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(infinity,r)-category > history | https://ncatlab.org/nlab/source/%28infinity%2Cr%29-category+%3E+history | < [[(infinity,r)-category]]
[[!redirects (infinity,r)-category -- history]] |
(n Γ k)-category | https://ncatlab.org/nlab/source/%28n+%C3%97+k%29-category |
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###Context###
#### Higher category theory
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[[!include higher category theory - contents]]
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#### Internal categories
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[[!include internal infinity-categories contents]]
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#Contents#
* table of contents
{:toc}
## Idea
An _$(n \times k)$-category_ (read "n-by-k category") is an [[n-category]] [[internalization|internal]] to a $k$-category. The term is "generic" in that it does not specify the level of strictness of the $n$-category and the $k$-category.
For example:
* A $(1 \times 0)$-category, as well as a $(0 \times 1)$-category, is precisely a [[category]]. More generally, $(n\times 0)$-categories and $(0\times n)$-categories are precisely $n$-categories.
* A $(1 \times 1)$-category is precisely a [[double category]] (either strict or weak).
* Generalizing to a 3rd axis, a $(1 \times 1 \times 1)$-category is precisely a [[triple category]], that is, a category internal to (categories internal to categories), i.e. a catgory internal to double categories, or a double category internal to categories --- which again could be strict or weak.
* An $(n \times 1)$-category is what [[Michael Batanin|Batanin]] calls a [[monoidal n-globular category]].
An $(n \times k)$-category has $(n + 1)(k + 1)$ kinds of cells.
Under suitable [[fibrant object|fibrancy]] conditions, a $(n \times k)$-category will have an underlying $(n + k)$-category (where here, $n + k$ is to be read arithmetically, rather than simply as notation). Fibrant $(1 \times 1)$-categories are known as [[framed bicategories]].
## Examples
* [[commutative ring|Commutative rings]], [[algebras]] and [[modules]] form a symmetric monoidal $(2 \times 1)$-category.
* [[conformal net|Conformal nets]] form a symmetric monoidal $(2 \times 1)$-category.
## Relationships
At least in some cases, if the structure is sufficiently strict or sufficiently fibrant, we can shift cells from $k$ to $n$. For instance:
* A sufficiently strict $(1 \times 2)$-category canonically gives rise to a $(2 \times 1)$-category. (Cor. 3.11 in **[DH10](#DH10)**)
* Any double category (i.e. a $(1\times 1)$-category) has an underlying 2-category.
* A sufficiantly fibrant $(2\times 1)$-category has an underlying tricategory (i.e. $(3\times 0)$-category).
## Related concepts
* [[internal category]]
* [[higher category]]
* [[n-fold category]]
* [[(n,r)-category]]
## References ##
* [[Mike Shulman]], Constructing symmetric monoidal bicategories, arXiv preprint arXiv:1004.0993 (2010)
* [[Michael Batanin]], _Monoidal globular categories as a natural environment for the theory of weak $n$-categories_ , Advances in Mathematics 136 (1998), no. 1, 39–103.
The following paper contains some discussion on the relationship between various (weak) $(n \times k)$-categories for $n, k \leq 3$.
* {#DH10} [[Chris Douglas]], and [[Andre Henriques]], _Internal bicategories_ ([arXiv:1206.4284](https://arxiv.org/abs/1206.4284)) (2012)
There is some discussion on [this n-Category CafΓ© post](https://golem.ph.utexas.edu/category/2010/04/symmetric_monoidal_bicategorie.html) as well as [this one](https://golem.ph.utexas.edu/category/2011/03/liang_kong_on_levinwen_models.html).
