properties discussed above in _[Direct sum, etc.](#DirectSumEtc)_ we have the following
+-- {: .num_prop }
###### Proposition
The category $Ab$ becomes a [[monoidal category]]
1. under [[direct sum]] $(Ab, \oplus, 0)$;
1. under [[tensor product of abelian groups]] $(Ab, \otimes, \mathbb{Z})$.
Indeed with both structures combined we have
* $(Ab, \oplus, \otimes, 0, \mathbb{Z})$
is a [[bimonoidal category]] (and can be made a [[bipermutative category]]).
=--
It's also easy to see that under direct sum or tensor product, Ab can be turned into a [[symmetric monoidal category]] by equipping it with the appropriate braiding map. For example, under $\oplus$, the braiding is $\sigma_{A, B}(a, b) = (b, a)$.
+-- {: .num_remark }
###### Remark
A [[monoid]] [[internalization|internal to]] $(Ab, \otimes, \mathbb{Z})$ is equivalently a [[ring]].
=--
+-- {: .num_remark }
###### Remark
A [[monoid]] in $(Ab, \oplus, 0)$ is equivalently just an abelian group again (since $\oplus$ is the [[coproduct]] in $Ab$, so every object has a unique monoid structure with respect to it).
=--
### Pointed objects
$Ab$ is a [[monoidal category]] with [[tensor unit]] $\mathbb{Z}$, so the [[pointed objects in a monoidal category|pointed objects]] in $Ab$ are the objects $A$ with a [[group homomorphism]] $\mathbb{Z} \to A$.
### Closed monoidal structure
Abelian groups are equivalently $\mathbb{Z}$-modules. Because the [[Mod|category of $R$-modules]] $Mod_R$ is closed monoidal for all [[commutative rings]] $R$, $Ab = Mod_{\mathbb{Z}}$ is also closed monoidal.
### Natural numbers object
The [[natural numbers object]] in $Ab$ is the [[free abelian group]] $\mathbb{Z}[\mathbb{N}] = \bigoplus_{n \in \mathbb{N}} \mathbb{Z}$ on the [[natural numbers]], and comes with group homomorphisms $z_0:\mathbb{Z} \to \mathbb{Z}[\mathbb{N}]$ and $z_s:\mathbb{Z}[\mathbb{N}] \to \mathbb{Z}[\mathbb{N}]$ such that for all abelian groups $A$ and group homomorphisms $f:\mathbb{Z} \to A$, $g: A \to A$, there is a unique group homomorphism $\phi_{f, g}:\mathbb{Z}[\mathbb{N}] \to A$ making the following diagram commute:
$$\array{
\mathbb{Z} & \stackrel{z_0}{\to} & \mathbb{Z}[\mathbb{N}] & \stackrel{z_s}{\leftarrow} & \mathbb{Z}[\mathbb{N}] \\
& \mathllap{f} \searrow & \downarrow \mathrlap{\phi_{f, g}} & & \downarrow \mathrlap{\phi_{f, g}} \\
& & A & \underset{g}{\leftarrow} & A
}$$
Abelian groups are $\mathbb{Z}$-[[modules]], so the free $\mathbb{Z}$-module $\mathbb{Z}[\mathbb{N}]$ has a function $v:\mathbb{N} \to \mathbb{Z}[\mathbb{N}]$ representing the [[basis of a free module|basis]] of $\mathbb{Z}[\mathbb{N}]$; it has the property that for all integers $m \in \mathbb{Z}$, $m \cdot v(0) = z_0(m)$ and for all $n \in \mathbb{N}$, $m \cdot v(s(n)) = z_s(m \cdot v(n))$, where $m \cdot v$ is the scalar multiplication of an element $v$ by an integer $m$ in a $\mathbb{Z}$-module.
The [[ring]] structure on $\mathbb{Z}[\mathbb{N}]$ is defined by double [[induction]] on $\mathbb{Z}[\mathbb{N}]$, we define
$$(-)(-):\mathbb{Z}[\mathbb{N}] \times \mathbb{Z}[\mathbb{N}] \to \mathbb{Z}[\mathbb{N}] \otimes \mathbb{Z}[\mathbb{N}] \to \mathbb{Z}[\mathbb{N}]$$
by
$$z_0(m)z_0(n) = z_0(m \cdot n) \qquad z_s(v)z_0(n) = z_s(v z_0(n))$$
$$z_0(m)z_s(w) = z_s(z_0(m) w) \qquad z_s(v)z_s(w) = z_s(z_s(vw))$$
for all $m, n \in \mathbb{Z}$ and $v, w \in \mathbb{Z}[\mathbb{N}]$ (recall the definition of [[addition]] in the [[natural numbers]], inductively defined by $0(p) + 0(q) = 0(p \cdot q)$, $s(m) + 0(p) = s(m + 0(p))$, $0(p) + s(n) = s(0(p) + n)$, and $s(m) + s(n) = s(s(m + n))$ for all $p, q \in \mathbb{1}$ and $m, n \in \mathbb{N}$). It is a [[commutative ring]] and represents multiplication in the [[polynomial ring]] $\mathbb{Z}[X]$; the group homomorphism $z_0$ represents the function which takes integers to constant polynomials, and $z_s$ represents the function which takes a polynomial and multiplies it by the indeterminate $X$.
### Enrichment over $Ab$
Categories [[enriched category|enriched]] over $Ab$ are called [[additive category|pre-additive categories]] or sometimes just additive categories. If they satisfy an extra exactness condition they are called [[abelian category|abelian categories]]. See at _[[additive and abelian categories]]_.
## Related concepts
* [[CMon]]
* [[Mod]]
* [[sAb]]
category: category
[[!redirects AbGrp]]
[[!redirects category of abelian groups]]
[[!redirects categories of abelian groups]]
[[!redirects AbelianGroups]]
[[!redirects ZMod]]
[[!redirects Z Mod]]
[[!redirects Z-Mod]]
[[!redirects category of Z-modules]]
[[!redirects categories of Z-modules]]
[[!redirects category of integer modules]]
[[!redirects categories of integer modules]]
[[!redirects ZModules]]
[[!redirects Z Modules]]
[[!redirects Z-Modules]]
[[!redirects IntegerModules]]
|
Ab-enriched category | https://ncatlab.org/nlab/source/Ab-enriched+category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
=--
=--
# $Ab$-enriched categories
* tic
{: toc}
## Idea
An _$Ab$-enriched category_ (or, if small, _ringoid_) is a [[category enriched]] over the [[monoidal category]] [[Ab]] of [[abelian groups]] with its usual [[tensor product]].
Sometimes they are called [[pre-additive category|pre-additive categories]], but sometimes that term also implies the existence of a [[zero object]].
## Definition
Explicitly, an __$Ab$-enriched category__ is a [[category]] $C$ such that
for all objects $a,b$ the [[hom-set]] $Hom_C(a,b)$
is equipped with the structure of an [[abelian group]]; and
such that for all triples $a,b,c$ of objects the
[[composition]] operation
$
\circ_{a,b,c} : Hom_C(a,b) \times Hom_C(b,c) \to Hom_C(a,c)
$
is bilinear. A __ringoid__ is small $Ab$-enriched category.
## Remarks
* $Ab$-enriched categories are called ringoids since the concept is a [[horizontal categorification]] (or 'oidification') of the concept of a [[ring]].
* There is a canonical forgetful functor $Ab \to Set_*$ from
abelian groups to [[pointed set]]s, which sends each group to its underlying set with point being the neutral element. Using this functor, every $Ab$-enriched category $C$ is in particular also a category that is enriched over pointed sets (that is, a category with [[zero morphisms]]). This is sufficient for there to be a notion of [[kernel]] and [[cokernel]] in $C$.
* In general, [[abelian category|abelian categories]] are the most important examples of $Ab$-enriched categories. See [[additive and abelian categories]].
## Finite products are absolute
One of the remarkable facts about $Ab$-enriched categories is that finite [[products]] (and [[coproducts]]) are [[absolute limit|absolute limits]]. This implies that finite products coincide with finite coproducts, and are preserved by _any_ $Ab$-enriched functor.
### Zero objects
In an $Ab$-enriched category $C$, any [[initial object]] is also a [[terminal object]], hence a [[zero object]], and dually. An object $a\in C$ is a zero object just when its identity $1_a$ is equal to the zero morphism $0:a\to a$ (that is, the identity element of the abelian group $\hom_C(a,a)$). Expressed in this way, it is easy to see that any $Ab$-enriched functor preserves zero objects.
### Biproducts
For $c_1, c_2 \in C$ two objects in an $Ab$-enriched category $C$, [[generalized the|the]] [[product]] $c_1 \times c_2$ coincides with [[generalized the|the]] [[coproduct]] $c_1 \sqcup c_2$ when either exists.
For example, if $c_1 \times c_2$ exists, with [[projection]] maps $p_i\colon c_1 \times c_2 \to c_i$, then according to the [[universal property]] of products, there are unique maps
$$
q_i\colon c_i \to c_1 \times c_2
$$
such that $p_i q_i = 1_{c_i}$ and $p_j q_i = 0$ for $j \neq i$, and these maps $q_i$ are the coproduct [[coprojections]], i.e., they realize $c_1 \times c_2$ as the coproduct of $c_1$ and $c_2$. Indeed, for any maps $r_1\colon c_1 \to e$ and $r_2\colon c_2 \to e$, it is easily checked that
$$r = r_1 p_1 + r_2 p_2\colon c_1 \times c_2 \to e$$
satisfies $r q_1 = r_1$ and $r q_2 = r_2$, and is the unique map satisfying these equations. The full argument is spelled out at *[[additive category]]*.
By a [[formal duality|dual]] argument, if the coproduct $c_1 \sqcup c_2$ exists, then it may also be realized as the product of $c_1$ and $c_2$. Either way, the product or coproduct is called a *[[biproduct]]* or (sometimes) a *[[direct sum]]* and is generally denoted
$$
c_1 \oplus c_2.
$$
It can be characterized diagrammatically as an object $c_1\oplus c_2$ equipped with morphisms $q_i \colon c_i\to c_1\oplus c_2$ and $p_i \colon c_1\oplus c_2 \to c_i$ such that $p_i q_j = \delta_{i j}$ and $q_1 p_1 + q_2 p_2 = 1_{c_1\oplus c_2}$. Expressed in this form, it is clear that any $Ab$-[[enriched functor]] [[preserved limit|preserves]] biproducts.
## As a generalisation of rings
When using the term 'ringoid', one often assumes a ringoid to be [[small category|small]].
Ringoids share many of the properties of (noncommutative) [[rings]]. For instance, we can talk about (left and right) [[modules]] over a ringoid $R$, which can be defined as $Ab$-enriched [[functors]] $R\to Ab$ and $R^{op}\to Ab$. [[bimodule|Bimodules]] over ringoids have a tensor product (the enriched [[tensor product of functors]]) under which they form a [[bicategory]], also known as the bicategory $Ab Prof$ of $Ab$-enriched [[profunctors]]. Modules over a ringoid also form an [[abelian category]] and thus have a [[derived category]].
One interesting operation on ringoids is the ($Ab$-enriched) [[Cauchy completion]], which is the completion under finite [[direct sums]] and [[split idempotents]]. In particular, the Cauchy completion of a ring $R$ is the category of [[finitely generated object|finitely generated]] [[projective object|projective]] $R$-modules (aka [[split monomorphism|split]] [[subobjects]] of finite-rank [[free object|free]] modules). Every ringoid is [[equivalence|equivalent]] to its Cauchy completion in the bicategory $Ab Prof$, and two ringoids are equivalent in $Ab Prof$ if and only if their Cauchy completions are [[equivalence of categories|equivalent]] as $Ab$-enriched categories. This sort of equivalence is naturally called [[Morita equivalence]].
See also [[dg-category]].
## Examples
* The category [[Ab]] is [[closed monoidal category|closed monoidal]] and hence canonically enriched over itself.
* An $Ab$-enriched category with one object is precisely a [[ring]].
* For any small $Ab$-enriched category $R$, the enriched [[presheaf category]] $[R^{op},Ab]$ is, of course, $Ab$-enriched. If $R$ is a ring, as above, then $[R^{op},Ab]$ is the category of $R$-modules.
## Related concepts
* [[hom-group]]
* [[additive category]]
## References
* John Baez, [Ringoids](http://golem.ph.utexas.edu/category/2006/09/ringoids.html), blog
* C. Weibel, [[An Introduction to Homological Algebra]], Cambridge Univ. Press
* [[Daniel Murfet]], _Localisation of ringoids_, [pdf](http://therisingsea.org/notes/LocalisationOfRingoids.pdf) 2006 notes
* N. Popescu, _Abelian categories with applications to rings and modules_, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375
[[!redirects Ab-enriched category]]
[[!redirects Ab-enriched categories]]
[[!redirects Ab-enriched]]
[[!redirects Ab-category]]
[[!redirects Ab-categories]]
[[!redirects ringoid]]
[[!redirects ringoids]]
|
abc conjecture | https://ncatlab.org/nlab/source/abc+conjecture |
#Contents#
* table of contents
{:toc}
## Idea
The _abc conjecture_ (or _ABC conjecture_) is a [[number theory|number theoretic]] conjecture due to ([Oesterlé-Masser 1985](#OesterléMasser)), which says that there are only [[finite set|finitely many]] integer solutions to the [[equation]]
$$
a + b = c \;\;\; for \;\;\; a,b,c \geq 1 \in \mathbb{N}
$$
(or instead $a+b+c= 0$)
if one requires the [[integer numbers]] $a,b,c$ to have no common factor as well as having "joint power" greater than a given bound.
Here the _power_ of $a,b,c$ is
$$
P(a,b,c) \coloneqq \frac{log|c|}{log(rad(a\cdot b\cdot c))}
\,,
$$
where the [[radical]] $rad(n)$ of an [[integer]] $n$ is the product of all its distinct [[prime factors]].
The precise form of the conjecture is:
+-- {: .un_theorem}
###### Conjecture
**(abc conjecture)**
For any number $\epsilon\gt 0$ there are only finitely many positive relatively prime ([[coprime integers|coprime]]) [[integer]] solutions $(a,b,c)$ to the [[equation]] $a + b = c$ with power $P(a,b,c)\geq 1+\epsilon$.
=--
According to ([Mazur](#Matzur)):
> The beauty of such a Conjecture is that it captures the intuitive sense that triples of numbers which satisfy a linear relation, and which are divisible by high perfect powers, are rare; the precision of the Conjecture goads one to investigate this rarity quantitatively. Its very statement makes an attractive appeal to perform a range of numerical experiments that would test the empirical waters. On a theoretical level, it is enlightening to understand its relationship to the constellation of standard arithmetic theorems, conjectures, questions, etc., and we shall give some indications of this below.
## Mason's Theorem
According to [Lang](#Lang), one important antecedent of the abc conjecture is a simple but at the time unexpected relation for the function field case, published in 1984. Consider polynomials $f \in k[t]$ over an algebraically closed field $k$ of characteristic $0$, and define $n_0(f)$ to be the number of _distinct_ roots of $f$, counted _without_ regard to multiplicity.
+-- {: .un_theorem}
###### Theorem ([Mason](#Mason))
Let $a, b, c \in k[t]$ be relatively prime polynomials, not all constant, such that $a + b = c$. Then $\max \{deg(a), deg(b), deg(c)\} \leq n_0(a b c) - 1$.
=--
+-- {: .proof}
###### Proof
Let $f = a/c$, $g = b/c$, so that $f + g = 1$. Taking the derivative, we obtain
$$\frac{f'}{f} f + \frac{g'}{g} g = 0$$
whence
$$b/a = g/f = -\frac{f'/f}{g'/g}.$$
Put
$$a(t) = c_1 \prod (t - \alpha_i)^{m_i}, \qquad b(t) = c_2 \prod (t - \beta_j)^{n_j}, \qquad c(t) = c_3 \prod (t - \gamma_k)^{p_k}.$$
Then
$$\frac{b}{a} = -\frac{f'/f}{g'/g} = -\frac{\sum \frac{m_i}{t - \alpha_i} - \sum \frac{p_k}{t - \gamma_k}}{\sum \frac{n_j}{t - \beta_j} - \sum \frac{p_k}{t - \gamma_k}} .$$
A common denominator for $f'/f$ and $g'/g$ is given by
$$N_0 = \prod (t - \alpha_i) \prod (t - \beta_j) \prod (t - \gamma_k)$$
whose degree is $n_0(a b c)$. We then have
$$\frac{b}{a} = -\frac{N_0 f'/f}{N_0 g'/g}$$
where the numerator and denominator on the right are polynomials. However, since $b$ and $a$ are relatively prime, the fraction $b/a$ is already in lowest terms. From this we conclude that $deg(b) \leq deg(N_0 f'/f) \leq n_0(a b c) - 1$, and similarly $deg(a) \leq deg(N_0 g'/g) \leq n_0(a b c) - 1$, which completes the proof.
=--
+-- {: .un_cor}
###### Corollary (FLT for polynomials)
Assume $x, y, z \in k[t]$ are relatively prime polynomials, not all constant, and suppose $x^n + y^n = z^n$. Then $n \leq 2$.
=--
+-- {: .proof}
###### Proof
From Mason's theorem, we conclude $n deg(x) = deg(x^n) \leq deg(x) + deg(y) + deg(z) - 1$, and similarly upon replacing $x$ by $y$ and $z$ on the left. Adding the results, we have
$$n(deg(x) + deg(y) + deg(z)) \leq 3(deg(x) + deg(y) + deg(z)) - 3$$
which is impossible if $n \geq 3$.
=--
Guided by analogies between the ring of integers and the ring of polynomials in one variable, and building on insights of Mason, Frey, Szpiro, and others, Masser and Oesterlé were led to formulate the abc conjecture for integers as follows. Again define $N_0(m)$ for $m$ a non-zero integer to be the number of distinct primes dividing $m$.
* Conjecture: For all $\epsilon \lt 0$ there exists $C(\epsilon) \lt 0$ such that for relatively prime integers $a, b, c$ satisfying $a + b = c$, we have
$$\max({|a|}, {|b|}, {|c|}) \leq C(\epsilon)N_0(a b c)^{1+\epsilon}.$$
Of course, this differs from the polynomial case because of the presence of $1+ \epsilon$ in the exponent, but this is a necessary evil. For example, for any $C \gt 0$, we can find relatively prime $a$, $b$, $c$ with $a + b = c$ and ${|a|} \gt C N_0(a b c)$: take $a = 3^{2^n}$, $b = -1$, and observe by repeated application of $x^2 - y^2 = (x-y)(x+y)$ that $c = a + b$ is of the form $2^n d$ for some integer $d$. Taking $n$ sufficiently large, we can easily derive the claimed inequality.
## Relation to other statements
The abc conjecture implies the [[Mordell conjecture]] ([Elkies](#Elkies)).
It is equivalent to the general form of [[Szpiro's conjecture]].
* [[Vojta's conjecture]]
## References
### General
The abc conjecture was stated in
* {#OesterléMasser} Joseph Oesterlé, David Masser (1985)
Mason's theorem was presented in
* {#Mason} R. C. Mason, Equations over function fields. In _Number Theory, Proceedings of the Noordwijkerhout_, Springer Lecture Notes 1068 (1984), 149-157.
Material on Mason's theorem and its relation to the abc conjecture was taken from
* {#Lang} Serge Lang, Algebra ($3^{rd}$ Edition), Addison-Wesley (1993), 194-196.
The relation to the [[Mordell conjecture]] is discussed in
* {#Elkies} [[Noam Elkies]], _ABC conjecture implies Mordell_, Int. Math. Research Notices 7 (1991) 99-109
The relation to [[Szpiro's conjecture]] is discussed in
* Matt Baker (notes taken by William Stein), _Elliptic curves, the ABC conjecture, and points of small canonical height_ ([pdf](http://modular.math.washington.edu/mcs/archive/Fall2001/notes/12-10-01/12-10-01.pdf))
See also
* [[Barry Mazur]], _Questions about number_, [pdf](http://www.math.harvard.edu/~mazur/papers/scanQuest.pdf) scan
{#Mazur}
* PolyMath, _[ABC conjecture](http://michaelnielsen.org/polymath1/index.php?title=ABC_conjecture)_
* Abderrahmane Nitaj, _[The ABC Conjecture Homepage](http://www.math.unicaen.fr/~nitaj/abc.html)_
* Wikipedia, _[abc conjecture](http://en.wikipedia.org/wiki/Abc_conjecture)_
### Alleged proof via IUT
[[Shinichi Mochizuki]] anounced the proof which the mathematical community perceives as a serious but unchecked claim. See the references at _[[inter-universal Teichmüller theory]]_.