[[!redirects n-by-k category]]
[[!redirects n-by-k categories]] |
(n+1,1)-category of n-truncated objects | https://ncatlab.org/nlab/source/%28n%2B1%2C1%29-category+of+n-truncated+objects | +-- {: .rightHandSide}
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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### 2-ategory theory
+--{: .hide}
[[!include 2-category theory - contents]]
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#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Given any [[object]] $X$ in any $(k,1)$-category $C$ for a [[natural number]] $k$, the $n$-[[truncated objects]] of $X$ for $n+1 \lt k$ form an $(n+1,1)$-category, an $(n+1)$-truncated $(k,1)$-category, (though in general it may be a [[large category|large]] one, cf. [[well-powered category]]). This is called the __$(n+1,1)$-category of $n$-truncated objects__ of $X$, or the __$n$-truncated object $(n+1,1)$-category__ of $X$.
## Properties
If $C$ is [[finitely complete category|finitely complete]], then the $n$-truncated objects form a finitely complete $(n+1,1)$-category, so we may speak of the __finitely complete $(n+1,1)$-category of $n$-truncated objects__.
In any [[coherent category|coherent]] $(k,1)$-category, then the $n$-truncated objects form a coherent $(n+1,1)$-category, so we may speak of the __coherent $(n+1,1)$-category of $n$-truncated objects__.
In any $(k,1)$-[[topos]], the $n$-truncated objects of $X$ form a $(n+1,1)$-topos, so we may speak of the __$(n+1,1)$-topos of $n$-truncated objects__.
The reader can probably think of other variations on this theme.
If $f : X \to Y$ is a morphism that has pullbacks along $n$-truncated morphisms, then pullback along $f$ induces a $(n+1,1)$-category morphism $f^* : Trunc_n(Y) \to Trunc_n(X)$. This is functorial in the sense that if $g : Y \to Z$ also has this property, then there is an inhabited equivalence $n+1$-groupoid $f^* \circ g^* \cong (g \circ f)^*$.
If $C$ has pullbacks of $n$-truncated morphisms, $Trunc_n$ is often used to denote the contravariant functor $C^{op} \to (n+1,1)Cat$ whose action on morphisms is $Trunc_n(f) = f^*$.
## Examples
* A [[(0,1)-category]] of (-1)-truncated objects of an object $X$ is a [[poset of subobjects]] of $X$.
* A [[well-pointed category|well-pointed]] [[Heyting category|Heyting]] [[pretopos]] of 0-truncated objects of the [[terminal object]] $1$ is a (possibly) [[predicative mathematics|predicative]] model of the [[category of sets]].
* A [[well-pointed category|well-pointed]] [[Heyting category|Heyting]] [[Ξ -pretopos]] of 0-truncated objects of the [[terminal object]] of $1$ is a (possibly) [[predicative mathematics|weakly predicative]] model of the [[category of sets]].
* A [[well-pointed category|well-pointed]] [[topos]] of 0-truncated objects of the [[terminal object]] $1$ is an [[predicative mathematics|impredicative]] model of the [[category of sets]].
* A [[well-pointed category|well-pointed]] [[W-topos]] of 0-truncated objects of the [[terminal object]] $1$ whose 0-truncated (-1)-[[connected object|connected]] [[morphisms]] all have [[sections]] is a model of [[ETCS]].
## Related concepts
* If one opts for the alternative definition that $n$-truncated objects _are_ $n$-truncated morphisms into the object (not equivalence large $n+1$-groupoids thereof), then one gets a $(n+1,1)$-[[precategory]] of $n$-truncated objects instead. In any case, the *$(n+1,1)$-category* of $n$-[[truncated objects]] $Trunc_n(X)$ in our sense is the $(n+1,1)$-categorical reflection of the $(n+1,1)$-precategory $TruncMor_n(X)$ of $n$-truncated objects in the alternative sense, and of course the reflection [[Rezk completion]] map $TruncMor_n(X) \to Trunc_n(X)$ is an [[equivalence]].
* [[poset of subobjects]]
* [[n-truncated object classifier]]
[[!redirects $(n+1)$-category of $n$-truncated objects]]
[[!redirects $(n+1)$-categories of $n$-truncated objects]]
[[!redirects $n$-truncated object $(n+1)$-category]]
[[!redirects $n$-truncated object $(n+1)$-categories]] |
(n,0)-category | https://ncatlab.org/nlab/source/%28n%2C0%29-category |
An _$(n,0)$-category_ is an [[(n,r)-category]] that is an [[n-groupoid]].