Comments on the proof are at
* _[[Mochizuki's proof of abc]]_.
* MathOverflow, _[Philosophy behind Mochizuki's work on the ABC conjecture](http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/106658#106658)_
A popular account of the problem of the math community checking the proof is in
* Caroline Chen, _The Paradox of the Proof_ ([web](http://projectwordsworth.com/the-paradox-of-the-proof/))
[[!redirects abc-conjecture]]
|
Abdelmalek Abdesselam | https://ncatlab.org/nlab/source/Abdelmalek+Abdesselam |
* [webpage](http://people.virginia.edu/~aa4cr/)
## related $n$Lab entries
* [[renormalization group]]
category: people |
abductive reasoning | https://ncatlab.org/nlab/source/abductive+reasoning | #Contents#
* table of contents
{:toc}
##General idea
**Abductive** [[reasoning]] is a process whereby one reasons to the truth of an explanation from its ability to account for what is observed. It is therefore sometimes also known as **inference to the best explanation**.
[[Charles Peirce]], the originator of the term, illustrated the differences between [[deductive reasoning|deduction]], [[inductive reasoning|induction]], and abduction by the following example.
* **Deduction**
* All beans in that bag are white.
* These beans are from that bag.
* Therefore, these beans are white.
* **Induction**
* These beans are from that bag.
* These beans are white.
* Therefore, all beans in that bag are white.
* **Abduction**
* These beans are white.
* All beans in that bag are white.
* Therefore, these beans are from that bag.
It is not completely clear what Peirce meant by abduction, which he also termed **retroduction**. Clearly the inference cannot be to just any possible explanation, e.g., in the case above, there might have been many other bags full of white beans. But before we decide what constitutes a **best** explanation, we had been inquire into the nature of explanation itself.
There is an extensive literature about explanation in the Philosophy of Science, for example, ([FourDecades](#FourDecades)). Clearly it is not merely a matter of devising propositions, perhaps a general law and a particular statement, which have the thing to be explained (explicandum) as a consequence. We want the proposed explanation to 'give the reason for' the observation. A thorough account of what constitutes the 'reason' for something is notoriously difficult to formulate. For some, it is a matter of subsuming the observation under a general covering law, while for others, it is a matter of giving a causal or mechanistic story with the observation as the outcome.
Note also that there is a growing literature now on the concept of 'explanatory proofs' in mathematics, it being felt that one may have proved a mathematical fact without understanding 'why' it is true.
For some, abduction also signifies the creative process of coming up with a good explanation. Otherwise, if it is merely a case of assessing a range of existing rival hypotheses as explanations, it may be possible to employ [[Bayesian reasoning]], generally taken to be a form of [[inductive reasoning]].
> If you really can find an explanation having sufficient probability to be worth consideration, you escape in great measure from reposing upon retroduction [abduction] and make your inference inductive. (Peirce, [Harvard Lectures, p. 193](#Harvard))
## Abduction as lifting
In Peirce's [Harvard lectures, p. 315](#Harvard), he describes the triad -- _deduction_, _induction_, _abduction_ -- in terms of the logical relations between three concepts, $M$, $P$ and $S$.
* Deduction strings together, say, $M$ is $P$ and $P$ is $S$ to give $M$ is $S$.
* Induction looks to generalise from $M$ is $S$, taking $M$ as a sample of $P$, to conclude that $P$ is $S$.
* Abduction looks to explain why $M$ is $S$, having noted that $P$ is $S$, by hypothesising that $M$ is $P$.
Seen from the point of view of category theory, this would seem rather like: composition, [[extension]], and [[lifting]]. Induction as a kind of extension seems quite reasonable. Abduction may account for an instance of some concept, $E$, by lifting to a concept, $C$, through a law connecting cause, $C$, to effect, $E$.
## References
* Wesley Salmon, 1989, _Four Decades of Scientific Explanation_, University of Pittsburgh Press. {#FourDecades}
* Proposed formalization as a functor between [categories of structures](structure+in+model+theory#categories_of_structures) can be found in Fernando Tohmé, Gianluca Caterina, Rocco Gangle, [_Abduction: A categorical characterization_](http://doi.org/10.1016/j.jal.2014.12.004), Journal of Applied Logic, Volume 13, Issue 1, March 2015, Pages 78-90
* Gerhard Schurz, 2008, _Patterns of abduction_, Synthese 164:201–234.
* {#Harvard} [[Charles Peirce]], 1992, _Reasoning and the logic of things_, Harvard University Press (lectures from 1898, [book](http://www.hup.harvard.edu/catalog.php?isbn=9780674749672))
* Peter Krause, _Presupposition and abduction in type theory_, In Working Notes of. CLNLP-95: Computational Logic and Natural Language Processing.
[[!redirects abduction]]
[[!redirects abductive reasoning]]
[[!redirects abductive inference]]
[[!redirects abductive logic]]
|
Abdus Salam | https://ncatlab.org/nlab/source/Abdus+Salam |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Abdus_Salam)
## Selected writings
Co-introducing the theory of the [[electroweak field]]:
* [[Abdus Salam]], [[John Clive Ward]], *Weak and electromagnetic interactions*, Nuovo Cimento. **11** 4 (1959) 568–577 [[doi:10.1007/BF02726525](https://doi.org/10.1007/BF02726525)]
Introducing [[super Yang-Mills theory]]:
* [[Abdus Salam]], [[John Strathdee]], _Super-symmetry and non-Abelian gauges_, Physics Letters B Volume 51, Issue 4, 19 August 1974, Pages 353-355 (<a href="https://doi.org/10.1016/0370-2693(74)90226-3">doi:10.1016/0370-2693(74)90226-3</a>)
Introducing [[superspace]] and [[superfields]] for [[supersymmetry|supersymmetric]] [[quantum field theory]]:
* [[Abdus Salam]], [[John Strathdee]], _Supergauge Transformations_, Nucl.Phys. B76 (1974) 477-482 ([spire:89208](http://inspirehep.net/record/89208))
* [[Abdus Salam]], [[John Strathdee]], _Superfields and Fermi-Bose symmetry_, Physical Review D11, 1521-1535 (1975) ([doi:10.1142/9789812795915_0051](https://doi.org/10.1142/9789812795915_0051))
On [[supergravity]]:
* [[Abdus Salam]], [[Ergin Sezgin]] (eds.), *Supergravities in Diverse Dimensions*, Elsevier & World Scientific (1990) [[doi:10.1142/0277](https://doi.org/10.1142/0277)]
## Related entries
* [[superspace]], [[superfield]]
* [[8-dimensional supergravity]]
* [[grand unified theory]]
* [[E11]]
category: people
|
Abel-Jacobi map | https://ncatlab.org/nlab/source/Abel-Jacobi+map |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[algebraic geometry|algebraic]]/[[complex geometry]] The term *Abel-Jacobi map* refers to various [[group homomorphisms]] from certain [[groups]] of [[algebraic cycles]] to some sorts of [[Jacobians]] or generalized Jacobians. Such maps generalize the classical Abel-Jacobi map from points of a complex algebraic curve to its [[Jacobian]], which answers the question of which divisors of degree zero arise from [[meromorphic functions]].
## Definition
### for curves
Let $X$ be a smooth projective [[complex curve]]. Recall that a [[Weil divisor]] on $X$ is a [[formal linear combination]] of [[closed points]]. Classically, the Abel-Jacobi map
$$
\alpha
\;\colon\;
\Div^0(X)
\longrightarrow
J(X)
,\,
$$
on the group of [[Weil divisors]] of degree zero, is defined by [[integration]]. According to [[Abel's theorem]], its [[kernel]] consists of the [[principal divisors]], i.e. the ones coming from [[meromorphic functions]].
### on Deligne cohomology
The cycle map to [[de Rham cohomology]] due to [Zein & Zucker (1981)](#ZeinZucker81) is discussed in [Esnault & Viehweg (1988), section 6](#EsnaultViehweg88), the refinement to [[Deligne cohomology]] in [Esnault & Viehweg (1988), section 6](#EsnaultViehweg88). By the characterization of [[intermediate Jacobians]] as subgroups of the [[Deligne complex]] (see *[intermediate Jacobian -- characterization as Hodge-trivial Deligne cohomology](intermediate+Jacobian#CharacterizationAsHodgeTrivialDeligneCohomology)*) this induces a map from cycles to [[intermediate Jacobians]]. This is the Abel-Jacobi map ([Esnault & Viehweg (1988), theorem 7.11](#EsnaultViehweg88)).
### on higher Chow groups
An Abel-Jacobi map on [[higher Chow groups]] is discussed in [K-L-MS 04](#KLMS04).
### via extensions of mixed Hodge structures
An alternate construction of the Abel-Jacobi map, via [[Hodge theory]], is due to Arapura-Oh.
By a theorem of Carlson, the [[Jacobian]] is identified with the following group of [[extensions]] in the [[abelian category]] of [[mixed Hodge structures]]:
$$ J(X) = Ext^1_{MHS}(\mathbf{Z}(-1), H^1(X, \mathbf{Z})) $$
where $\mathbf{Z}(-1)$ is the Tate Hodge structure.
Given a [[divisor]] $D$ of degree zero, one can associate to it a certain class in the above extension group.
This gives a map
$$ \alpha : Div^0(X) \longrightarrow J(X) $$
which is called the Abel-Jacobi map.
The Abel theorem says that its [[kernel]] is precisely the subgroup of [[principal divisors]], i.e. divisors which come from invertible rational functions.
See ([Arapura-Oh, 1997](#ArapuraOh97)) for details of this construction.
## Related concepts
* [[Hodge-filtered differential cohomology]]
## References
* {#ZeinZucker81} Fouad El Zein and Steven Zucker, _Extendability of normal functions associated to algebraic cycles_, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Ann. of Math. Stud., vol. 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 269–288. [MR 756857](http://www.ams.org/mathscinet-getitem?mr=756857)
* {#EsnaultViehweg88} [[Hélène Esnault]], [[Eckart Viehweg]], _Deligne-Beilinson cohomology_, in: [[Michael Rapoport]], [[Norbert Schappacher]], [[Peter Schneider]] (eds.), _[[Beilinson's Conjectures on Special Values of L-Functions]]_, Perspectives in Mathematics **4**, Academic Press, Inc. (1988) [ISBN:978-0-12-581120-0, [[EsnaultViehweg-DeligneBeilinsonCohomology.pdf:file]]]
* {#Voisin02} [[Claire Voisin]], section 12 of _[[Hodge theory and Complex algebraic geometry]] I,II_, Cambridge Stud. in Adv. Math. __76, 77__, 2002/3
* {#ArapuraOh97} Donu Arapura, Kyungho Oh. _On the Abel-Jacobi map for non-compact varieties_. Osaka Journal of Mathematics 34 (1997), no. 4, 769--781. [Project Euclid](http://projecteuclid.org/euclid.ojm/1200787781).
* {#KLMS04} Matt Kerr, James Lewis, Stefan Müller-Stach, _The Abel-Jacobi map for higher Chow groups_, 2004, [arXiv:0409116](http://arxiv.org/abs/math/0409116).
* Wikipedia, _[Abel-Jacobi map](http://en.wikipedia.org/wiki/Abel–Jacobi_map)_
Remarks on generalization to the more general context of [[anabelian geometry]] are in
* [[Minhyong Kim]], _Galois Theory and Diophantine geometry, 2009 ([pdf](http://www.ucl.ac.uk/~ucahmki/cambridgews.pdf))
Refinement of the [[Abel-Jacobi map]] to [[Hodge filtration|Hodge filtered]] [[differential cobordism cohomology theory|differential]] [[MU]]-[[cobordism cohomology theory]]:
* [[Gereon Quick]], *An Abel-Jacobi invariant for cobordant cycles*,
Documenta Mathematica **21** (2016) 1645–1668 [[arXiv:1503.08449](https://arxiv.org/abs/1503.08449)]
* [[Knut B. Haus]], [[Gereon Quick]], *Geometric Hodge filtered complex cobordism* [[arXiv:2210.13259](https://arxiv.org/abs/2210.13259)]
Introduction and survey:
* [[Gereon Quick]], *Geometric Hodge filtered complex cobordism*, [talk at](Center+for+Quantum+and+Topological+Systems#QuickMar2023) *[[CQTS]]* (March 2023) [video:[YT](https://www.youtube.com/watch?v=pMu0gT5kIBo)]
[[!redirects Abel-Jacobi maps]]
[[!redirects Abel-Jacobi theorem]]
[[!redirects Abel-Jacobi theorems]] |
abelian 2-category | https://ncatlab.org/nlab/source/abelian+2-category | [[abelian category|Abelian categories]] are a common generalization of the category of abelian groups, categories of $R$-modules where $R$ is a ring, and the categories of sheaves of $\mathcal{O}_X$-modules where $\mathcal{O}_X$ is a sheaf of rings on a topological space $X$. Similarly, in the categorical dimension 2, one wants to define an __abelian 2-category__ as a generalization of symmetric 2-groups and possibly some 2-related categories. There are a few different proposals in the literature.
* Mathieu Dupont, _Abelian categories in dimension 2_, [arxiv/0809.1760](http://arxiv.org/abs/0809.1760)
* Hiroyuki Nakaoka, _Comparison of the definitions of Abelian 2-categories_, [arxiv/0904.0078](http://arxiv.org/abs/0904.0078)
[[!redirects Abelian 2-category]] |
abelian 3-group | https://ncatlab.org/nlab/source/abelian+3-group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
An _abelian 3-group_ is an [[abelian infinity-group]] which is a [[3-group]].
## Examples
If $(\mathcal{C}, \otimes)$ is a [[symmetric monoidal 2-category]] then its [[Picard 3-group]] is an abelian 3-group.
## Related concepts
* [[∞-group]], [[k-tuply groupal n-groupoid]]
* [[braided ∞-group]],
* [[braided 2-group]]
* [[braided 3-group]]
* [[sylleptic ∞-group]]
* [[sylleptic 3-group]]
* [[abelian ∞-group]]
* [[abelian group]]
* [[abelian 2-group]]
* **abelian 3-group**
[[!redirects abelian 3-group]]
|
abelian 4-group | https://ncatlab.org/nlab/source/abelian+4-group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
An _abelian 4-group_ is an [[abelian ∞-group]] which is a [[4-group]].
## Related concepts
* [[∞-group]], [[k-tuply groupal n-groupoid]]
* [[braided ∞-group]],
* [[braided 2-group]]
* [[braided 3-group]]
* [[sylleptic ∞-group]]
* [[sylleptic 3-group]]
* [[abelian ∞-group]]
* [[abelian group]]
* [[abelian 2-group]]
* **abelian 3-group**
[[!redirects abelian 4-groups]]
|
abelian 7-group | https://ncatlab.org/nlab/source/abelian+7-group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
An _abelian 7-group_ is an [[abelian ∞-group]] which is a [[7-group]].
## Related concepts
* [[∞-group]], [[k-tuply groupal n-groupoid]]
* [[braided ∞-group]],
* [[braided 2-group]]
* [[braided 3-group]]
* [[sylleptic ∞-group]]
* [[sylleptic 3-group]]
* [[abelian ∞-group]]
* [[abelian group]]
* [[abelian 2-group]]
* **abelian 3-group**
[[!redirects abelian 7-groups]]
|
Abelian categories with applications to rings and modules | https://ncatlab.org/nlab/source/Abelian+categories+with+applications+to+rings+and+modules |
This page compiles pointers related to:
* [[Nicolae Popescu]]:
\linebreak
__Abelian categories with applications to rings and modules__
\linebreak
London Mathematical Society Monographs **3*
Academic Press (1973)
[MR0340375](http://www.ams.org/mathscinet-getitem?mr=0340375)
on the [[algebra]] of [[rings]] and [[modules]] via [[abelian categories]].
\linebreak
Contents:
__Note to the reader__
Some terminology, notations and conventions used throughout the book.
__Categories and functors__
The notion of a category; examples. Special objects and morphisms. Functors. Inductive and
projective limits. Adjoint functors; equivalence of categories.
__Abelian categories__
Preadditive and additive category. The canonical factorization of a morphism; preabelian categories.
Abelian categories. Fibred products and fibred sums. Basic lemmas on abelian categories. The
isomorphism theorems. Direct sums of subobjects. Inductive limits; the conditions Ab.
__Additive functors__
Additive functors. Exactness of functors; injective and projective objects. The injective and
projective objects in the category Ab. Categories of additive functors; modules. Special objects in
abelian categories. Tensor products; a characterization of functor categories. Tensor products of
modules. Flat modules. Some remarks on projective modules. Injective envelopes. Semisimple
rings.
__Localization__
Categories of fractions; calculus of fractions. The spectral category of an abelian category. The
quotient category of an abelian category relative to a dense subcategory. The section functor.
Localization in categories with injective envelopes. Localization in Ab 3 categories. Sheaves over a
topological space. Torsion theories. Localizing subcategories in categories of modules. Localization
in categories of modules. Left exact functors; the embedding theorem. The study of the localization
ring of a ring. The complete ring of quotients. Some remarks on Grothendieck categories.
Finiteness conditions on localizing systems. Flat epimorphism of rings. Left quasi-orders of a ring.
Rings of fractions. Left orders. Left spectrum of a ring. Bilocalizing subcategories.
__The Krull-Remak-Schmidt theorem and decomposition theories__
The classical Krull-Remak-Schmidt theorem. The structure of spectral categories. Local
coirreducible categories. Decomposition theory. Semi-noetherian categories. Semi-artinian
categories. Noetherian and artinian categories. Locally noetherian categories. Noetherian and
artinian rings. Primary decomposition theory. Decomposition theories on locally noetherian categories.
__Duality__
Linearly compact subcategories. Topologically linearly compact rings. The duality theorem for
Grothendieck categories. Duality theory for l.n.- and l.f.-categories. Colocalization.
__Bibliography__
__Index__
category: reference
|
abelian category | https://ncatlab.org/nlab/source/abelian+category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
##Idea
The notion of _abelian category_ is an abstraction of basic properties of the category [[Ab]] of [[abelian groups]], more generally of the category $R$[[Mod]] of [[modules]] over some [[ring]], and still more generally of categories of [[sheaves]] of abelian groups and of modules. It is such that much of the [[homological algebra]] of [[chain complexes]] can be developed inside every abelian category.
The concept of abelian categories is one in a sequence of notions of [[additive and abelian categories]].
While additive categories differ significantly from [[toposes]], there is an intimate relation between abelian categories and toposes. See _[[AT category]]_ for more on that.
##Definition
Recall the following fact about [[pre-abelian categories]] from [this proposition](pre-abelian+category#DecompositionOfMorphisms), discussed there:
+-- {: .num_prop #DecompositionOfMorphisms}
###### Proposition
Every [[morphism]] $f \colon A\to B$ in a [[pre-abelian category]] has a canonical de-[[composition]]
\[
\label{PreImageFactorization}
A
\overset{\; p \;}{\twoheadrightarrow}
\coker\big(\ker f\big)
\xrightarrow{\; \overline{f} \;}
\ker\big(\coker(f)\big)
\xhookrightarrow{\; i \;}
B
\]
where:
* $p$ is a [[cokernel]], hence an [[epimorphism]],
* $i$ is a [[kernel]], hence a [[monomorphism]].
=--
+-- {: .num_defn #AbelianCategory}
###### Definition
An **abelian category** is a [[pre-abelian category]] satisfying the following equivalent conditions.
1. For every [[morphism]] $f$, the canonical morphism $\bar{f} \colon coker(ker(f)) \to ker(coker(f))$ (eq:PreImageFactorization) from prop. \ref{DecompositionOfMorphisms} is an [[isomorphism]] (hence providing an [[image]] factorization $A \twoheadrightarrow im(f) \hookrightarrow B$).
1. Every [[monomorphism]] is a [[kernel]] and every [[epimorphism]] is a [[cokernel]].
=--
+-- {: .num_prop}
###### Proposition
These two conditions are indeed equivalent.
=--
+-- {: .proof}
###### Proof
The first condition implies that if $f$ is a [[monomorphism]] then $f \cong \ker(\coker(f))$ (in the category of objects over $B$) so $f$ is a kernel. Dually if $f$ is an [[epimorphism]] it follows that $f \cong coker(ker(f))$. So (1) implies (2).
The converse can be found in, among other places, Chapter VIII of ([MacLane](#MacLane)).
=--
## Properties
### General
{#PropertiesGeneral}
+-- {: .num_remark}
###### Remark
The notion of abelian category is self-dual: [[opposite category|opposite]] of any abelian category is abelian.
=--
+-- {: .num_remark #RegularEpisAndMonos}
###### Remark
By the second formulation of the definition \ref{AbelianCategory}, in an abelian category
* every [[monomorphism]] is a [[regular monomorphism]];
* every [[epimorphism]] is a [[regular epimorphism]].
It follows that every abelian category is a _[[balanced category]]_.
=--
\begin{prop}\label{PullbackPreservesEpimorphisms}
In an [[abelian category]], [[pullback]] preserves [[epimorphisms]] and [[pushout]] preserves [[monomorphisms]].
\end{prop}
Because every abelian category is a [[regular category]]. For an explicit proof see, e.g., [Selick, Prop. 1.3.13](#Selick).
### Factorization of morphisms
{#FactorizationOfMorphisms}
+-- {: .num_prop}
###### Proposition
In an abelian category every morphism decomposes [[generalized the|uniquely up to a unique isomorphism]] into the composition of an [[epimorphism]] and a [[monomorphism]], via prop \ref{DecompositionOfMorphisms} combined with def. \ref{AbelianCategory}.
Since by remark \ref{RegularEpisAndMonos} every monic is [[regular monomorphism|regular]], hence [[strong monomorphism|strong]], it follows that $(epi, mono)$ is an [[orthogonal factorization system]] in an abelian category; see at _[[(epi, mono) factorization system]]_.
=--
+-- {: .num_remark}
###### Remark
Some references claim that this property characterizes abelian categories among pre-abelian ones, but it is not clear to the authors of this page why this should be so, although we do not currently have a counterexample; see [this discussion](http://nforum.mathforge.org/discussion/4094/?Focus=33415#Comment_33415).
=--
### Canonical $Ab$-enrichment
{#CanonicalAbEnrichment}
The $Ab$-enrichment of an abelian category need not be specified a priori. If an arbitrary (not necessarily pre-additive) [[locally small category|locally small]] category $C$ has a [[zero object]], binary products and coproducts, kernels, cokernels and the property that every monic is a kernel arrow and every epi is a cokernel arrow (so that all monos and epis are [[normal monomorphism|normal]]), then it can be equipped with a unique addition on the morphism sets such that composition is bilinear and $C$ is abelian with respect to this structure. However, in most examples, the $Ab$-enrichment is evident from the start and does not need to be constructed in this way. (A similar statement is true for [[additive categories]], although the most natural result in that case gives only enrichment over abelian [[monoids]]; see [[semiadditive category]].)
The last point is of relevance in particular for [[infinity-category|higher categorical]] generalizations of additive categories. See for instance [remark 2.14, p. 5](http://www.math.harvard.edu/~lurie/papers/DAG-I.pdf#page=5) of [[Jacob Lurie]]'s [[Stable Infinity-Categories]].
### Relation to exactness properties of toposes
{#RelationToToposes}
The [[exactness properties]] of abelian categories have many features in common with exactness properties of [[toposes]] or of [[pretoposes]].
[Freyd (1999)](AT+category#Freyd99) gave a sharp description of the properties shared by these categories, introducing a new concept called _[[AT categories]]_ (for "abelian-topos"), and showing convincingly that the difference between the A and the T can be concentrated precisely in the difference of the behavior of the initial object.
### Embedding theorems
{#EmbeddingTheorems}
Not every [[abelian category]] is a [[concrete category]] such as [[Ab]] or $R$[[Mod]]. But for many proofs in [[homological algebra]] it is very convenient to have a concrete abelian category, for that allows one to check the behaviour of morphisms on actual _elements_ of the sets underlying the [[objects]].
The following _embedding theorems_, however, show that under good conditions an abelian category can be _embedded_ into [[Ab]] as a [[full subcategory]] by an [[exact functor]], and generally can be embedded this way into $R Mod$, for some ring $R$. This is the celebrated _[[Freyd-Mitchell embedding theorem]]_ discussed [below](#FreydMitchellEmbedding).
This implies for instance that proofs about [[exact sequence|exactness of sequences]] in an abelian category can always be obtained by a naive argument on elements -- called a "[[diagram chasing|diagram chase]]" -- because that does hold true after such an embedding, and the exactness of the embedding means that the notion of exact sequences is preserved by it.
Alternatively, one can reason with [[generalized elements]] in an abelian category, without explicitly embedding it into a larger concrete category, see at _[[element in an abelian category]]_. But under suitable conditions this comes down to working subject to an embedding into $Ab$, see the discussion at _[Embedding into Ab](#EmbeddingIntoAb)_ below.
#### Counterexamples
First of all, it's easy to see that not every abelian category is equivalent to $R$[[Mod]] for some ring $R$. The reason is that $R Mod$ has all [[small category]] [[limits]] and [[colimits]]. For a [[Noetherian ring]] $R$ the category of [[finitely generated module|finitely generated]] $R$-modules is an abelian category that lacks these properties.
#### Embedding into $Ab$
{#EmbeddingIntoAb}
(...)
([Bergman 1974](#Bergman))
(...)
#### Freyd-Mitchell embedding into $R Mod$
{#FreydMitchellEmbedding}
+-- {: .num_thm}
###### Mitchell's Embedding Theorem
Every small abelian category admits a [[full functor|full]], [[faithful functor|faithful]] and [[exact functor|exact]] functor to the category $R Mod$ for some ring $R$.
=--
+-- {: .proof}
###### Proof
This result can be found as Theorem 7.34 on page 150 of Peter Freyd's book [Abelian Categories](http://www.emis.de/journals/TAC/reprints/articles/3/tr3.pdf#page=176). His terminology is a bit outdated, in that he calls an abelian category "fully abelian" if admits a full and faithful exact functor to a category of $R$-modules. See also the [Wikipedia article](http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem) for the idea of the proof.
=--
For more see at _[[Freyd-Mitchell embedding theorem]]_.
We can also characterize which abelian categories _are_ equivalent to a category of $R$-modules:
+-- {: .num_theorem}
###### Theorem
Let $C$ be an abelian category. If $C$ has all [[small category|small]] [[coproducts]] and has a [[compact object|compact]] [[projective object|projective]] [[generator]], then $C \simeq R Mod$ for some ring $R$. In fact, in this situation we can take $R = C(x,x)^{op}$ where $x$ is any compact projective generator. Conversely, if $C \simeq R Mod$, then $C$ has all small coproducts and $x = R$ is a compact projective generator.
=--
+-- {: .proof}
###### Proof
This theorem, minus the explicit description of $R$, can be found as Exercise F on page 103 of Peter Freyd's book [Abelian Categories](http://www.emis.de/journals/TAC/reprints/articles/3/tr3.pdf#page=132).
The first part of this theorem can also be found as Prop. 2.1.7. of Victor Ginzburg's [Lectures on noncommutative geometry](http://arxiv.org/PS_cache/math/pdf/0506/0506603v1.pdf#page=4). Conversely, it is easy to see that $R$ is a compact projective generator of $R Mod$.
=--
One can characterize functors between categories of $R$-modules that are either (isomorphic) to functors of the form $B \otimes_R -$ where $B$ is a bimodule or those which look as Hom-modules. For the characterization of the tensoring functors see [[Eilenberg-Watts theorem]].
Going still further one should be able to obtain a nice theorem describing the image of the embedding of the weak 2-category of
* rings
* bimodules
* bimodule homomorphisms
into the strict 2-category of
* abelian categories
* right exact functors
* natural transformations.
For more discussion see the [$n$-Cafe](http://golem.ph.utexas.edu/category/2007/08/questions_about_modules.html).
## Examples
* Of course, [[Ab]] is abelian,
* the [[category of modules|category $R$Mod of (left) modules]] over any [[ring]] $R$ is abelian
* Therefore in particular the category [[Vect]] of vector spaces over any field is an abelian category
* The full subcategory of $R$Mod whose objects are the Noetherian left $R$-modules is abelian, since it contains any submodule or quotient module of any of its objects (see Theorem 2.3.8 p.103 of Berrick and Keating in the textbook references below).
* Similarly, the full subcategory of $R$Mod whose objects are the Artinian left $R$-modules is abelian, since it contains any submodule or quotient module of any of its objects (loc. cit.).
* Also similarly, the full subcategory of $R$Mod whose objects are the Artinian semisimple modules is abelian, since it contains any submodule or quotient module of any of its objects (loc. cit.) .
* as is the [[category of representations]] of a [[group]] (e.g. [here](https://unapologetic.wordpress.com/2008/12/15/the-category-of-representations-is-abelian/))
* The [[category of sheaves|category of]] [[sheaves of abelian groups]] on any [[site]] is abelian.
Counter-examples:
* The category of [[torsion subgroup|torsion-free]] abelian groups is pre-abelian, but not abelian: the monomorphism $2:\mathbb{Z}\to\mathbb{Z}$ is not a kernel.
## Related concepts
* [[additive and abelian categories]]
* [[abelian subcategory]]
* [[Deligne tensor product of abelian categories]]
* [[pseudo-abelian category]]
* [[quasi-abelian category]]
* [[semi-abelian category]]
* [[length of an object]]
## References
Maybe the first reference on abelian categories, then still called _exact categories_ is
* D. A. Buchsbaum, _Exact categories and duality_, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 ([JSTOR](http://www.jstor.org/stable/1993003))
Further foundations of the theory were then laid in
* [[Alexander Grothendieck]], _[[Tohoku|Sur quelques points d'algèbre homologique]], Tôhoku Math. J. vol 9, n.2, 3, 1957.
Other classic references:
* [[Pierre Gabriel]], *[[Des Catégories Abéliennes]]*, Bulletin de la Société Mathématique de France **90** (1962) 323-448 [[numdam:BSMF_1962__90__323_0](http://www.numdam.org/item?id=BSMF_1962__90__323_0)]
* {#Freyd64} [[Peter Freyd]], _Abelian Categories -- An Introduction to the theory of functors_, originally published by Harper and Row, New York(1964), Reprints in Theory and Applications of Categories **3** (2003) [[tac:tr3](http://www.emis.de/journals/TAC/reprints/articles/3/tr3abs.html), [pdf](https://www.emis.de/journals/TAC/reprints/articles/3/tr3.pdf)]
Textbook accounts:
* [[Nicolae Popescu]], _[[Abelian categories with applications to rings and modules]]_, London Math. Soc. Monographs __3__, Academic Press (1973) [[MR0340375](http://www.ams.org/mathscinet-getitem?mr=0340375)]
* [[Saunders MacLane]], Chapter VIII of: _[[Categories for the Working Mathematician]]_ (1978)
* A. J. Berrick and M. E. Keating, *Categories and Modules, with K-theory in View*, Cambridge Studies in Advanced Mathematics **67**, Cambridge University Press (2000
* [[Masaki Kashiwara]], [[Pierre Schapira]], Section 8 of: *[[Categories and Sheaves]]*, Grundlehren der Mathematischen Wissenschaften **332**, Springer (2006) [[doi:10.1007/3-540-27950-4](https://link.springer.com/book/10.1007/3-540-27950-4), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf)]
* {#EGNO15} [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], Chapter 1 of: *Tensor Categories*, AMS Mathematical Surveys and Monographs **205** (2015) [[ISBN:978-1-4704-3441-0](https://bookstore.ams.org/surv-205), [pdf](http://www-math.mit.edu/~etingof/egnobookfinal.pdf)]
Further review:
* Rankey Datta, _An introduction to abelian categories_ (2010) ([pdf](http://www-bcf.usc.edu/~lauda/teaching/rankeya.pdf))
* {#Selick} Paul Selick, *Homological Algebra Notes* ([pdf](www.math.toronto.edu/selick/mat1352/1350notes.pdf),[[Selick_HomologicalAlgebra.pdf:file]])
On [[diagram chasing]] and [[elements in an abelian category]]:
* {#Bergman} [[George Bergman]], _A note on abelian categories -- translating element-chasing proofs, and exact embedding in abelian groups_ (1974) [[pdf](http://math.berkeley.edu/~gbergman/papers/unpub/elem-chase.pdf), [[Bergman-ElementChasing.pdf:file]]]
For more discussion of the _[[Freyd-Mitchell embedding theorem]]_ see there.
The proof that $R Mod$ is an abelian category is spelled out for instance in
* Rankeya Datta, _The category of modules over a commutative ring and abelian categories_ ([pdf](http://www.math.columbia.edu/~ums/pdf/Rankeya_R-mod_and_Abelian_Categories.pdf))
A discussion about to which extent abelian categories are a general context for [[homological algebra]] is archived at nForum [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2052&Focus=17680#Comment_17680).
See also the [catlist 1999 discussion](http://www.mta.ca/~cat-dist/catlist/1999/atcat) on comparison between abelian categories and topoi ([[AT categories]]).
Formalization of abelian categories as [[univalent categories]] in [[univalent foundations of mathematics]] ([[homotopy type theory]]):
* [[unimath]], *Abelian Categories* [[UniMath.CategoryTheory.Abelian](https://unimath.github.io/doc/UniMath/4dd5c17/UniMath.CategoryTheory.Abelian.html)]
[[!redirects abelian categories]]
[[!redirects Abelian category]]
|
abelian differential graded Lie algebra | https://ncatlab.org/nlab/source/abelian+differential+graded+Lie+algebra | An abelian [[differential graded Lie algebra]] is a [[cochain complex]] seen as a differential graded Lie algebra with trivial [[Lie algebra|Lie bracket]]. In the particular case when the cochain complex is concentrated in degree zero, one recovers the notion of [[abelian Lie algebra]]. |
abelian group | https://ncatlab.org/nlab/source/abelian+group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
An _abelian group_ (named after [[Niels Henrik Abel]]) is a [[group]] $A$ where the multiplication satisfies the commutative law: for all elements $x, y\in A$ we have
$$
x y = y x
\,.
$$
The [[category]] with abelian groups as [[object|objects]] and group homomorphisms as [[morphism|morphisms]] is called [[Ab]].
Every abelian group has the canonical structure of a [[module]] over the [[commutative ring]] $\mathbf{Z}$. That is, [[Ab]] = $\mathbf{Z}$-[[Mod]].
### With subtraction and unit only
This definition of abelian group is based upon [[Toby Bartels]]'s definition of an [[associative quasigroup]]:
An **abelian group** is a [[pointed set]] $(A, 0)$ with a binary operation $(-)-(-):A \times A \to A$ called **subtraction** such that
* for all $a \in A$, $a - a = 0$
* for all $a \in A$, $0 - (0 - a) = a$
* for all $a \in A$ and $b \in A$, $a - (0 - b) = b - (0 - a)$
* for all $a \in A$, $b \in A$, and $c \in A$, $a - (b - c) = (a - (0 - c)) - b$
For every element $a \in A$, the inverse element is defined as $-a \coloneqq 0 - a$ and addition is defined as $a + b \coloneqq a - (-b)$.
Addition is commutative:
$$a + b = a - (0 - b) = b - (0 - a) = b + a$$
and associative
$$(a + b) + c = (a - (0 - b)) - (0 - c)$$
$$(a + b) + c = (b - (0 - a)) - (0 - c)$$
$$(a + b) + c = b - ((0 - c) - a)$$
$$(a + b) + c = b - ((0 - c) - (0 - (0 - a)))$$
$$(a + b) + c = b - ((0 - a) - (0 - (0 - c)))$$
$$(a + b) + c = b - ((0 - a) - c)$$
$$(a + b) + c = (b - (0 - c)) - (0 - a)$$
$$(a + b) + c = a - (0 - (b - (0 - c)))$$
$$(a + b) + c = a + (b + c)$$
and has left identities
$$0 + a = 0 - (0 - a) = a$$
and right identities
$$a + 0 = 0 + a = a$$
and has left inverses
$$-a + a = (0 - a) - (0 - a) = 0$$
and right identities
$$a + (-a) = -a + a = 0$$
Thus, these axioms form an abelian group.
## Properties
### In homotopy theory
From the [[nPOV]], just as a [[group]] $G$ may be thought of as a ([[pointed object|pointed]]) [[groupoid]] $\mathbf{B}G$ with a single object -- as discussed at [[delooping]] -- an abelian group $A$ may be understood as a (pointed) [[2-groupoid]] $\mathbf{B}^2 A$ with a single object and a single morphism: the delooping of the delooping of $A$.
$$
\mathbf{B}^2 A
=
\left\{
\array{
& \nearrow \searrow^{\mathrlap{Id}}
\\
\bullet
&\Downarrow^{a \in A}&
\bullet
\\
& \searrow \nearrow_{\mathrlap{Id}}
}
\right\}
\,.
$$
The [[exchange law]] for the composition of [[2-morphisms]] in a [[2-category]] forces the product on the $a \in A$ here to be commutative. This reasoning is known as the [[Eckmann-Hilton argument]] and is the same as the reasoning that finds that the second [[homotopy group]] of a space has to be abelian.
So the identitfication of abelian groups with one-object, one-morphism 2-groupoids may also be thought of as an identification with 2-[[truncated]] and 2-[[connected]] [[homotopy types]].
### Relation to other concepts
A [[monoid]] in [[Ab]] with its standard [[monoidal category]] structure, equivalently a ([[pointed object|pointed]]) [[Ab]]-[[enriched category]] with a single object, is a [[ring]].
## Generalizations
Generalizations of the notion of abelian group in [[higher category theory]] include
* abelian [[group object in an (infinity,1)-category|group objects in an (∞,1)-category]]
* notably abelian [[simplicial groups]]
* and [[spectrum|spectra]].
An abelian group may also be seen as a [[discrete category|discrete]] [[compact closed category]].
## Related entries
* [[commutative magma]]
* [[commutative invertible semigroup]]
* [[tensor product of abelian groups]], [[direct sum of abelian groups]]
* [[free abelian group]], [[finite abelian group]]
* [[theory of abelian groups]]
* [[abelianization]]
* [[anabelian group]]
* [[module]], [[ring]]
* [[commutative ring]], [[commutative algebra]]
* [[super abelian group]], [[super module]]
* [[quadratic abelian group]]
* [[tensor ring]], [[Clifford ring]]
* [[Ab]]
* [[sheaf of abelian groups]]
* [[Ab-enriched category]], [[abelian category]]
* [[abelian ∞-group]]
* [[symmetric 2-group]]
* [[symmetric 3-group]]
* [[halving group]]
* [[symmetric midpoint group]]
## References
Textbook account:
* [[László Fuchs]], *Abelian Groups*, Springer (2015) [[doi:10.1007/978-3-319-19422-6](https://doi.org/10.1007/978-3-319-19422-6)]
Formalization of abelian groups in [[univalent foundations of mathematics]] ([[homotopy type theory]] with the [[univalence axiom]]):
* [[Univalent Foundations Project]], Section 6.11 of: *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)
* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]], Section 4.12 of: **[[Symmetry]]** (2021)
[[!redirects abelian groups]]
[[!redirects Abelian group]]
[[!redirects Abelian groups]] |
abelian infinity-group | https://ncatlab.org/nlab/source/abelian+infinity-group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An ordinary [[group]] is either an [[abelian group]] or not. For an [[∞-group]] there is an infinite tower of notions ranging from completely general non-abelian [[∞-groups]], over [[braided infinity-group|braided $\infty$-groups]], [[sylleptic infinity-groups|sylleptic $\infty$-groups]] ..., to ever more abelian groups By an _abelian ∞-group_ (not an established term) one may want to mean an [[∞-group]] which is maximally abelian, in this sense.
Technically, the level of abelianness may be encoded (see at *[[May recognition theorem]]*) by the [[En-operads|$E_n$-operads]] as $n$ ranges from 1 to $\infty$: On the non-abelian end, a general [[∞-group]] is equivalently a groupal [[algebra over an operad|algebra]] over $E_1$, also known as the [[associative operad]], hence is a groupal [[A-∞ algebra]]; while at the abelian end a groupal [[E-infinity space|$E_\infty$-space]] is an *[[infinite loop space]]* or *[[connective spectrum]]*. See also the *[periodic table](k-tuply+groupal+n-groupoid#PeriodicTable)* of [[k-tuply monoidal n-groupoid|$k$-tuply monoidal $n$-groupoids]].