By the general rules of $(n,r)$-[[(n,r)-category|categories]], an __$(n,0)$-category__ is an $\infty$-[[infinity-category|category]] such that
* any $j$-morphism is an [[equivalence]], for $j \gt 0$;
* any two parallel $j$-morphisms are equivalent, for $j \gt n$.
You can start from any notion of $\infty$-category, strict or weak; up to [[equivalence of categories|equivalence]], the result is the same as an [[n-groupoid]] with a corresponding level of strictness.
[[!redirects (n,0)-categories]] |
(n,1)-category | https://ncatlab.org/nlab/source/%28n%2C1%29-category |
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###Context###
#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
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#### Higher category theory
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[[!include higher category theory - contents]]
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#Contents#
* table of contents
{:toc}
## Idea
The special case of an [[(n,r)-category]] for $r = 1$.
## Definition
An __$(n,1)$-category__, is an $n$-[[n-category|category]] $C$ that is __locally $(n-1)$-groupoidal__; that is, for any [[object]]s $x$ and $y$, the $(n-1)$-category $C(x,y)$ is an $(n-1)$-[[n-groupoid|groupoid]].
Equivalently it is an $(\infty,1)$-category for which the mapping spaces are all $(n-1)$-[[n-truncated object of an (infinity,1)-category|truncated]].
## Special cases:
* A $(1,1)$-category is the same as a $1$-[[1-category|category]], which is an ordinary [[category]].
* A $(2,1)$-category is a [[locally groupoidal 2-category|locally groupoidal]] $2$-[[2-category|category]].
* An $(\infty,1)$-[[(infinity,1)-category|category]] can be understood as a [[quasi-category]] or in many other ways.
## Extra stuff, structure, property
* An $(n,1)$-category with the analogous properties of a [[topos]] is an [[(n,1)-topos]].
## Examples
The canonical example of an $(n+1,1)$-category is [[nGrpd]].
## Related concepts
* [[0-category]], [[(0,1)-category]]
* [[category]]
* [[2-category]]
* [[3-category]]
* [[n-category]]
* [[(β,0)-category]]
* **(n,1)-category**
* [[(β,1)-category]]
* [[(β,2)-category]]
* [[(β,n)-category]]
* [[(n,r)-category]]
## References
In Section 11 of
* [[Charles Rezk]], _A cartesian presentation of weak n-categories_, [arXiv:0901.3602](https://arxiv.org/abs/0901.3602)
the author describes a presentation of $(n,1)$-categories by a [[left Bousfield localization]] of the [[model structure]] presenting [[complete Segal spaces]].
[[!redirects (n,1)-categories]] |
(n,1)-sheaf | https://ncatlab.org/nlab/source/%28n%2C1%29-sheaf |
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###Context###
#### $(\infty,1)$-Topos Theory
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[[!include (infinity,1)-topos - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}
## Definition
For $n \in \mathbb{Z}$, $-2 \leq n \lt \infty$,
an _$(n,1)$-sheaf_ is an [[n-truncated]] [[(β,1)-sheaf]]: an [[(β,1)-sheaf]] with values in [[n-groupoids]].
This is also sometimes called an _$(n-1)$-stack_.
## Examples
* [[(0,1)-sheaf]] = [[ideal]] (of a [[poset]]/[[proset]])
* [[(1,1)-sheaf]] = [[sheaf]]
* [[(2,1)-sheaf]] = [[stack]] (of groupoids)
## Related concepts
* [[n-truncated object in an (β,1)-category]]
* [[model structure for homotopy n-types]]
[[!redirects (n,1)-sheaves]]
[[!redirects (n+1,1)-sheaf]]
[[!redirects n-stack]]
[[!redirects n-stacks]]
|
(n,1)-topos | https://ncatlab.org/nlab/source/%28n%2C1%29-topos |
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###Context###
#### $(\infty,1)$-Topos theory
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[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A (Grothendieck) $(n,1)$-topos is the [[(n,r)-category|(n,1)-category]] version of a [[Grothendieck topos]]: a collection of [[n-groupoid|(n-1)-groupoid]]-valued sheaves on an $(n,1)$-categorical site.