Notice that referring to [[connective spectra]] as "abelian $\infty$-groups" (which is not standard) matches the established terminology for *[[non-abelian cohomology]]* (which is standard): The qualifier "non-abelian" in [[non-abelian cohomology]] is in contrast to [[Whitehead-generalized cohomology theories]] which are [[Brown representability theorem|represented]] by [[spectra]].
In a more restrictive sense one may say that plain abelian [[cohomology]] is just *[[ordinary cohomology|ordinary]]* cohomology theory, subsuming only those [[Whitehead-generalized cohomology theories]] which are [[Brown representability theorem|represented]] specifically by [[Eilenberg-MacLane spectra]]. Under the [[Dold-Kan correspondence]] these are [[equivalence of (infinity,1)-categories|equivalently]] [[chain complexes]] of [[abelian groups]]. One may think of these as being yet more commutative than general spectra and might want to reserve the term "abelian $\infty$-group" for them.
## Proposition
### Relation to commutative $\infty$-rings
+-- {: .num_defn #GroupOfUnitsFunctor}
###### Definition
Write
$$
gl_1 \; \colon \; CRing_\infty \to AbGrp_\infty
$$
for the [[(∞,1)-functor]] which sends a [[E-∞ ring|commutative ∞-ring]] to its [[∞-group of units]].
=--
+-- {: .num_defn}
###### Definition
The [[∞-group of units]] [[(∞,1)-functor]] of def. \ref{GroupOfUnitsFunctor} is a right-[[adjoint (∞,1)-functor]] (or at least a [[right adjoint]] on [[homotopy category of an (∞,1)-category|homotopy categories]])
$$
CRing_\infty
\stackrel{\overset{\mathbb{S}[-]}{\leftarrow}}{\underset{gl_1}{\to}}
AbGrp_\infty
\,.
$$
=--
This is ([ABGHR 08, theorem 2.1](#ABGHR08)).
## Examples
* A [[0-truncated]] abelian $\infty$-group is equivalently an [[abelian group]].
* A [[1-truncated]] abelian $\infty$-group is equivalently a [[symmetric 2-group]].
* A [[2-truncated]] abelian $\infty$-group is equivalently a [[symmetric 3-group]].
## Related concepts
[[!include k-monoidal table]]
* [[∞-group]], [[k-tuply groupal n-groupoid]]
* [[braided ∞-group]],
* [[braided 2-group]]
* [[braided 3-group]]
* [[sylleptic ∞-group]]
* [[sylleptic 3-group]]
* **abelian ∞-group**
* [[abelian group]]
* [[abelian 2-group]]
* [[abelian 3-group]]
## References
General discussion is in section 5 of
* [[Jacob Lurie]], _[[Higher Algebra]]_
Discussion in the context of [[E-∞ rings]] and [[twisted cohomology]] is in
* {#ABGHR08} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _Units of ring spectra and Thom spectra_ ([arXiv:0810.4535](http://arxiv.org/abs/0810.4535))
[[!redirects abelian infinity-groups]]
[[!redirects k-monoidal ∞-group]]
[[!redirects abelian ∞-group]]
[[!redirects abelian ∞-groups]]
|
abelian Lie algebra | https://ncatlab.org/nlab/source/abelian+Lie+algebra |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A [[Lie algebra]] $\mathfrak{g}$ is __abelian__ if its bracket is identically 0, in that for all $x,y \in \mathfrak{g}$ we have
$$
[x,y] = 0
\,.
$$
## Examples
Every [[vector space]] has a (necessarily unique) abelian Lie algebra structure. As such, we may identify an abelian Lie algebra with its underlying vector space.
A $0$-dimensional or $1$-dimensional Lie algebra must be abelian. The $0$-dimensional Lie algebra is the [[trivial Lie algebra]]. The $1$-dimensional Lie algebra is a [[simple object]] in [[LieAlg]], but it is traditionally *not* considered a [[simple Lie algebra]].
## Lie integration
Under [[Lie integration]] abelian Lie algebras integrate to [[abelian Lie group]]s.
[[!redirects abelian Lie algebra]]
[[!redirects abelian Lie algebras]]
[[!redirects Abelian Lie algebra]]
[[!redirects Abelian Lie algebras]]
|
abelian model category | https://ncatlab.org/nlab/source/abelian+model+category | ## Idea
An abelian model category is an [[abelian category]] with a compatible [[model structure]].
## Definition
An **abelian model category** is an [[abelian category]] $\mathcal{A}$ that is [[complete category|complete]] and [[cocomplete category|cocomplete]], together with a [[model structure]] such that
* (AMC1) A [[morphism]] $i : A \to B$ is a [[cofibration]] if and only if it is a [[monomorphism]] with [[cofibrant object|cofibrant]] [[cokernel]].
* (AMC2) A [[morphism]] $p : X \to Y$ is a [[fibration]] if and only if it is an [[epimorphism]] with [[fibrant object|fibrant]] [[kernel]].
## Relation to cotorsion pairs
[Hovey](#Hovey) has shown that, roughly speaking, [[model structures]] on [[abelian categories]] correspond to [[cotorsion pairs]].
In one direction we have
+-- {: .num_prop}
###### Proposition
Let $\mathcal{A}$ be an [[abelian model category]], i.e. an [[abelian category]] with a compatible [[model structure]]. Let $\mathcal{C}$, $\mathcal{F}$, and $\mathcal{W}$ denote the classes of cofibrant, fibrant, and trivial objects, respectively.
Then $(\mathcal{C} \cap \mathcal{W}, \mathcal{F})$ and $(\mathcal{C}, \mathcal{F} \cap \mathcal{W})$ are complete [[cotorsion pairs]].
=--
And under some more assumptions we have a converse
+-- {: .num_theorem}
###### Theorem
Let $\mathcal{A}$ be an [[abelian category]] that is [[complete category|complete]] and [[cocomplete category|cocomplete]]. Let $\mathcal{C}$, $\mathcal{F}$, and $\mathcal{W}$ denote three classes of objects in $\mathcal{A}$, such that $\mathcal{W}$ is a [[thick subcategory]], and $(\mathcal{C} \cap \mathcal{W}, \mathcal{F})$ and $(\mathcal{C}, \mathcal{F} \cap \mathcal{W})$ are complete [[cotorsion pairs]].
Then there exists a unique [[abelian model structure]] on $\mathcal{A}$ such that $\mathcal{C}$, $\mathcal{F}$, $\mathcal{W}$ are the classes of [[cofibrant]], [[fibrant]], and trivial objects, respectively.
=--
Under certain assumptions on the cotorsion pair we can further guarantee that the associated model structure is [[monoidal model category|monoidal]].
+-- {: .num_theorem}
###### Theorem
Under the assumptions of the previous theorem, suppose further that $\mathcal{A}$ is [[closed monoidal category|closed]] [[symmetric monoidal category|symmetric monoidal]], and that
* Every object in the class $\mathcal{C}$ is [[flat]] ($X \otimes \cdot$ is an [[exact functor]]).
* For any two objects $X$ and $Y$ in $\mathcal{C}$, the tensor product $X \otimes Y$ is also in $\mathcal{C}$. If one of $X$ and $Y$ is further in $\mathcal{W}$ then $X \otimes Y$ is also in $\mathcal{W}$.
* The unit is in $\mathcal{C}$.
Then $\mathcal{A}$ is a [[monoidal model category]] (with the model structure given by the previous theorem).
=--
## References
* [[Mark Hovey]], _Cotorsion pairs, model category structures, and representation theory_, Math. Z. 241 (2002), no. 3, 553–592. MR 2003m:55027
{#Hovey}
An overview is in
* [[Mark Hovey]], _Cotorsion pairs and model categories_, 2006 ([pdf](http://homepages.math.uic.edu/~bshipley/hovey.pdf))
[[!redirects abelian model categories]]
[[!redirects abelian model structure]]
[[!redirects abelian model structures]] |
abelian sheaf | https://ncatlab.org/nlab/source/abelian+sheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
> see also _[[sheaf of abelian groups]]_
#Contents#
* table of contents
{:toc}
## Idea
An **abelian sheaf** is a [[sheaf]] with values in an [[abelian category]] which usually is itself, or is taken to be embedded in, a category of [[complex]]es in an [[abelian category]].
In light of the [[Dold-Kan correspondence]] this means that abelian sheaves can usefully be regarded as special cases of [[simplicial presheaf|simplicial presheaves]] and in particular the corresponding [[derived category]] of abelian sheaves, traditionally mainly investigated in terms of [[sheaf cohomology]], is analogous to the [[homotopy category]] of abelian [[infinity-stack homotopically|infinity-stacks]]. In this way, via [[Dold-Kan correspondence|Dold-Kan]], plain abelian sheaves already go a long way towards (abelian) [[Higher Topos Theory]], which is one way of understanding the relevance of the concept of abelian sheaves.
For instance [[Deligne cohomology]], which classifies higher abelian [[gerbe]]s (certain [[infinity-stack]]s) with [[connection]]), is the [[sheaf cohomology]] of a certain class of sheaves with values in abelian complexes. This is understood conceptually by realizing that after embedding complexes of abelian sheaves -- via [[Dold-Kan correspondence|Dold-Kan]] -- into general [[simplicial presheaf|simplicial sheaves]], a complex of abelian sheaves becomes an abelian $\infty$-prestack and the computation of its [[sheaf cohomology]] corresponds to passing to its [[infinity-stackification]].
## Properties
### Projective objects
See at _[[projective object]]_ the section _[Existence of enough projectives](projective+object#ExistenceOfEnoughProjectives)_.
## Related concepts
* [[sheaf on a topological space]]
* [[abelian group]]
* [[abelian sheaf cohomology]]
* an abelian sheaf of [[torsion groups]] is called a _[[torsion sheaf]]_
## References
A basic textbook introduction begins for instance around Definition 1.5.6 of
* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_
A detailed textbook discussion is in section 18 of
* [[Masaki Kashiwara]], [[Pierre Schapira]], _Categories and Sheaves_, Grundlehren der Mathematischen Wissenschaften __332__, Springer (2006)
category: sheaf theory
[[!redirects abelian sheaves]]
|
abelian sheaf cohomology | https://ncatlab.org/nlab/source/abelian+sheaf+cohomology |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
##Idea##
### General
The [[cohomology]] $H^n(X,F)$ of a [[topological space]] $X$ with values in a [[sheaf of abelian groups]] / [[abelian sheaf]] $F$ was originally defined as the value of the right [[derived functor]] of the [[global section]] functor, the [[derived direct image]] functor.
But by embedding sheaves with values in abelian groups as special cases of [[simplicial presheaf|simplicial sheaves]] into the more general context of [[(∞,1)-sheaves|∞-groupoid-valued sheaves]] via the [[Dold-Kan correspondence]] and thus the abelian sheaf cohomology into the more general context of the intrinsic [[nonabelian cohomology]] of an [[(∞,1)-topos]] $\mathbf{H} = Sh_{(\infty,1)}(C)$, this definition becomes equivalent to a special case of the general notion of [[nonabelian cohomology]] defined simply as the set of homotopy classes of maps
$$
H^n(X,F) = \pi_0 \mathbf{H}(X,\mathbf{B}^n F)
$$
from the space $X$ regarded a ("nonabelian"!) sheaf, to the [[Eilenberg-MacLane object]] in degree $n$, defined by $F$.
The relation of this more conceptual and more general point of view on abelian sheaf cohomology to the original definition was originally clarified in
* [[Kenneth Brown]], [[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf Cohomology]]
(whose proof is reproduced below).
Brown constructed effectively the [[homotopy category of an (∞,1)-category|homotopy category]] of $\mathbf{H}$ using a model of a [[category of fibrant objects]] paralleling the [[model structure on simplicial presheaves]] as a [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-category of (∞,1)-sheaves]]. This says that ordinary abelian sheaf cohomology in fact computes the equivalence classes of the [[∞-stackification]] of a sheaf with values in [[chain complexes]] of [[abelian groups]].
The general [[(∞,1)-topos]]-theoreric perspective on cohomology is described in more detail at [[cohomology]]. The details on how to realize abelian sheaf cohomology as an example of this are discussed below.
### More details on this idea ###
Using the [[Dold-Kan correspondence]] in [[higher topos theory]], [[complex]]es of [[abelian sheaf|abelian sheaves]] can be understood as a generalization of [[topological space]]s, in a precise sense recalled below.
This induces a notion of cohomology of [[complex]]es of [[abelian sheaf|abelian sheaves]] from the familiar notion of cohomology of spaces.
Which is a simple one: recall that the [[cohomology]] of one [[topological space]] $X$ with coefficients in another space $A$ is nothing but the space of morphisms (continuous maps) $H(X,A) := [X,A]$ from $X$ to $A$ -- or, in a more restrictive sense traditionally adopted, the set $\Pi_0[X,A]$ of connected path components of this space.
Similarly, when considering [[chain complexes]] of [[abelian sheaves]] in their natural [[higher topos theory|higher topos theoretic]] home, the cohomology of a complex of sheaves $A$ on a space $X$ is nothing but the hom-space $H(X,A) = [X,A]$ -- where the space $X$ itself is regarded as a special case of a sheaf.
One reason this conceptually simple picture is not usually presented is that the space $X$ is typically not represented by an _abelian_ complex of sheaves, so that the full simplicity of the story becomes manifest only in general [[nonabelian cohomology]].
More precisely, via the [[Dold-Kan correspondence]] (non-negatively graded) [[complex]]es of [[abelian sheaf|abelian sheaves]] -- which are equivalently [[sheaf|sheaves]] with values in (non-negatively graded) [[category of chain complexes|categories of chain complexes]] -- can be regarded as special cases of [[simplicial presheaf|simplicial sheaves]]. But thanks to the [[model structure on simplicial presheaves|model category structure]] on the category of [[simplicial presheaf|simplicial sheaves]], this in turn is a model for the [[(infinity,1)-topos]] of [[space and quantity|generalized spaces]] called [[infinity-stack]]s. The very point of $(\infty,1)$-[[(infinity,1)-topos|topoi]] is that they are [[(infinity,1)-category|(infintiy,1)-categories]] which behave in all structural aspects relevant for [[homotopy theory]] as the archetypical example [[Top]] does. In particular, as in [[Top]], the notion of [[cohomology]] in any [[(infinity,1)-topos]] simply coincides with that of [[hom-space]]s.
In the 1-categorical [[model category|model theoretic models]] these hom-spaces are computed technically by [[derived functor]]s. More precisely, the Hom-space $[X,A]$ for $X$ an ordinary space computes the [[global section]]s $\Gamma(X,A)$ of the complex of [[abelian sheaf|abelian sheaves]] $A$ which is computed by the right [[derived functor]] of the [[global section]] $R \Gamma(X,-)$ of the [[global section functor]] $\Gamma(X,-)$, which does exist entirely within the abelian context.
This, then, is the definition of sheaf cohomology as usually presented: the cohomology of the complex $R \Gamma(X,A)$.
## Properties
### As intrinsic (∞,1)-topos cohomology
{#AsIntrinsicInfinityToposCohomology}
Under the ([[stable Dold-Kan correspondence|stable]]) [[Dold-Kan correspondence]] we have the following
identification of sheaves taking values in [[chain complexes]]
with sheaves taking values in [[infinity-groupoids]] and
[[spectrum|spectra]], crucial for a conceptual understanding of
abelian sheaf cohomology:
let $X$ be a [[site]]
* the category $Sh(X, Ch_+(Ab))$ of non-negatively graded
[[chain complex]]es of [[abelian sheaf|abelian sheaves]] is
[[homotopical functor|homotopically]] [[equivalence|equivalent]]
to the category $Sh(X, sAb)$ of [[sheaf|sheaves]] with values in
simplicial abelian groups (i.e. [[Kan complex]]es with strict abelian group structure);
* the category $Sh(X, Ch(Ab))$ of unbounded
[[chain complex]]es of [[abelian sheaf|abelian sheaves]] is
[[equivalence of categories|equivalent]] to $Sh(X, Sp(Ab))$, the category
of sheaves with values in [[combinatorial spectrum|combinatorial spectra]]
internal to abelian groups.
Let how $F \in Sh(X,Ab)$ be a [[sheaf]] on a [[site]] $X$ with values in the category [[Ab]] of abelian groups.
For $n \in \mathbb{N}$ write $B^n F \in Sh(X, Ch_+(Ab))$ for the [[complex]] of [[sheaf|sheaves]] with values in abelian groups which is trivial everywhere except in degree $n$, where it is given by $F$.
By the [[Dold-Kan correspondence]] we can regard $B^n F$ equivalently as a complex of sheaves of abelian groups as well as sheaf with values in [[infinity-groupoid]]s.
Write $H$ for the [[(infinity,1)-category]] of
[[simplicial presheaf|simplicial sheaves]] on $X$ and $H_{ab}$ for the
[[(infinity,1)-category]] of complexes of [[abelian sheaf|abelian sheaves]] on $X$.
Write $X$ for the terminal sheaf of $X$, i.e. for the [[sheaf]] that corresponds to the space $X$ itself.
Then
$$
H^n(X,A) := \pi_0 H(X,\mathbf{B}^n F)
$$
is the degree $n$ [[cohomology]] class of $X$ with values in $F$, regarded as computed in [[nonabelian cohomology]].
Now write $\mathbb{Z}[X]$ for the free abelianization of the sheaf $X$. This is the sheaf constant on the abelian group $\mathbb{Z}$ of integers. Then the above cohomology set, which of course happens to be a cohomology group here, due to the abelianness of $F$, is canonically isomorphic to the cohomology set
$$
\cdots \simeq \pi_0 H_{ab}(\mathbb{Z}[X], \mathbf{B}^n F)
$$
which can be regarded as the [[hom-set]] in the [[derived category]] of [[complex]]es of [[abelian sheaf|abelian sheaves]]. This, in turn, is the same as the traditional expression
$$
\cdots \simeq R^n \Gamma(X,F)
$$
giving the $n$th [[derived functor]] of the [[global section functor]] of the [[abelian sheaf]] $F$.
This, finally, is the same group as obtained by choosing any [[complex]] $I_F$ of [[abelian sheaf|abelian sheaves]] that is [[injective complex|injective]] and [[quasi-isomorphism|quasi-isomorphic]] to $F$ regarded as a complex concentrated in degree 0 and then computing the $n$ [[homology]] group of the complex $\Gamma(X,I_F)$ of global sections of $F$:
$$
\cdots \simeq H_n(\Gamma(X,I_F))
\,.
$$
Historically the development of abelian sheaf cohomology was precisely in reverse order to this derivation from the general $(\infty,1)$-[[(infinity,1)-category|categorical]] [[cohomology]].
+-- {: .num_theorem}
###### Theorem (K. Brown, 1973)
Let $X$ be a [[topological space]], $F$ a [[sheaf]] on (the [[category of open subsets]] of) $X$ with values in abelian groups, and $\mathbf{B}^n F = K(F,n)$ the image of the complex of abelian sheaves $F[n]$ ($F$ in degree $n$, trivial elsewhere) under the [[Dold-Kan correspondence]] in sheaves with values in [[Kan complex]]es
$$
\Gamma \circ (-) : Sh(X,Ch_+(Ab)) \to Sh(X, AbSimpGrp)
$$
$$
F[n] \mapsto K(F,n) =: \mathbf{B}^n F
\,.
$$
Then we have the following natural isomorphism of cohomologies:
$$
H^n(X,F) \simeq H(X, \mathbf{B}^n F)
$$
where
* on the left we have ordinary abelian sheaf cohomology defined as the right [[derived functor]] of the global sections functor
$$
H^n(X,F) := (R \Gamma_X)(F)
\,;
$$
* on the right we have [[nonabelian cohomology]], namely the hom-set in the [[homotopy category]] of Kan complex valued [[model structure on simplicial presheaves|simplicial sheaves]]
$$
H(X, \mathbf{B}^n F) := Ho_{Sh(X,\infty Grpd)}(X,\mathbf{B}^n X)
\,.
$$
=--
+-- {: .proof}
###### Proof
This is the first four steps in the proof of theorem 2 in [[BrownAHT]].
The proof proceeds along the following four steps, which we describe in more detail below:
$$
\begin{aligned}
H^n(X,F) & \simeq
Ho_{Sh(X,Ch(Ab))}(\mathbb{Z}, F[n])
\\
& \simeq
Ho_{Sh(X,Ch_+(Ab))}(\mathbb{Z}, F[n])
\\
& \simeq
Ho_{Sh(X,AbSimpGrp)}(\mathbb{Z}X, K(F,n))
\\
& \simeq
Ho_{Sh(X,\infty Grpd)}(X, K(F,n))
\end{aligned}
$$
1. By the [[derived functor]] definition of sheaf cohomology, $H^n(X,F)$ is the cohomology of any complex of sheaves $I^\bullet \in Sh(X,Ch(Ab))$ that is [[injective object|injective]] and weakly equivalent to $F[n]$, $F[n] \stackrel{\simeq}{\to} I^\bullet$:
$$
H^n(X,F) \simeq H^0(I^\bullet(X))
\,.
$$
On the other hand, by the general formula for hom-sets in [[homotopy category|homtotopy categories]] obtained by localizing at the [[calculus of fractions|multiplicative system]] given by [[quasi-isomorphism]]s of complexes (e.g. def. 13.1.2 in [[Categories and Sheaves|CaS]]) we have
$$
Ho_{Sh(X,Ch(Ab))}(\mathbb{Z}, F[n])
\simeq
colim_{Y^\bullet \stackrel{\simeq}{\to} \mathbb{Z}} Hom_{K(Sh(X,Ab))}(Y, I^\bullet)
\,.
$$
But due to the injectiveness of $I^\bullet$, the integrand on the right is constant (lemma 14.1.5 in [[Categories and Sheaves|CaS]]) and hence the colimit is isomorphic to $\cdots \simeq Hom_{K(Sh(X,Ab))}(\mathbb{Z}, I^\bullet) \simeq H^0(I^\bullet(X))$, as desired.
1. The second step uses that the inclusion functor
$$
Ho_{Sh(X,Ch_+(Ab))} \hookrightarrow Ho_{Sh(X,Ch(Ab))}
$$
is [[full and faithful functor|full and faithful]].
This in turn follows from
* first observing that the inclusion
$S : Sh(X,Ch_+(Ab)) \hookrightarrow Sh(X, Ch(Ab))$ of
chain complexes concentrated in non-negative degree
into all complexes of sheaves is
[[full and faithful functor|full and faithful]] and
has the obvious [[right adjoint]]
$T : Sh(X,Ch(Ab)) \to Sh(X, Ch_+(Ab))$ obtained by
**t**runcating a complex.
* By inspection, or else by the general properties
of [[adjoint functor]]s (see the list of properties
given there) this implies that $Id \to T \circ S$
is an [[isomorphism]]. This implies that also
$Id \to Ho T \circ Ho S$ is an [[isomorphism]].
* But by the adjoint functor lemma for homotopical
categories, $Ho S$ is also left adjoint to $Ho T$
(since both preserve weak equivalences). So that
once again with the general properties of
[[adjoint functor]]s it follows that
$Ho S$ is
[[full and faithful functor|full and faithful]].
1. The third step uses that the [[normalized chain complex]] functor $Sh(X,AbSimpGrp) \to Sh(X, Ch_+(Ab))$ is an [[equivalence of categories]] that preserves the respective weak equivalences and homotopies.
1. The fourth step finally uses that the [[stuff, structure, property|forgetful functor]] $Sh(X, SimpAbGrp) \to Sh(X, \infty Grpd)$ that only remembers the [[Kan complex]] underlying a [[simplicial group]] has a [[left adjoint]], the free abelian group functor $\mathbb{Z} : Sh(X,\infty Grpd) \to Sh(X, AbSimpGrp)$ (see [[Dold-Kan correspondence]] for details), and that preserves weak equivalences (see the discussion at [[simplicial group]] for more on that).
=--
### Relation to derived direct images
+-- {: .num_prop}
###### Proposition
Let $f^{-1} \colon Y \to X$ be a [[morphism of sites]].
Then the $q$th [[derived functor]] $R^q f_\ast$ of the induced [[direct image]] functor sends $\mathcal{F} \in Ab(Sh(X_{et}))$ to the [[sheafification]] of the [[presheaf]]
$$
U_Y
\mapsto
H^q(f^{-1}(U_Y), \mathcal{F})
\,,
$$
where on the right we have the degree $q$ [[abelian sheaf cohomology]] [[cohomology group|group]] with [[coefficients]] in the given $\mathcal{F}$.
=--
(e.g. [Tamme, I (3.7.1), II (1.3.4)](#Tamme), [Milne, 12.1](#Milne)).
+-- {: .proof}
###### Proof
We have a [[commuting diagram]]
$$
\array{
Ab(PSh(X)) &\stackrel{(-)\circ f^{-1}}{\longrightarrow}& Ab(PSh(Y))
\\
\uparrow^{\mathrlap{inc}} && \downarrow^{L}
\\
Ab(Sh(X)) &\stackrel{f_\ast}{\longrightarrow}& Ab(Sh(Y))
}
\,,
$$
where the right vertical morphism is [[sheafification]]. Because $(-) \circ f^{-1}$ and $L$ are both [[exact functors]] it follows that for $I^\bullet \to \mathcal{F}$ an [[injective resolution]] that
$$
\begin{aligned}
R^p f_\ast(\mathcal{F})
& :\simeq
H^p( f_\ast I)
\\
& = H^p(L I^\bullet(f^{-1}(-)))
\\
& = L (H^p(I^\bullet)(f^{-1}(-)))
\end{aligned}
$$
=--
## Examples
* [[de Rham cohomology]]
* [[Dolbeault cohomology]]
* [[Spencer cohomology]]
* [[Deligne cohomology]]
* [[etale cohomology]]
* [[crystalline cohomology]]
* [[syntomic cohomology]]
## Related concepts
* [[sheaf]], [[cohomology]]
* **abelian sheaf cohomology**
* [[model structure on chain complexes]]
* [[resolutions]]: [[soft sheaf]], [[fine sheaf]], [[flabby sheaf]]
* [[triangulated category of sheaves]]
* [[Verdier duality]]
## References
Textbook account:
* [[Jean Gallier]], [[Jocelyn Quaintance]], *Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry*, World Scientific (2022) [[doi:10.1142/12495](https://doi.org/10.1142/12495), [webpage](https://www.cis.upenn.edu/~jean/gbooks/sheaf-coho.html)]
Original texts:
* {#Godement58} [[Roger Godement]], *Topologie algébrique et theorie des faisceaux*, Hermann, Paris (1958) [[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), [[Godement-TopologieAlgebrique.pdf:file]]]
The traditional definition of sheaf cohomology in terms of the right [[derived functor]] of the [[global sections]] functor:
* [[Ugo Bruzzo]], _Derived Functors and Sheaf Cohomology_, Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes: Volume 2 ([doi:10.1142/11473](https://doi.org/10.1142/11473))
* Chênevert, Kassaei, _Sheaf Cohomology_ ([pdf](http://www.math.mcgill.ca/goren/SeminarOnCohomology/Sheaf_Cohomology.pdf))
* Cibotaru, _Sheaf cohomology_ ([pdf](http://www.nd.edu/~lnicolae/sheaves_coh.pdf))
* [[Patrick Morandi]], _Sheaf cohomology_ ([pdf](http://sierra.nmsu.edu/morandi/notes/SheafCohomology.pdf))
Its discussion in the more general [[nonabelian cohomology]] and [[infinity-stack]] context emphasized above is due to
* [[Kenneth Brown]], _[[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf Cohomology]]_
This uses homotopical structures of a [[category of fibrant objects]] on complexes of abelian sheaves. Discussion of actual [[model structure on chain complexes]] of abelian sheaves is in
* [[Mark Hovey]], _Model category structures on chain complexes of sheaves_, Trans. Amer. Math. Soc. 353, 6 ([pdf](http://www.mathaware.org/tran/2001-353-06/S0002-9947-01-02721-0/S0002-9947-01-02721-0.pdf))
A discussion of the [[Čech cohomology]] description of sheaf cohomology along the above lines is in
* [[Tibor Beke]], _Higher Čech Theory_ ([web](http://www.math.uiuc.edu/K-theory/0646/), [pdf](http://www.math.uiuc.edu/K-theory/0646/cech.pdf))
See also:
* {#Duskin79} [[John Duskin]], _Higher-dimensional torsors and the cohomology of topoi: the abelian theory_, p. 255-279 in: _Applications of sheaves_, Lecture Notes in Mathematics **753**, Springer (1979) [[doi:10.1007/BFb0061822](https://doi.org/10.1007/BFb0061822)]
* {#Tamme} [[Günter Tamme]], section II 1 of _[[Introduction to Étale Cohomology]]_
* {#Milne} [[James Milne]], section 7 of _[[Lectures on Étale Cohomology]]_
[[!redirects chain complex of sheaves]]
[[!redirects chain complexes of sheaves]]
[[!redirects sheaf of chain complexes]]
[[!redirects sheaves of chain complexes]]
[[!redirects abelian presheaf]]
[[!redirects abelian presheaves]] |
abelian subcategory | https://ncatlab.org/nlab/source/abelian+subcategory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Notions of subcategory
+-- {: .hide}
[[!include notions of subcategory]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
For $\mathcal{A}$ an [[abelian category]] an _abelian subcategory_ is a [[subcategory]] $\mathcal{B} \to \mathcal{A}$ such that $\mathbf{B}$ is also [[abelian category|abelian]] and such that every [[exact sequence]] in $\mathcal{B}$ is also exact in $\mathcal{A}$.
## References
Section 1.2 of
* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_
[[!redirects abelian subcategories]]
|
abelian variety | https://ncatlab.org/nlab/source/abelian+variety |
#Contents#
* table of contents
{:toc}
## Idea
Abelian varieties are higher dimensional analogues of [[elliptic curve]]s (which are included) -- they are [[varieties]] equipped with a structure of an [[abelian group]], hence abelian [[group schemes]], whose multiplication and inverse are regular maps.
## Definition
In his book _Abelian Varieties_, David Mumford defines an **abelian variety** over an algebraically closed field $k$ to be a [[complete algebraic variety|complete]] [[algebraic group]] over $k$. Remarkably, any such thing is an _abelian_ algebraic group. The assumption of connectedness is necessary for that conclusion.
## Automatic abelianness
David Mumford gives at two proofs that every complete algebraic group over an algebraically closed field is automatically abelian. One of them uses a 'rigidity lemma' which has an interesting category-theoretic interpretation. We outline this here:
In simple terms, the rigidity lemma says that under certain circumstances "a 2-variable function $f(x,y)$ that is independent of $x$ for one value of $y$ is independent of $x$ for all values of $y$."
More precisely, in a category with products, say a morphism $f: X \to Y$ is <b>constant</b> if it factors through the unique morphism $X \to 1$. Say a morphism $f : X \times Y \to Z$ is <b>independent of $X$</b> if it factors through the projection $X \times Y \to Y$.
Say a <b>point</b> of $Y$ is a morphism $p: 1 \to Y$. Say a morphism $f: X \times Y \to Z$ is <b>independent of $X$ at some point $p$ of $Y$</b> if $f \circ (1_X \times p) : X \to Z$ is constant.
**Definition.** A category with finite products **obeys the rigidity lemma** if any morphism $f: X \times Y \to Z$ that is independent of $X$ at some point of $Y$ is in fact independent of $X$.
**Theorem 1.** The category of complete algebraic varieties over an algebraically complete field $k$ has finite products and obeys the rigidity lemma.
+-- {: .proof}
######Proof
The hard part, the rigidity lemma, is proved for complete algebraic varieties on page 43 of Mumford's _Abelian Varieties_. Mumford mentions in a footnote that complete algebraic varieties are automatically irreducible, and he later seems to assume without much explanation that they are connected: these points could use some clarification, at least for amateurs.
=--
**Theorem 2.** Suppose $G,H$ are group objects in a category $C$ with finite products obeying the rigidity lemma. Suppose $f : G \to H$ is any morphism in $C$ preserving the identity. Then $f$ is a homomorphism.
+-- {: .proof}
######Proof
The idea is this: suppose $C$ is a [[concrete category]] and look at the function $k : G \times G \to H$ given by
$$k(g,g') = f(g\cdot g')\cdot (f(g) \cdot f(g'))^{-1} $$
Assume $f(1) = 1$. Then $k(1,g') = 1$ for all $g' \in G$, so by the rigidity lemma $k(g,g')$ is independent of $g'$ and we can write $k(g,g') = r(g)$. Furthermore $k(g,1) = 1$ for all $g \in G$ so $k(g,g')$ is independent of $g$. This means that $r(g)$ is independent of $g$, but $r(1) = k(1,1) = 1$ so $r(g) = 1$ for all $g$. This says that $f(g \cdot g') = f(g) \cdot f(g')$, so $f$ preserves multiplication. This in turn implies that $f$ preserves inverses, so $f$ is a group homomorphism.
In fact a version of this argument works in any category with finite products obeying the rigidity lemma. The expression $f(g\cdot g')\cdot (f(g) \cdot f(g'))^{-1}$ compiles to a particular morphism $G\times G \xrightarrow{k} H$. The fact that $f$ preserves the identity $e_G:1\to G$ implies that both composites $G \xrightarrow{(id,e_G)} G\times G \xrightarrow{k} H$ and $G \xrightarrow{(e_G,id)} G\times G \xrightarrow{k} H$ are constant at the identity $e_H:1\to H$. (This is a straightforward calculation in the internal logic of a category with products, or alternatively a slightly tedious diagram chase.)
In particular, $k$ is independent of the first $G$ in its domain at the point $e_G : 1\to G$ of the second $G$ in its domain. So by the rigidity lemma, there exists a morphism $r : G\to H$ such that the composite $G\times G \xrightarrow{\pi_2} G \xrightarrow{r} H$ is equal to $k$. Now precompose both of these with $G \xrightarrow{(e_G,id)} G\times G$: the first gives $r \circ \pi_2 \circ (e_G,id) = r$ and the second gives $k \circ (e_G,id) = e_H \circ !$. Thus, $r$ is constant at the identity of $H$, and hence so is $k$. This implies $f$ preserves multiplication, and thus also inverses (again, by a calculation in internal logic or a diagram chase).
=--
**Corollary 1.** If $C$ is a category with finite products obeying the rigidity lemma, any group object in $C$ is abelian.
+-- {: .proof}
######Proof
If $G$ is a group object in $C$, the inverse map $inv: G \to G$ preserves the identity, so by the above theorem it is a group homomorphism. This in turn implies that $G$ is abelian.
=--
**Corollary 2.** Let $Var_*$ be the category of pointed complete varieties over an algebraically closed field $k$, and let $AbVar$ be the category of abelian varieties over $k$. Then the forgetful functor $U : AbVar \to Var_*$ is full.
+-- {: .proof}
######Proof
This follows immediately from the two theorems above.
=--
A consequence of Corollary 2 is that if $Alb : Var_* \to AbVar$ is the left adjoint to $U$, sending any connected pointed projective variety to its Albanese variety, the monad $T = U \circ Alb$ is an [[idempotent monad]]. For more on this see [[Albanese variety]].
## Related concepts
* [[theta function]]
* [[Cartier duality]]
* [[dual abelian group scheme]]
* [[Albanese variety]]
## Literature
* C. Bartocci, [[Ugo Bruzzo]], D. Hernandez Ruiperez, _Fourier-Mukai and Nahm transforms in geometry and mathematical physics_, Progress in Mathematics 276, Birkhauser 2009.
* [[M. Demazure]], [[P. Gabriel]], _Groupes algebriques_, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970 – has functor of points point of view; for review see Bull. London Math. Soc. (1980) 12 (6): 476-478, [doi](http://dx.doi.org/10.1112/blms/12.6.476b)
* [[Daniel Huybrechts]], _Fourier-Mukai transforms in algebraic geometry_, Oxford Mathematical Monographs. 2006. 307 pages.
* J. S. Milne, _Abelian varieties_, course notes, [pdf](http://www.jmilne.org/math/CourseNotes/AV.pdf)
* David Mumford, _Abelian varieties_, Oxford Univ. Press 1970.
* [[Alexander Polishchuk]], _Abelian varieties, theta functions and the Fourier transform_, Cambridge Univ. Press 2003.
* Goro Shimura, _Abelian varieties with complex multiplication and modular functions_, Princeton Univ. Press 1997.
* [[André Weil]], _Courbes algébriques et variétés abéliennes_, Paris: Hermann 1971
In [[E-infinity geometry]]:
* {#LurieI} [[Jacob Lurie]], _Elliptic Cohomology I: Spectral Abelian Varieties_ ([pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-I.pdf))
For a discussion of how the rigidity lemma gives 'automatic abelianness' see:
* [Two miracles in algebraic geometry](https://golem.ph.utexas.edu/category/2016/08/the_magic_of_algebraic_geometr.html), _The n-Category Café_.
[[!redirects abelian varieties]]
[[!redirects abelian group scheme]]
[[!redirects abelian group schemes]]
|
abelianization | https://ncatlab.org/nlab/source/abelianization |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
Abelianisation is the process of [[free functor|freely]] making an [[algebraic structure]] 'abelian'. There are several notions of abelianizations for various algebraic structures, notably there is the abelianization of [[non-abelian groups]] to [[abelian groups]].
There is also [[Verdier's abelianization functor]] from a [[triangulated category]] to an [[abelian category]] with some universal property; this is treated in a separate entry.
## For groups
### Definition
+-- {: .num_defn }
###### Definition
For $G$ a [[group]], its **abelianization** $G^{ab} \in $ [[Grp]] is the [[quotient]] of $G$ by its [[commutator subgroup]]:
$$
G^{ab} \coloneqq G/[G,G]
\,,
$$
=--
A group whose abelianization is the [[trivial group]] is called a _[[perfect group]]_.
The abelianization is an [[abelian group]]. Indeed, it is the [[universal construction|universal]] abelian group induced by $G$, in the following sense:
+-- {: .num_prop }
###### Proposition
Abelianization extends to a [[functor]] $(-)^{ab} \colon $ [[Grp]] $\to$ [[Ab]] and this functor is [[left adjoint]] to the [[forgetful functor]] $U \colon Ab \to Grp$ from abelian groups to group.
=--
Hence abelianization is the _[[free construction]]_ of an abelian group from a group.
### Examples
#### Homotopy groups
+-- {: .num_example }
###### Example
Given a [[pointed space|pointed]] [[connected topological space]] $(X,a)$, its first [[singular homology|singular]] [[homology group]] with coefficients in the [[integers]] is the abelianisation of its [[fundamental group]]:
$$ H_1(X,\mathbb{Z}) \cong \pi_1(X,a)^{ab} .$$
This is a [[natural isomorphism]] filling the following diagram of [[functors]]:
$$ \array {
Top_{\geq 1}^{*/} & \overset{\pi_1}\longrightarrow & Grp \\
\llap{U}\downarrow & & \downarrow\rlap{ab} \\
Top & \underset{H_1({-},\mathbb{Z})}\longrightarrow & Ab Grp
} $$
(where $U$ [[forgetful functor|forgets]] the point).
This example can also be done starting with an arbitrary pointed topological space and letting $U$ take the [[connected component]] of the point. Of course, this example really lives in [[∞ Grpd]] and so we could start with a (pointed, maybe connected) [[simplicial set]], [[Kan complex]], etc.
=--
For more discussion of this see at _[[singular homology]]_ the section _[Relation to homotopy groups](singular%20homology#RelationToHomotopyGroups)_.
#### Free abelian groups
A [[free abelian group]] on a set $S$ is the abelianization of the [[free group]] on $S$.
In other words, if $F \colon Set \to Grp$ is the [[free group]]-functor and $F_{Ab} \colon Set \to Ab$ is the [[free abelian group]] functor, then
$$
\array{
Set &&\stackrel{F_{ab}}{\to}&& Ab
\\
& {}_{\mathllap{F_{grp}}}\searrow && \nearrow_{\mathrlap{(-)^{ab}}}
\\
&& Grp
}
$$
commutes up to a canonical isomorphism. This is because we have a corresponding commutative diagram of forgetful functors
$$
\array{
Set &&\stackrel{U_{ab}}{\leftarrow}&& Ab
\\
& {}_{\mathllap{U_{grp}}}\nwarrow && \swarrow_{\mathrlap{U}}
\\
&& Grp
}
$$
and $(-)^{ab} \circ F_{grp}$ is left adjoint to $U_{grp} \circ U$.
## For monoids etc
Abelianisation of monoids works pretty much like abelianisation of groups.
We can also do abelianisation of [[monoid objects]] in many [[monoidal categories]] (or [[closed categories]] or more generally [[multicategories]]). For example, we can form abelianisations of [[rings]], which are monoid objects in [[Ab]].
We can even form abelianisations of [[semigroups]] or [[magmas]].
## For Lie algebras
Lie algebras are not monoid objects in any category, but one still considers [[abelian Lie algebras]], which may be identified with their underlying [[vector spaces]]. These are so called because they correspond to abelian [[Lie groups]]. Lie algebras also can be abelianised.
[[!redirects abelianization]]
[[!redirects abelianizations]]
[[!redirects abelianisation]]
[[!redirects abelianisations]]
|
Abhijit Champanerkar | https://ncatlab.org/nlab/source/Abhijit+Champanerkar |
* [webpage](https://www.math.csi.cuny.edu/abhijit/)
## Selected writings
On [[knot theory]], [[knot complements]], etc.:
* [[Abhijit Champanerkar]], _The geometry of knot complements_ ([pdf](https://www.math.csi.cuny.edu/~abhijit/talks/knot-geometry-h.pdf), [[ChampanerkarKnotComplements.pdf:file]])
category: people
|
Abhijit Gadde | https://ncatlab.org/nlab/source/Abhijit+Gadde |
* [Institute page](http://qst.theory.tifr.res.in/StringTheoryStud/faculty1.php)
* [InSpire page](https://inspirehep.net/authors/1065230)
## Selected writings
On [[KK-compactification]] of [[D=6 N=(2,0) SCFT]] on [[4-manifolds]] to [[2d CFTs]]:
* [[Abhijit Gadde]], [[Sergei Gukov]], [[Pavel Putrov]], *Fivebranes and 4-manifolds*, in: *Arbeitstagung Bonn 2013*, Progress in Mathematics **319**, Birkhäuser (2016) [[arXiv:1306.4320](https://arxiv.org/abs/1306.4320), [doi:10.1007/978-3-319-43648-7_7](https://doi.org/10.1007/978-3-319-43648-7_7)]
On [[E-strings]] and emergence of [[SU(2)]] [[flavor (particle physics)|flavor]]-[[chiral symmetry|symmetry]] on [[M5-branes]] in [[heterotic M-theory]] (in the [[D=6 N=(1,0) SCFT]] on [[small instantons]] in [[heterotic string theory]]):
* {#GHKKLV15} [[Abhijit Gadde]], [[Babak Haghighat]], [[Joonho Kim]], [[Seok Kim]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _6d String Chains_, JHEP **1802** (2018) 143 [[arXiv:1504.04614](https://arxiv.org/abs/1504.04614)]
category: people
|
Abhishek Banerjee | https://ncatlab.org/nlab/source/Abhishek+Banerjee |
* [personal page](https://sites.google.com/site/abhishekb1313/)
* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=132991)
## Selected writings
Discussion of [[opposite categories]] of [[commutative monoids in a symmetric monoidal category]] regarded as categories of generalized [[affine schemes]]:
* [[Abhishek Banerjee]], *The relative Picard functor on schemes over a symmetric monoidal category*, Bulletin des Sciences Mathématiques **135** 4 (2011) 400-419 [[doi:10.1016/j.bulsci.2011.02.001](https://doi.org/10.1016/j.bulsci.2011.02.001)]
* [[Abhishek Banerjee]], *On integral schemes over symmetric monoidal categories* [[arXiv:1506.04890](https://arxiv.org/abs/1506.04890)]
* [[Abhishek Banerjee]], *Noetherian Schemes over abelian symmetric monoidal categories*, International Journal of Mathematics **28** 07 (2017) 1750051 [[doi:1410.3212](https://arxiv.org/abs/1410.3212), [doi:10.1142/S0129167X17500513](https://doi.org/10.1142/S0129167X17500513)]
category: people |
ABJM theory | https://ncatlab.org/nlab/source/ABJM+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _ABJM model_ ([ABJM 08](#ABJM08)) is an $\mathcal{N} = 6$ [[3d superconformal gauge field theory]] involving [[Chern-Simons theory]] with [[gauge group]] [[special unitary group|SU(N)]] and [[minimal coupling|coupled]] to [[matter fields]]. For [[level (Chern-Simons theory)|Chern-Simons level]] $k$ it is supposed to describe the [[worldvolume]] theory of $N$ coincident [[black brane|black]] [[M2-branes]] at an $\mathbb{Z}/k$-[[cyclic group]] [[orbifold]] [[singularity]] with [[near-horizon geometry]] $AdS_4 \times S^7/(\mathbb{Z}/k)$ (see at _[M2-branes -- As a black brane](M2-brane#AsABlackBrane)_).
[[!include superconformal symmetry -- table]]
For $k = 2$ the supersymmetry of the ABJM model increases to $\mathcal{N} = 8$. For $k = 2$ and $N = 2$ the ABJM model reduces to the [[BLG model]] ([ABJM 08, section 2.6](#ABJM08)).
Due to the matter coupling, the ABJM model is no longer a [[topological field theory]] as pure Chern-Simons is, but it is still a [[conformal field theory]]. As such it is thought to correspond under [[AdS-CFT]] duality to [[M-theory]] on [[anti-de Sitter spacetime|AdS4]] $\times$ [[7-sphere|S7]]/$\mathbb{Z}/k$ (see also [MFFGME 09](#MFFGME09)).
Notice that the worldvolume $SU(N)$ gauge group enhancement at an $\mathbb{Z}_k$-[[ADE singularity]] is akin to the [[enhanced gauge symmetry|gauge symmetry enhancement]] of the [[effective field theory]] for [[M-theory on G2-manifolds]] at the same kind of singularities (see at _[M-theory on G2-manifolds -- Nonabelian gauge groups](M-theory+on+G2-manifolds#EnhancedGaugeGroups)_).
More generally, classification of the [[near horizon geometry]] of smooth (i.e. non-[[orbifold]]) $\geq \tfrac{1}{2}$ [[BPS state|BPS]] [[black brane|black]] [[M2-brane]]-solutions of the [[equations of motion]] of [[11-dimensional supergravity]] shows that these are the [[Cartesian product]] $AdS_4 \times (S^7/G)$ of 4-[[dimension|dimensional]] [[anti de Sitter spacetime]] with a 7-[[dimension|dimensional]] [[spherical space form]] $S^7/{\widehat{G}}$ with [[spin structure]] and $N \geq 4$, for $\widehat{G}$ a [[finite subgroup of SU(2)]] ([MFFGME 09](#MFFGME09), see [here](spherical+space+form#7DSphericalSpaceFormsWithSpinStructure)).
[[!include 7d spherical space forms -- table]]
## Properties
### AdS/CFT duality
Under [[holographic duality]] supposed to be related to [[11-dimensional supergravity|M-theory]] on $AdS_4 \times S^7 / \mathbb{Z}_k$.
### Boundary conditions
Discussion of [[boundary conditions]] of the BLG model, leading to [[brane intersection]] with [[M-wave]], [[M5-brane]] and [[MO9-brane]] is in ([Chu-Smith 09](#ChuSmith09), [BPST 09](#BPST09)).
## Related concepts
* [[super Chern-Simons theory]]
* [[BLG model]]
* [[membrane matrix model]]
[[!include superconformal symmetry -- table]]
## References
### Precursors
Precursor considerations in
* [[John Schwarz]], _Superconformal Chern-Simons Theories_ ([arXiv:arXiv:hep-th/0411077](https://arxiv.org/abs/hep-th/0411077)
The lift of [[Dp-D(p+2)-brane bound states]] in [[string theory]] to [[M2-M5-brane bound states]]/[[E-strings]] in [[M-theory]], under [[duality between M-theory and type IIA string theory]]+[[T-duality]], via generalization of [[Nahm's equation]] (this eventually motivated the [[BLG-model]]/[[ABJM model]]):
* [[Anirban Basu]], [[Jeffrey Harvey]], _The M2-M5 Brane System and a Generalized Nahm's Equation_, Nucl.Phys. B713 (2005) 136-150 ([arXiv:hep-th/0412310](https://arxiv.org/abs/hep-th/0412310))
* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], Section 2.2.1 of _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))
This inspired the [[BLG model]]:
* {#BaggerLambert06} [[Jonathan Bagger]], [[Neil Lambert]], _Modeling Multiple M2's_, Phys. Rev. D75, 045020 (2007). ([hep-th/0611108](http://arxiv.org/abs/hep-th/0611108)).
* [[Jonathan Bagger]], [[Neil Lambert]], _Gauge Symmetry and Supersymmetry of Multiple M2-Branes_, Phys. Rev. D77, 065008 (2008). ([arXiv:0711.0955](http://arXiv.org/abs/0711.0955)).
### General
The original article on the $N=6$-case is
* {#ABJM08} [[Ofer Aharony]], [[Oren Bergman]], [[Daniel Jafferis]], [[Juan Maldacena]], _$N=6$ superconformal Chern-Simons-matter theories, M2-branes and their gravity duals_, JHEP 0810:091,2008, [DOI:10.1088/1126-6708/2008/10/091](http://iopscience.iop.org/article/10.1088/1126-6708/2008/10/091/meta;jsessionid=FCE6764D4E19F3038C9530E50B057A56.c3.iopscience.cld.iop.org) ([arXiv:0806.1218](http://arxiv.org/abs/0806.1218))
and for [[discrete torsion]] in the [[supergravity C-field]] in
* [[Ofer Aharony]], [[Oren Bergman]], [[Daniel Jafferis]], _Fractional M2-branes_, JHEP 0811:043, 2008 ([arXiv:0807.4924](https://arxiv.org/abs/0807.4924))
(on [[fractional M2-brane|fractional M2-branes]])
inspired by the $N=8$-case of the [[BLG model]] ([Bagger-Lambert 06](#BaggerLambert06))
The $N=5$-case is discussed in
* {#HLLLP08} Kazuo Hosomichi, Ki-Myeong Lee, Sangmin Lee, Sungjay Lee, [[Jaemo Park]], _$\mathcal{N}=5,6$ Superconformal Chern-Simons Theories and M2-branes on Orbifolds_, JHEP 0809:002, 2008 ([arXiv:0806.4977](https://arxiv.org/abs/0806.4977))
* {#BHRSS08} [[Eric Bergshoeff]], [[Olaf Hohm]], Diederik Roest, [[Henning Samtleben]], [[Ergin Sezgin]], _The Superconformal Gaugings in Three Dimensions_, JHEP0809:101, 2008 ([arXiv:0807.2841](https://arxiv.org/abs/0807.2841))
* {#AharonyBergmanJafferis08} [[Ofer Aharony]], [[Oren Bergman]], [[Daniel Jafferis]], _Fractional M2-branes_, JHEP 0811:043, 2008 ([arXiv:0807.4924](https://arxiv.org/abs/0807.4924))
The $N=4$-case is discussed in
* {#HLLLP08b} Kazuo Hosomichi, Ki-Myeong Lee, Sangmin Lee, Sungjay Lee, [[Jaemo Park]], _\mathcal{N}=4 Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets_, JHEP 0807:091,2008 ([arXiv:0805.3662](https://arxiv.org/abs/0805.3662))
* {#ChenWu10} Fa-Min Chen, Yong-Shi Wu, _Superspace Formulation in a Three-Algebra Approach to D=3, N=4,5 Superconformal Chern-Simons Matter Theories_, Phys.Rev.D82:106012, 2010 ([arXiv:1007.5157](https://arxiv.org/abs/1007.5157))
More on the role of [[discrete torsion]] in the [[supergravity C-field]] is in
* [[Mauricio Romo]], _Aspects of ABJM orbifolds with discrete torsion_, J. High Energ. Phys. (2011) 2011 ([arXiv:1011.4733](https://arxiv.org/abs/1011.4733))
Discussion of [[boundary conditions]] leading to [[brane intersection]] laws with the [[M-wave]], [[black brane|black]] [[M5-brane]] and [[MO9]] is in
* {#ChuSmith09} [[Chong-Sun Chu]], Douglas J. Smith, _Multiple Self-Dual Strings on M5-Branes_, JHEP 1001:001, 2010 ([arXiv:0909.2333](https://arxiv.org/abs/0909.2333))
* {#BPST09} [[David Berman]], [[Malcolm J. Perry]], [[Ergin Sezgin]], [[Daniel C. Thompson]], *Boundary Conditions for Interacting Membranes*, JHEP 1004:025, 2010 [[arXiv:0912.3504](https://arxiv.org/abs/0912.3504), <a href="https://doi.org/10.1007/JHEP04(2010)025">doi:10.1007/JHEP04(2010)025</a>]
As a [[matrix model]],:
* {#MohammedMuruganNastase10} Asadig Mohammed, Jeff Murugan, [[Horatiu Nastase]], _Looking for a Matrix model of ABJM_, Phys. Rev. D82:086004, 2010 ([arXiv:1003.2599](https://arxiv.org/abs/1003.2599))
Review includes
* {#KlebanovTorri10} [[Igor Klebanov]], Giuseppe Torri, _M2-branes and AdS/CFT_, Int.J.Mod.Phys.A25:332-350, 2010 ([arXiv;0909.1580](https://arxiv.org/abs/0909.1580))
* Neil B. Copland, _Introductory Lectures on Multiple Membranes_ ([arXiv:1012.0459](https://arxiv.org/abs/1012.0459))
* {#Lambert12} [[Neil Lambert]], _M-Theory and Maximally Supersymmetric Gauge Theories_, Annual Review of Nuclear and Particle Science, Vol. 62:285-313 ([arXiv:1203.4244](https://arxiv.org/abs/1203.4244), [doi:10.1146/annurev-nucl-102010-130248](https://doi.org/10.1146/annurev-nucl-102010-130248))
* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 2013 ([arXiv:1203.3546](https://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))
* {#Lambert18} [[Neil Lambert]], _Lessons from M2's and Hopes for M5's_, _Proceedings of the [LMS-EPSRC Durham Symposium](http://www.maths.dur.ac.uk/lms/):_ _[[Higher Structures in M-Theory 2018]]_ Fortschritte der Physik, 2019 ([arXiv:1903.02825](https://arxiv.org/abs/1903.02825), [slides pdf](http://www.maths.dur.ac.uk/lms/109/talks/1877lambert.pdf), [video recording](http://www.maths.dur.ac.uk/lms/109/movies/1877lamb.mp4))
Discussion of [[Montonen-Olive duality]] in [[D=4 super Yang-Mills theory]] via [[ABJM-model]] as [[D3-brane]] model after [[double dimensional reduction]] followed by [[T-duality]]:
* [[Koji Hashimoto]], Ta-Sheng Tai, [[Seiji Terashima]], _Toward a Proof of Montonen-Olive Duality via Multiple M2-branes_, JHEP 0904:025, 2009 ([arxiv:0809.2137](https://arxiv.org/abs/0809.2137))
Discussion of extension to [[boundary field theory]] (describing [[M2-branes]] ending on [[M5-branes]]) includes
* {#BermanThomson09} [[David Berman]], Daniel Thompson, _Membranes with a boundary_, Nucl.Phys.B820:503-533,2009 ([arXiv:0904.0241](http://arxiv.org/abs/0904.0241))
A kind of [[double dimensional reduction]] of the ABJM model to something related to [[type II superstrings]] and [[D1-branes]] is discussed in
* [[Horatiu Nastase]], Constantinos Papageorgakis, _Dimensional reduction of the ABJM model_, JHEP 1103:094,2011 ([arXiv:1010.3808](http://arxiv.org/abs/1010.3808))
Discussion of the ABJM model in [[Horava-Witten theory]] and reducing to [[heterotic strings]] is in
* {#Lambert15} [[Neil Lambert]], _Heterotic M2-branes_, Physics Letters B Volume 749, 7 October 2015, Pages 363-367 ([arXiv:1507.07931](http://arxiv.org/abs/1507.07931))
Discussion of the model as a [[higher gauge theory]] (due to its coupling to the [[supergravity C-field]]) is in
* Sam Palmer, [[Christian Saemann]], section 2 of _M-brane Models from Non-Abelian Gerbes_, JHEP 1207:010, 2012 ([arXiv:1203.5757](http://arxiv.org/abs/1203.5757))
* Sam Palmer, [[Christian Saemann]], _The ABJM Model is a Higher Gauge Theory_, IJGMMP 11 (2014) 1450075 ([arXiv:1311.1997](http://arxiv.org/abs/1311.1997))
Classification of the possible [[superpotentials]] via [[representation theory]] is due to
* {#MFFME09} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], _Superpotentials for superconformal Chern-Simons theories from representation theory_, J. Phys. A 42:485204,2009 ([arXiv:0908.2125](http://arxiv.org/abs/0908.2125))
and derived from this a classification of the possible [[orbifold|orbifolding]] (see at _[[spherical space form]]: [7d with spin structure](spherical+space+form#7DSphericalSpaceFormsWithSpinStructure)_) is in
* {#MFFGME09} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Sunil Gadhia]], [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))
* Paul de Medeiros, [[José Figueroa-O'Farrill]], _Half-BPS M2-brane orbifolds_, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761), [eujclid:atmp/1408561553](https://projecteuclid.org/euclid.atmp/1408561553))
* [[José Figueroa-O'Farrill]], _M2-branes, ADE and Lie superalgebras_, talk at IPMU 2009 ([pdf](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/Hongo.pdf))
Discussion via the [[conformal bootstrap]]:
* Nathan B. Agmon, [[Shai Chester]], Silviu S. Pufu, _The M-theory Archipelago_ ([arXiv:1907.13222](https://arxiv.org/abs/1907.13222))
* Damon J. Binder, [[Shai Chester]], Max Jerdee, Silviu S. Pufu, _The 3d $\mathcal{N}=6$ Bootstrap: From Higher Spins to Strings to Membranes_ ([arXiv:2011.05728](https://arxiv.org/abs/2011.05728))
See also
* Nadav Drukker, [[Marcos Marino]], Pavel Putrov, _From weak to strong coupling in ABJM theory_ ([arXiv:1007.3837](http://arxiv.org/abs/1007.3837))
* [[Shai Chester]], Silviu S. Pufu, Xi Yin, _The M-Theory S-Matrix From ABJM: Beyond 11D Supergravity_ ([arXiv:1804.00949](https://arxiv.org/abs/1804.00949))
Computation of [[black hole entropy]] in 4d via [[AdS4-CFT3 duality]] from [[holographic entanglement entropy]] in the ABJM theory for the [[M2-brane]] is discussed in
* Jun Nian, Xinyu Zhang, _Entanglement Entropy of ABJM Theory and Entropy of Topological Black Hole_ ([arXiv:1705.01896](https://arxiv.org/abs/1705.01896))
Discussion of [[higher curvature corrections]] in the abelian case:
* Shin Sasaki, _On Non-linear Action for Gauged M2-brane_, JHEP 1002:039, 2010 ([arxiv:0912.0903](https://arxiv.org/abs/0912.0903))
### Mass deformation
The [[Myers effect]] in [[M-theory]] for [[M2-branes]] polarizing into [[M5-branes]] of ([[fuzzy 3-sphere|fuzzy]]) [[3-sphere]]-shape ([[M2-M5 brane bound states]]):
* [[Iosif Bena]], _The M-theory dual of a 3 dimensional theory with reduced supersymmetry_, Phys. Rev. D62:126006, 2000 ([arXiv:hep-th/0004142](https://arxiv.org/abs/hep-th/0004142))
* Masato Arai, [[Claus Montonen]], Shin Sasaki, _Vortices, Q-balls and Domain Walls on Dielectric M2-branes_, JHEP 0903:119, 2009 ([arXiv:0812.4437](https://arxiv.org/abs/0812.4437))
* [[Iosif Bena]], [[Mariana Graña]], Stanislav Kuperstein, Stefano Massai, _Tachyonic Anti-M2 Branes_, JHEP 1406:173, 2014 ([arXiv:1402.2294](https://arxiv.org/abs/1402.2294))
With emphasis on the role of the [[Page charge]]/[[Hopf WZ term]]:
* [[Krzysztof Pilch]], Alexander Tyukov, [[Nicholas Warner]], _Flowing to Higher Dimensions: A New Strongly-Coupled Phase on M2 Branes_, JHEP11 (2015) 170 ([arXiv:1506.01045](https://arxiv.org/abs/1506.01045))
Via the [[mass-deformed ABJM model]]:
* [[Jaume Gomis]], Diego Rodriguez-Gomez, [[Mark Van Raamsdonk]], [[Herman Verlinde]], _A Massive Study of M2-brane Proposals_, JHEP 0809:113, 2008 ([arXiv:0807.1074](https://arxiv.org/abs/0807.1074))
* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], Section 6.4 of: _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))
The corresponding D2-NS5 bound state under [[duality between M-theory and type IIA string theory]]:
* [[Iosif Bena]], Aleksey Nudelman, _Warping and vacua of $(S)YM_{3+1}$_, Phys. Rev. D62 (2000) 086008 ([arXiv:hep-th/0005163](https://arxiv.org/abs/hep-th/0005163))
On [[Wilson line]]-[[quantum observables]] and [[bosonization]] in the ABJM model:
* [[Amit Sever]], _Line Operators in Chern-Simons-Matter Theories and Bosonization in Three Dimensions_, talk at *[[Strings 2022]]* [[indico:4940843](https://indico.cern.ch/event/1085701/contributions/4940843), [slides](https://indico.cern.ch/event/1085701/contributions/4940843/attachments/2482058/4261226/Slides_Sever.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=a9f7872c-1603-4954-9030-192a72e4f456)]
[[!redirects ABJM model]]
[[!redirects ABJM-model]]
[[!redirects D=3 SCFT]]
[[!redirects mass-deformed ABJM model]]
|
AbMon > history | https://ncatlab.org/nlab/source/AbMon+%3E+history | [[!redirects AbMon]]
The [[category]] of commutative (abelian) [[monoids]].
category: category
[[!redirects Ab Mon]]
|
Abner Shimony | https://ncatlab.org/nlab/source/Abner+Shimony |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Abner_Shimony)
* Alisa Bokulich and Don Howard: *[Abner Shimony (1928-2015)](https://www.bu.edu/cphs/about/abner-shimony/)*
## Selected writings
On [[Bell's theorem]] in [[quantum physics]]:
* [[John F. Clauser]], [[Abner Shimony]], *Bell's theorem. Experimental tests and implications*, Rep. Prog. Phys. **41** (1978) 1881 [[doi:10.1088/0034-4885/41/12/002](https://iopscience.iop.org/article/10.1088/0034-4885/41/12/002)]
category: people |
About | https://ncatlab.org/nlab/source/About |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Contents### {: .clickToReveal}
###Contents### {: .clickToHide tabindex="0"}
+--{: .hide}
[[!include contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## History
{#History}
The [[HomePage|nLab]] is a _collaborative [wiki](http://en.wikipedia.org/wiki/Wiki)_. It grew out of the desire ([I](http://golem.ph.utexas.edu/category/2007/09/towards_a_higherdimensional_wi.html), [II](http://golem.ph.utexas.edu/category/2008/11/beyond_the_blog.html)) to have a place for development (the "Lab" in "$n$Lab") and indexed archives of the ideas and concepts surrounding the discussions at the [The $n$-Category Café](http://golem.ph.utexas.edu/category/). These discussions primarily are about [[mathematics]], [[physics]] and [[philosophy]] from the [[higher structures]] [[n-point of view|point of view]] of [[homotopy theory]]/[[algebraic topology]], [[homotopy type theory|homotopy]] [[type theory]], [[higher category theory|higher]] [[category theory]] and [[higher algebra|higher]] [[categorical algebra]] (the "$n$" in "$n$Lab" and "[[nPOV]]"), as well as from the perspective of [[string theory]] for physics related articles.
The $n$Lab was [created](http://golem.ph.utexas.edu/category/2008/11/nlab.html) in November 2008 by [[Urs Schreiber]], using software provided and set up by [[Jacques Distler]]. The name "$n$Lab" (short for "$n$-Category Lab", but perhaps less restrictive) was dreamt up by Lisa Raphals ([[John Baez]]'s wife).
Discussion of and around edits to the $n$Lab happen on the [$n$Forum](https://nforum.ncatlab.org/) which roughly plays the role of the "talk"-pages at Wikipedia.
## What the $n$Lab is
The most apt analogy for the $n$Lab is of a **group lab book**. A lab book for a research scientist is a place where they write down anything that they consider relevant for their work; a group lab book is one that several researchers with common interests use. The material recorded in a lab book can include:
* Notes from seminars
* Notes from papers and books
* Summaries of known work
* Observations and results from experiments
* Ideas for future work
Apart from not having actual physical experiments, this matches the content of the $n$Lab fairly well. The key difference between the $n$Lab and an "ordinary" lab book is that it is public. By making it public we hope to achieve two things:
1. To enable others to benefit from our work while it is still being done.
2. To benefit from the work of others while we are doing it.
There is much more to a research project than that which appears in the final paper and we believe that since our results and our final proofs will be public, then both we and those who might read such a paper can benefit from beginning the interaction much earlier in its development.
Thus by making our lab book public, we hope that casual passers-by will stop, read, and scribble something on it that will help us do our research.
There are three things to say about this that are worth making very clear.
1. Our reasons for having the $n$Lab are ultimately selfish. We have set it up and run it to make it easier for us to do our research.
2. But to make it _work_ properly in public, we need to make it attractive to others so that they will stop by, read our jottings, and scribble something of their own.
3. We aim to achieve a system whereby those who put most in are also those who get most out.
For more recent impressions of the nature of this wiki see also at: *[[schreiber:What is... the nLab]]*.
## Who Are We?
The $n$Lab is not a community project set up for the education and improvement of Humanity. It originally grew out of the [The $n$-Category Café](http://golem.ph.utexas.edu/category/) and the loose-knit community surrounding that blog. It is not possible to give a precise description of who is in that community, and the community involved in the $n$Lab has itself evolved from that starting point. Moreover, even if it were possible to describe that community we would not wish to. We are not a closed group, we are always keen for others to join us. The two simple rules are:
1. To join, you need to be willing to interact with the group and material already present, from the [[n-point of view]] or the [[string theory]] point of view for material on [[physics]]. Specifically: is there something already on the $n$Lab that you are interested in and is there someone already working in the $n$Lab that you would like to talk to about it?
2. We will ask you to leave if what you are doing has little connection with what the rest of the group is doing and if it is interfering with others trying to do their work. This is of course unpleasant, since most people who come to contribute do so with honorable intentions. To help minimize possible areas of friction, it may help to read [[writing in the nLab]].
##In Practice?
The way that this is being worked out in practice is as follows. Currently it seems that common motivations for contributing to the $n$Lab are
* to **assemble information** on [[mathematics]], [[physics]] and [[philosophy]] in a modern unified way, from the perspective of [[higher algebra]], [[homotopy theory]], [[type theory]], [[category theory]], and [[higher category theory]], as well as from the perspective of [[string theory]] for physics related articles
* to **provide exposition** of this information which is useful to a wide range of readers of differing expertise;
* to jointly **develop ideas** and research on applying higher algebraic, homotopical, type theoretic, categorical, and higher categorical concepts and tools to issues in [[mathematics]], [[physics]] and [[philosophy]], and [[string theory]] concepts and tools to issues in [[physics]].
To some extent this involves
* collecting in principle "well known" definitions and facts in an encyclopaedic fashion, to the extent that these are not readily available, or not in the desired form, for instance on [Wikipedia](http://en.wikipedia.org/wiki/Main_Page). Depending on the enthusiasm of those who decide to contribute, this _could_ eventually develop into something like a modern version of a [Bourbaki](http://en.wikipedia.org/wiki/Bourbaki)-like project. The potential for this is conceivably there, but of course this is an ambitious idea. Time will show if the $n$Lab can live up to this goal.
But on top of this encyclopaedic function, and hopefully in parallel to that, the $n$Lab is
* intended as providing a _laboratory_ for collaborative development of ideas -- for research. Within the community out of which the $n$Lab grew exists the feeling that there is considerable potential for the fruitful application and development of higher algebraic, homotopical, type theoretic, categorical, and higher categorical concepts and tools to various areas in [[mathematics]], [[physics]] and [[philosophy]], as well as the fruitful application and development of [[string theory]] concepts and tools to various areas in [[physics]]. Several contributors to the $n$Lab are actively involved in research along these lines.
The $n$Lab is meant as a place to collect, develop and present such research.
These points together imply that
* on the $n$Lab we do not hesitate to provide non-traditional perspectives, definitions and explanations of terms and phenomena for mathematics, physics, and philosophy, if we feel that these are the _right_ perspectives, definitions and explanations from a modern unified higher algebraic, homotopical, type theoretic, categorical, and higher categorical perspective, and we also do not hesitate to provide non-traditional perspectives, definitions and explanations of terms and phenomena for physics if we feel that these are the _right_ perspectives, definitions and explanations from a string theory perspective;
* at the same time we want to indicate clearly which part of an entry is traditional common material, which is a modern but widely-accepted reformulation and which is genuinely the result of original research by a contributor or by several contributors;
* we will intersperse possibly controversial but always constructive **discussion** into entries if we feel the need. While in general the [nforum](https://nforum.ncatlab.org/) is where we have chat and discussion and the $n$Lab is where we compile material, few entries on the $n$Lab are or can be meant as representing a final truth. The $n$Lab will be the better the more we discuss its contents. For more on how to inject discussion into entries see the [[HowTo]] page.
##What the $n$Lab is not
Most importantly the $n$Lab is
* **not complete** and **not meant to be complete**. Neither its general structure nor each single entry are meant to be optimal in their current state. Many existing entries, possibly all of them, deserve to be and are meant to be eventually improved and expanded on.
Many entries are [stubs](http://en.wikipedia.org/wiki/Wikipedia:Stub)!
**If you find yourself annoyed by the state any given entry is in**, for whatever reason, please feel encouraged to edit it in order to improve the situation.
**Notice**: an entry being in a pitiful state is usually more a sign of nobody having spared the time and energy to work on it, than of our joint incompetence to write a decent entry if we were being paid for doing it. So if you find your eyebrows raised by some entry, don't turn away to be the next one _not_ to work on it. Instead, improve it. We all do this voluntarily. Most of us have other duties to attend. So don't be annoyed with "us", help us.
So if you feel existing material needs to be changed, you can do so. If you feel further material needs to be added, different perspectives need to be amplified, you can add a new paragraph, headed by a suitable headline. **Be bold**: The $n$Lab will be the better the more people decide to contribute to it.
See the [[HowTo]] page for information on how to contribute to the $n$Lab and edit and/or create its entries.
##Organization of the $n$Lab
* see [[organization of the nLab]]
## Technology and Support
Historically, the nlab was powered by [Instiki](https://golem.ph.utexas.edu/wiki/instiki/show/HomePage), but now runs on a [custom fork](https://github.com/ncatlab/nlab). All bug reports or other software issues/requests for the nLab are currently best raised in the category [nLab Technical Matters](https://nforum.ncatlab.org/21/) at the nForum, but can also be posted [at github](https://github.com/ncatlab/nlab/issues).
The physical server running the nLab lives at [Carnegie Mellon University](http://www.cmu.edu), and is supported by the [[homotopy type theory]] MURI award FA9550-15-1-0053 from the United States Air Force Office of Scientific Research. Any opinions, findings and conclusions or recommendations expressed on the nLab are those of the authors and do not necessarily reflect the views of the AFOSR.
category: meta |
Abraham Adrian Albert | https://ncatlab.org/nlab/source/Abraham+Adrian+Albert |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Abraham_Adrian_Albert)
## related $n$Lab entries
* [[Albert algebra]]
category:people
[[!redirects Abraham Albert]]
[[!redirects Adrian Albert]] |
Abraham Fraenkel | https://ncatlab.org/nlab/source/Abraham+Fraenkel | Abraham Fraenkel was a mathematician who worked on [[set theory]] and [[foundations]]. He is responsible for the 'F' in [[ZFC]], but also did work using [[ZFA]].
Among other things, he also introduced the [[basic Fraenkel model|basic]] and [[second Fraenkel model|second Fraenkel models]], early examples of [[forcing]] using what is now known as a [[permutation model]].
[[!redirects Fraenkel]]
category:people |
Abraham Robinson | https://ncatlab.org/nlab/source/Abraham+Robinson | Abraham Robinson was a German born mathematician, who studied in Jerusalem, before taking up various positions in France then in Cranfield (the staff college of the RAF in the UK, during the second world war), Israel, Canada and finally in the USA at UCLA and Yale. He is the 'father' of Nonstandard Analysis, the first full formalisation of the intuitive infinitesimal calculus, used historically in mathematics, and still in use in physics. He died of cancer in 1974.
* [Abraham Robinson in Wikipedia](http://en.wikipedia.org/wiki/Abraham_Robinson)
* [Abraham Robinson's biography](http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Robinson.html) on the St. Andrews History website.
category: people |
Abraham Westerbaan | https://ncatlab.org/nlab/source/Abraham+Westerbaan |
* [personal page](https://bram.westerbaan.name/home/about.html)
* [GoogleScholar page](https://scholar.google.com/citations?user=6Hz9H7sAAAAJ&hl=en)
## Selected writings
On [[categorical semantics]] of [[quantum programming languages]] on [[von Neumann algebras]] for the [[quantum lambda-calculus]]:
* [[Kenta Cho]], [[Abraham Westerbaan]], *Von Neumann Algebras form a Model for the Quantum Lambda Calculus*, QPL 2016 [[arXiv:1603.02133](https://arxiv.org/abs/1603.02133), [pdf](https://www.cs.ru.nl/K.Cho/papers/model-qlc.pdf), slides:[pdf](http://qpl2016.cis.strath.ac.uk/pdfs/2cho-final.pdf), [[Cho-vNQuantumCalculus.pdf:file]]]
category: people
|
abrupt category | https://ncatlab.org/nlab/source/abrupt+category | Each [[category with star-morphisms]] gives rise to a category (<b>abrupt
category</b>, see a remark below why I call it "abrupt"), as described below.
Below for simplicity I assume that the set $M$ and the set of our indexed
families of functions are disjoint. The general case (when they are not
necessarily disjoint) may be easily elaborated by the reader.
% Objects are indexed (by $\operatorname{arity}m$ for some
$m \in M$) families of objects of the category $C$ and an (arbitrarily
choosen) object $\operatorname{None}$ not in this set
% There are the following disjoint sets of morphisms:
* indexed (by $\operatorname{arity} m$ for some $m \in M$) families of morphisms of $C$
* elements of $M$
* the identity morphism $\operatorname{id}_{\operatorname{None}}$ on $\operatorname{None}$
% Source and destination of morphims are defined by the formulas:
* $\operatorname{Src}f = \lambda i \in
\operatorname{dom}f : \operatorname{Src}f_i$;
* $\operatorname{Dst}f = \lambda i \in
\operatorname{dom}f : \operatorname{Dst}f_i$
* $\operatorname{Src}m =\operatorname{None}$
* $\operatorname{Dst}m
=\operatorname{Obj}_m$.
% Compositions of morphisms are defined by the formulas:
* $g \circ f = \lambda i \in \operatorname{dom}f : g_i
\circ f_i$
* $f \circ m =\operatorname{StarProd} \left( m ; f
\right)$
* $m \circ \operatorname{id}_{\operatorname{None}} = m$
* $\operatorname{id}_{\operatorname{None}} \circ \operatorname{id}_{\operatorname{None}} = \operatorname{id}_{\operatorname{None}}$
% Identity morphisms for an object $X$ are:
* $\lambda i \in X : \operatorname{id}_{X_i}$ if $X \neq
\operatorname{None}$
* $\operatorname{id}_{\operatorname{None}}$
if $X =\operatorname{None}$
We need to prove it is really a category.
**Proof** We need to prove:
* Composition is associative
* Composition with identities complies with the identity law.
Really:
* $\left( h \circ g \right) \circ f = \lambda i \in \operatorname{dom} f :
\left( h_i \circ g_i \right) \circ f_i = \lambda i \in \operatorname{dom} f : h_i
\circ \left( g_i \circ f_i \right) = h \circ \left( g \circ f \right)$;
$g \circ \left( f \circ m \right) = \operatorname{StarComp} \left( \operatorname{StarComp}
\left( m ; f \right) ; g \right) = \operatorname{StarComp} \left( m ; \lambda i \in
\operatorname{arity} m : g_i \circ f_i \right) = \operatorname{StarComp} \left( m ; g \circ
f \right) = \left( g \circ f \right) \circ m$;
$f \circ \left( m \circ \operatorname{id}_{\operatorname{None}} \right) = f \circ m = \left(
f \circ m \right) \circ \operatorname{id}_{\operatorname{None}}$.
* $m \circ \operatorname{id}_{\operatorname{None}} = m$; $\operatorname{id}_{\operatorname{Dst} m} \circ
m = \operatorname{StarComp} \left( m ; \lambda i \in \operatorname{arity} m :
\operatorname{id}_{\operatorname{Obj}_m i} \right) = m$.
**Remark** I call the above defined category <b>abrupt category</b> because (excluding identity morphisms) it allows composition with an $m \in M$ only on the left (not on the right) so that the morphism $m$ is "abrupt" on the right.
[[category with star-morphisms|Categories with star-morphisms]] and **abrupt categories** arise in research of [[cross-composition product]]. |
absolute coequalizer | https://ncatlab.org/nlab/source/absolute+coequalizer |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
An **absolute coequalizer** in a [[category]] $C$ is a [[coequalizer]] which is preserved by *any* [[functor]] $F\colon C \to D$. This is a special case of an [[absolute colimit]].
## Characterization
Intuitively, an absolute coequalizer is a diagram that is a coequalizer "purely for diagrammatic reasons." The most common example is a [[split coequalizer]]. A trivial example of an absolute coequalizer that is not split is a diagram of the form
$$ X \; \underoverset{f}{f}{\rightrightarrows}\; Y \overset{1_Y}{\to} Y $$
whenever $f$ is not a [[split epimorphism]].
In fact, split coequalizers and "trivial" absolute coequalizers are the cases $n=1$ and $0$ of a general characterization of absolute coequalizers, which we now describe. Suppose that
$$ X\; \underoverset{f_1}{f_0}{\rightrightarrows}\; Y \overset{e}{\to} Z $$
is an absolute coequalizer. Then it must be preserved, in particular, by the hom-functor $hom(Z,-)\colon C \to Set$; that is, we have a coequalizer diagram
$$ hom(Z,X)\; \underoverset{f_1\circ -}{f_0\circ -}{\rightrightarrows}\; hom(Z,Y) \overset{e\circ -}{\to} hom(Z,Z)$$
in $Set$. In particular, that means that $e\circ -$ is surjective, and so in particular there exists some $s\colon Z\to Y$ such that $e s = 1_Z$. In other words, $e$ is [[split epimorphism|split epic]].
Now the given coequalizer must also be preserved by the hom-functor $hom(Y,-)$, so we have another coequalizer diagram
$$ hom(Y,X)\; \underoverset{f_1\circ -}{f_0\circ -}{\rightrightarrows}\; hom(Y,Y) \overset{e\circ -}{\to} hom(Y,Z)$$
in $Set$. We also have two elements $1_Y$ and $s e$ in $hom(Y,Y)$ with the property that $e \circ 1_Y = e = e \circ s e$ (since $e s = 1_Z$).
However, a coequalizer of two functions $h_0,h_1\colon P\to Q$ in $Set$ is constructed as the [[quotient set]] of $Q$ by the [[equivalence relation]] generated by the image of $(h_0,h_1)\colon P\to Q\times Q$. That means that we set
$q\sim q'$ iff there is a finite sequence $p_1,\dots, p_n$ of elements of $P$ and a finite sequence $\varepsilon_0,\dots,\varepsilon_n$ with $\varepsilon_i\in\{0,1\}$, such that $h_{\varepsilon_1}(p_1)=q$,
$h_{1-\varepsilon_i}(p_i)=h_{\varepsilon_{i+1}}(p_{i+1})$, and $h_{1-\varepsilon_n}(p_n)=q'$. We consider $q=q'$ as the case $n=0$.
Therefore, since $1_Y$ and $s e$ are in the same class of the equivalence relation on $hom(Y,Y)$ generated by $f_0$ and $f_1$, they must be related by such a finite chain of elements of $hom(Y,X)$. That is, we must have morphisms $t_1,\dots, t_n\colon Y\to X$ and a sequence of binary digits $\varepsilon_1,\dots,\varepsilon_n$ such that $f_{\varepsilon_1} t_1=1_Y$, $f_{1-\varepsilon_i} t_i = f_{\varepsilon_i} t_{i+1}$, and $f_{1-\varepsilon_n}t_n=s e$. (Note that if $n=1$ then this says precisely that we have a split coequalizer, and if $n=0$ it is the trivial case above.) Conversely, it is easy to check that given $s$ and $t_1,\dots, t_n$ satisfying these equations, the given fork must be a coequalizer, for essentially the same reason that any split coequalizer is a coequalizer. Thus we have a complete characterization of absolute coequalizers.
This characterization is essentially a special case of the characterization of [[absolute colimits]] (in unenriched categories).
## Examples
* [[Beck coequalizer]]
## References
* [[Robert Pare]], *Absolute coequalizers*, Lecture Notes in Math. 86 (1969), 132-145.
[[!redirects absolute coequalizers]]
|
absolute cohomology | https://ncatlab.org/nlab/source/absolute+cohomology |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
## Contents
* table of contents
{:toc}
## Idea
Roughly something like a combination of a "[[geometry|geometric]]" and an [[arithmetic geometry|arithmetic]] concept of [[cohomology]].
## Examples
* [[absolute Hodge cohomology]]
* [[syntomic cohomology]]
* [[motivic cohomology]]
## References
* Roy Joshua, _K-Theory and Absolute Cohomology for algebraic stacks_ ([KTheory](http://www.math.uiuc.edu/K-theory/0732/))
* [[Cohomology Theory Database]], _[Absolute cohomology](http://notes.andreasholmstrom.org/ct.php?n=Absolute+cohomology)_
* Jan Nekovar, section 3 of _Beilinson's Conjectures_ ([pdf](http://people.math.jussieu.fr/~nekovar/pu/mot.pdf))
|
absolute colimit | https://ncatlab.org/nlab/source/absolute+colimit |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An **absolute colimit** is a [[colimit]] which is [[preserved limit|preserved]] by any [[functor]] whatsoever. In general this happens because the colimit is a colimit for purely "diagrammatic" reasons. The notion is most important in [[enriched category theory]].
Of course, there is a dual notion of **absolute limit**, but it is used less frequently.
## Definitions
The term "absolute colimit" is actually used for two closely related, but distinct, notions.
### Particular absolute colimits
+-- {: .num_defn #DefParticular}
###### Definition
A particular [[colimit]] diagram in a particular [[category]] $C$ is an **absolute colimit** if it is [[preserved limit|preserved]] by every [[functor]] with domain $C$.
=--
This definition makes sense also in [[enriched category theory]]: for any $V$, a [[weighted colimit]] in a particular $V$-[[enriched category]] $C$ is an **absolute colimit** if it is preserved by every $V$-functor with domain $C$.
Note, however, that a [[conical colimit]] in a $V$-category $C$ may be preserved by all $V$-functors without being preserved by all unenriched functors on the underlying ordinary category $C_o$.
Generalizing in a different direction, absolute colimits in $Set$-enriched categories can be regarded as the particular case of [[postulated colimit]]s in [[sites]] where the site has the [[trivial topology]].
### Weights for absolute colimits
+-- {: .num_defn #DefWeight}
###### Definition
For a given $V$, a [[weighted colimit|weight]] $\Phi\colon D^{op} \to V$ for colimits is an **absolute weight**, or a **weight for absolute colimits**, if $\Phi$-weighted colimits in *all* $V$-categories are preserved by all $V$-functors.
=--
Absolute colimits of this sort are also called **Cauchy colimits**. A $V$-[[category]] which admits all absolute colimits --- that is, all $V$-weighted colimits whose weights are absolute -- is called [[Cauchy complete category|Cauchy complete]]. By the characterization below, it is equivalent to admit *limits* weighted by all weights for absolute limits.
## Characterizations
Both types of absolute colimits admit pleasant characterizations.
### Particular absolute colimits
For a particular [[cocone]] $\mu \colon F \to \Delta A$ under a functor $F\colon I\to C$ (all in the Set-enriched world), the following are equivalent:
* $\mu$ is an absolute colimiting cocone.
* $\mu$ is a colimiting cocone which is is preserved by the [[Yoneda embedding]] $C \hookrightarrow [C^{op},Set]$.
* $\mu$ is a colimiting cocone which is preserved by the representable functors $C(F(i),-)\colon C\to Set$ (for all $i\in I$) and $C(A,-)\colon C\to Set$.
* There exists $i_0\in I$ and $d_0\colon A\to F(i_0)$ such that
1. For every $i\in I$, $d_0 \circ \mu_i$ and $1_{F(i)}$ are in the same connected component of the [[comma category]] $(F(i) / F)$.
1. $\mu_{i_0} \circ d_0 = 1_{A}$.
The equivalence of the first two is basically because the Yoneda embedding is the [[free cocompletion]] of $C$. The third clearly follows from the second. The fourth follows from the third by inspecting exactly what preservation by those representables means in terms of colimits in [[Set]] (as is explained in more detail in the special case of [[absolute coequalizers]]). Finally, it is straightforward to check that the fourth implies that $\mu$ is colimiting, and it is clearly a property preserved by any functor.
Note that in particular, the fourth condition implies that $A$ is a [[retract]] of $F(i_0)$. Also, the first half of the fourth condition by itself characterizes absolute [[weak colimits]].
It is also possible to prove directly that the third condition implies the first two, without extracting the fourth condition along the way. Namely, Let $B$ be the full subcategory of $C$ consisting of the objects $F(i)$ and $A$. Then $F$ defines a functor $I\to B$; call it $F'$. Note that $A$ is also the colimit of $F'$ in $B$. Moreover, by the equivalence of the first two conditions, $A$ is an *absolute* colimit of $F'$, since by hypothesis it is preserved by all representable functors out of $B$. Therefore, this colimit is in particular preserved by the inclusion $B\hookrightarrow C$, along with its composite with any functor out of $C$; so $A$ is an absolute colimit of $F$.
### Weights for absolute colimits
{#CharacterizationsWeightsForAbsoluteColimits}
Let $V$ be a Bénabou [[cosmos]] and $J\colon K ⇸ A$ a $V$-[[profunctor]]. Then the following are equivalent:
* $J$ is a weight for absolute colimits (i.e. $J$-weighted colimits in any $V$-category are preserved by all $V$-functors)
* $J$ has a [[right adjoint]] $J^*$ in the [[bicategory]] $V$-[[Prof]].
* There is a weight $J^*\colon A ⇸ K$ such that $J$-weighted colimits coincide naturally with $J^*$-weighted limits.
## Examples
{#Examples}
### Particular absolute colimits
Of course, every colimit weighted by a weight for absolute colimits is itself a particular absolute colimit. But it may also happen that a particular colimit may be absolute without all colimits of that shape being absolute. For example (in ordinary category theory, with $V=Set$):
* [[split coequalizers]] are absolute, and figure in Beck's [[monadicity theorem]].
* More generally (of course), [[absolute coequalizers]] are absolute.
* [[absolute pushouts]] appear in [[elegant Reedy categories]].
We can also say something about non-examples.
* Initial objects (in $Set$-enriched categories) are *never* absolute. If $0$ is an initial object, then it is never preserved by the representable functor $C(0,-)\colon C \to Set$.
* Similarly, coproducts in $Set$-enriched categories are never absolute.
### Weights for absolute colimits
{#WExampleseightsForAbsoluteColimits}
In ordinary $Set$-enriched category theory there are not very many weights for absolute colimits, but we have
* [[split idempotents]].
In fact, this example is "universal," in that an ordinary category is Cauchy complete iff it has split idempotents, although not every absolute colimit "is" the splitting of an idempotent. More precisely, the class of absolute $Set$-limits is the [[saturated class of limits|saturation]] of idempotent-splittings.
In [[enriched category theory]] there can be more types of absolute colimits. For instance:
* in categories with [[zero morphisms]] (that is, enriched over [[pointed sets]]), [[initial objects]] are absolute.
* in [[Ab-enriched categories]] (or, more generally, categories enriched over commutative [[monoids]]), finite [[biproducts]] are absolute. Finite biproducts and splitting of idempotents together are "universal" absolute colimits for Ab-enrichment.
* in [[SupLat]]-enriched categories, arbitrary small biproducts are absolute, and together with splitting of idempotents these generate all absolute colimits.
* in [[dg-categories]] (or more generally, categories enriched over [[graded set|graded sets]]), *shifts/suspensions* and [[mapping cones]] are absolute.
* in Lawvere [[metric spaces]], limits of Cauchy sequences are absolute. This is the origin of the name "Cauchy colimit."
* in [[posets]], suprema of subsets with a greatest element are absolute.
* in categories [[bicategory-enriched category|enriched]] over the [[bicategory]] (or [[double category]]) of relations in a [[site]], *gluings* are absolute. In this case the enriched categories can roughly be identified with [[separated presheaves]] and the Cauchy-complete ones with [[sheaves]].
* in categories enriched over [[rational number|rational]] [[vector spaces]], quotients by finite [[group actions]] are absolute.
New kinds of absolute (co)limits also arise in [[higher category theory]].
* for [[(∞,1)-categories]] (enriched over $\infty$-groupoids), splitting of idempotents is a universal absolute colimit.
* in [[stable (∞,1)-categories]] (which are enriched, in the $(\infty,1)$-categorical sense, over the $(\infty,1)$-category of [[spectra]]), initial objects and [[pushouts]] are absolute, and therefore so are all finite colimits.
## Generalisations
If $K : A \to C$ is a functor, then a colimit in $A$ is **$K$-absolute** if it is preserved by the [[nerve]] $N_K : C \to [A^{\mathrm{op}}, \mathscr{V}]$ of $K$. This generalises absoluteness, taking $K$ to be the [[identity functor]].
## References
* [[Ross Street]], *Absolute colimits in enriched categories*, [Cahiers 1983](http://www.numdam.org/item/CTGDC_1983__24_4_377_0/)
* [[Robert Pare]], *On absolute colimits*, J. Alg. 19 (1971), 80-95.
* [[Max Kelly]], _Basic Concepts of Enriched Category Theory_, Cambridge University Press, Lecture Notes in Mathematics 64 (1982)
* [[Mike Shulman]]'s answer to [*What are _all_ of the exactness properties enjoyed by stable ∞-categories?*](https://mathoverflow.net/a/267324)
[[!redirects absolute colimits]]
[[!redirects absolute limit]]
[[!redirects absolute limits]]
[[!redirects Cauchy colimit]]
[[!redirects Cauchy colimits]]
|
absolute conclusion | https://ncatlab.org/nlab/source/absolute+conclusion | [[!redirects agar.io]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
[[!redirects Absolute conclusion]]
[[!redirects the absolute conclusion]]
[[!redirects Absoluter Schluss]]
[[!redirects der absolute Schluss]]
## Idea
The **absolute conclusion**, or in German, _der absolute Schluss_, is a key concept in the speculative philosophy of the German philosopher [[Georg Hegel|G. W. F. Hegel]].
The name of the concepts plays on the triple sense of _Schluss_ in German evoking _end_ , as well as 'Zusammenschluss' - _union_ and _logical deduction_. The primary sense is that the absolute conclusion unites three deductions into a deductive relation to conclude the development of a concept or part of the philosophical system. It is called _absolute_ because its ' premises' and 'conclusion' are themselves deductions that relate to other deductions, whence, it is a _deduction between deductions_.
The main idea is that this deductive pattern improves on ordinary 'relative' deductions that hinge on unmediated premises for their mediation, by elevating each premise into conclusive positions and vice versa, thereby solving a 'contradiction' well known in logic since Aristotle's _Analytica Posteriora_, namely that the desired proofs require unproved axioms.[^nat]
[^nat]: That it is also possible to proceed from (retractable) assumptions instead of (true) axioms was shown by the end of the 1920s by G. Gentzen and S. Jaskowski (cf. [[natural deduction]]).
Hegel's model for a logical deduction is an Aristotelian [[syllogism]] consisting of two premises and a conclusion that involve a subject notion $S$ and a predicate notion $P$ and a middle term $M$. Hegel first identifies then $S$ with a _general_ notion, $P$ with a _particular_ notion, and $M$ with a _singular_ notion. The absolute conclusion then consists of three syllogisms where the general, the particular and the singular change position in the syllogistic figure such that each takes the place thereby expressing the deep-structural identity between the three notions.
The _objectivity_ and _reality_ of an entity is viewed as the **absolute mediation** of the general, particular and singular parts involved in its notion as expressed by such an absolute conclusion. E.g. in the philosophy of religion Hegel ([Enzyklopädie III](#Hegel10), pp.374ff) interprets the Christian trinity as an absolute conclusion where the conceptions of god as father (generality), is united with the conception of god as a son (particularity) and the conception as holy spirit (singularity). Once the identity of these concepts is expressed, the trinitarian concept of god realizes itself as communital spirit in the church - i.e. once rightly understood god lives in and as the spirit of community in human society.
This might seem to be a somewhat speculative use of the absolute conclusion but a similar passage from a completed mediation to objectivity occurs in the subjective logic of "Wissenschaft der Logik" when the subjective notion transforms into 'objectivity'. That a mediation of moments yields to objectivity sounds rather strange but the background here is Hegel's reinterpretation of Kant's concept of 'transcendental apperception' where the integration of perceptions under the notion of an object is accompanied by the 'I think' i.e. the unity of perception gets correlated with the unity of a self. Hegel equates the unity of the self with the complete mediation of the concept's moments from which the thought of an object springs: the self that flows around such a circle of mediations is viewed as the generic invariant that imparts its invariance to objects.
Beyond all metaphysical speculation one can view Hegel's concept as a general proposal for the **architecture of foundational theories** that 'escapes' the Münchhausen problem of a foundational (vicious) circle by entangling three circles into a circle. Under this interpretation one would view it as the suggestion that foundations should ideally consists of three fundamental systems that interpret each other and thereby corroborate each other.
To get an intuition one might think of the three different notions of [[computability]] as Turing machines, recursive functions and lambda calculus whose mutual equivalence back the Church-Turing hypothesis resulting in an 'absolute' concept of algorithm. In a similar vein is [[Bob Harper|R. Harper's]] [[computational trinitarianism]]. Also relevant is the possibility in [[categorical logic]] to interpret classes of categories via an external 'objective' description, then as models of a 'subjective' internal language and as models of appropriate [[sketch|sketches]] at an intermediate level.
Notice the proximity of the triads involved in such foundations to the mathematical rendering of the triadic structure involving the imaginary, the real, and the symbolic in Lacan's theory of subjectivity as proposed in [[René Guitart|Guitart's]] theory of _Borromean objects_ ([2009](#Guitart09)).
To sum up: the concept of the absolute conclusion enjoys currently little popularity among philosophers not even among Hegelian philophers despite the importance Hegel attached to it. It might nevertheless be worthwhile to probe into this concept from the perspective of contemporary foundational theories in the mathematical sciences.
## Some background
Hegel apparently introduced the concept in his philosophy around 1800 in or shortly before the Jena period. He purloined it from Plato's [[Timaeus dialogue|Timaios dialogue]] (**7**, 32c):
>But it isn't possible to combine two things well all by themselves, without a third; there has to be some bond between the two that unites them. Now the best bond is one that really and truly makes a unity of itself together with the things bonded by it, and this in the nature of things is accomplished by proportion. For whenever of three numbers which are either solids or squares the middle term between any two of them is such that what the first term is to it, it is to the first, then, since the middle term turns out to be both first and and last, and the last and first likewise both turn out to be middle terms, they will all of necessity turn out to have the same relationship to each other, and, given this, will all be unified.[^cite]
[^cite]: Translation by D. J. Zeyl, p.1237 in [Cooper (ed.) 1997](#Plato). Note that 'most harmonious bond' is closer to the Greek text than 'best bond'. Hegel quotes it as 'der Bande schönstes' in German.
Hegel cites this passage already in _'Differenz des Fichteschen und Schellingschen Systems der Philosophie'_ ([1801](#Hegel2), p.97). A similar configuration with triangles instead of circles occurs also in his (lost) fragment on the _'divine triangle'_ as reported and dated to 1804 by [[Karl Rosenkranz|Rosenkranz]] (cf. [Hegel Werke 2](#Hegel2), pp.534-539) which is somewhat reminiscent of an adjoint triple $L\dashv M\dashv R$ as an [[adjoint triple|adjunction between adjunctions]].
The conception seems to have been the subject of the discussions between Hegel and Schelling during their collaboration in Jena, since the latter alludes to the display of the absolute by the three judgements in two texts of the epoch, namely the dialogue _'Bruno oder Über das göttliche und natürliche Prinzip der Dinge'_ ([1802](#Bruno)) and _'Philosophie und Religion'_ ([1804](#SchellSchrift)).
The mature Hegel comments the Timaios passage extensively and with emphatic approval in his [[Lectures on the History of Philosophy]] (cf. [Hegel Werke 19](#Hegel19), pp.89-91).
## Related entries
* [[Aufhebung]]
* [[Science of Logic]]
* [[infinite judgement]]
* [[construction in philosophy]]
* [[computational trinitarianism]]
* [[syllogism]]
* [[Timaios]]
## References
* {#Plato} J. M. Cooper (ed.), _Plato - Complete Works_ , Hackett Indianapolis 1997.
* N. Füzesi, _Hegels drei Schlüsse_ , Alber München 2004.
* {#Guitart09}[[René Guitart| R. Guitart]], _Klein's Group as a Borromean Object_ , Cah. Top. Géo. Diff. Cat. **L** no.2 (2009) pp.144-155. ([pdf](http://rene.guitart.pagesperso-orange.fr/textespublications/Guitart%202009%20G168CTGDC.pdf))
* {#Hegel2} [[Georg Hegel|G. W. F. Hegel]], _Jenaer Schriften 1801-1807 - Werke 2_ , Suhrkamp Frankfurt a. M 1986.
* {#Hegel8} [[Georg Hegel|G. W. F. Hegel]], _Enzyklopädie der philosophischen Wissenschaften I - Werke 8_ , Suhrkamp Frankfurt a. M 1986[1817, rev. 1830]. (§ 187 Zusatz, p.339f.)
* {#Hegel10} [[Georg Hegel|G. W. F. Hegel]], _Enzyklopädie der philosophischen Wissenschaften III - Werke 10_ , Suhrkamp Frankfurt a. M 1986[1817, rev. 1830]. (§§ 569-571, pp.376f; §§ 575-77, pp.393f)
* {#Hegel19} [[Georg Hegel|G. W. F. Hegel]], _Vorlesungen über die Geschichte der Philosophie II - Werke 19_ , Suhrkamp Frankfurt a. M 1986. (ch. III A. 2, pp.89-91)
* W. Jaeschke, _Die Schlüsse der Philosophie (§§ 574-577)_ , pp.479-486 in Drüe et al. (eds.), _Hegels 'Enzyklopädie der philosophischen Wissenschaften' (1830) - Ein Kommentar zum Systemgrundriß_ , Suhrkamp Frankfurt a. M. 2000.
* G. Sans, _Die Realisierung des Begriffs - Eine Untersuchung zu Hegels Schlusslehre_ , Akademie Verlag Berlin 2004.
* {#SchellSchrift}F. W. J. Schelling, _Schriften 1804-1812_ , Union Berlin 1982. (sect. 1, p.48)
* {#Bruno}F. W. J. Schelling, _Bruno oder Über das göttliche und natürliche Prinzip der Dinge_ , Reclam Leipzig 1986. (sect. IIB 7b, p.88)
* G. Werckmeister, _Hegels absoluter Schluss als logische Grundstruktur der Objektivität_ , PhD TU Kaiserslautern 2009. ([link](https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2160)) |
absolute convergence | https://ncatlab.org/nlab/source/absolute+convergence |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
=--
=--
\tableofcontents
## Definition
### For series in the real numbers
In [[real analysis]], given a sequence $a:\mathbb{N} \to \mathbb{R}$ in the [[real numbers]], the series $\sum_{n = 0}^\infty a_n$ is **absolutely convergent** if the [[sequence]] of [[partial sums]] is a [[Cauchy sequence]], and the [[sequence]] of [[partial sums]] of the series $\sum_{n = 0}^\infty \vert a_n \vert$ is also a Cauchy sequence.
### For series in Banach spaces
In [[functional analysis]], given a sequence $a:\mathbb{N} \to B$ in a [[Banach space]] $B$, the series $\sum_{n = 0}^\infty a_n$ is **absolutely convergent** if the [[sequence]] of [[partial sums]] is a [[Cauchy sequence]] in $B$, and the [[sequence]] of [[partial sums]] of the series $\sum_{n = 0}^\infty \Vert a_n \Vert$ is a Cauchy sequence in the real numbers.
## See also
* [[conditional convergence]]
* [[Banach space]]
## External links
* Wikipedia, *[Absolute convergence](https://en.wikipedia.org/wiki/Absolute_convergence)*
[[!redirects converges absolutely]]
[[!redirects absolutely converges]]
[[!redirects absolutely convergent]]
[[!redirects absolutely convergent series]] |