Notice that an [[β-stack]] on an ordinary (1-categorical) [[site]] that takes values in [[β-groupoid]]s which happen to by 0-[[truncated]], i.e. which happen to take values just in [[Set]] $\hookrightarrow$ [[βGrpd]] is the same as an ordinary [[sheaf]] of sets.
This generalizes: every $(n,1)$-topos arises as the full [[sub-quasi-category|(β,1)-subcategory]] on $(n-1)$-[[truncated]] objects in an [[(β,1)-topos]] of $\infty$-stacks on an [[(n,1)-category]] site.
## Definition
Recall that
* a 1-[[Grothendieck topos]] is precisely an [[accessible functor|accessible]] [[geometric embedding]] into a [[category of presheaves]] $PSh(C)$ on some [[small category]] $C$
$$
Sh(C) \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} PSh(C)
\,.
$$
* a [[(β,1)-topos]] (of [[β-stack]]s/[[(β,1)-category of (β,1)-sheaves|(β,1)-sheaves]]) is precisely an [[accessible (β,1)-functor|accessible]] [[reflective (β,1)-subcategory|geometric embedding]] into a [[(β,1)-category of (β,1)-presheaves]] $PSh_{(\infty,1)}(C)$ on some small [[(β,1)-category]] $C$:
$$
Sh_{(\infty,1)}(C)
\stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow}
PSh_{(\infty,1)}(C)
\,.
$$
Accordingly now,
+-- {: .un_defn}
###### Definition
An **$(n,1)$-topos** $\mathcal{X}$ is an accessible left exact [[localization of an (β,1)-category|localization]] of the full [[sub-quasi-category|(β,1)-subcategory]] $PSh_{\leq n-1}(C) \subset PSh_{(\infty,1)}(C)$ on $(n-1)$-[[truncated]] objects in an [[(β,1)-category of (β,1)-presheaves]] on a small [[(β,1)-category]] $C$:
$$
\mathcal{X}
\stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow}
PSh_{\leq n-1}(C)
\,.
$$
=--
This appears as [[Higher Topos Theory|HTT, def. 6.4.1.1]].
## Properties
Write [[(β,1)-Topos]] for the [[(β,1)-category]] of [[(β,1)-topos]] and [[(β,1)-geometric morphisms]]. Write $(n,1)Topos$ for the [[(n,1)-category|(n+1,1)-category]] of $(n,1)$-toposes and geometric morphisms between these.
The following proposition asserts that when passing to the $(n,1)$-topos of an [[(β,1)-topos]] $\mathcal{X}$, only the [[n-localic (β,1)-topos|n-localic]] "Postnikov stage" of $\mathcal{X}$ matters.
+-- {: .un_prop}
###### Proposition
Every $(n,1)$-topos $\mathcal{Y}$ is the [[(n,1)-category]] of $(n-1)$-[[truncated]] objects in an [[n-localic (β,1)-topos]] $\mathcal{X}_n$
$$
\tau_{n-1} X_n \stackrel{\simeq}{\to} \mathcal{Y}
\,.
$$
=--
This is ([[Higher Topos Theory|HTT, prop. 6.4.5.7]]).
+-- {: .un_prop}
###### Proposition
For any $0 \leq m \leq n \leq \infty$, $(m-1)$-truncation induces a [[localization of an (β,1)-category|localization]]
$$
Topos_{(m,1)} \stackrel{\overset{\tau_{m-1}}{\leftarrow}}{\hookrightarrow}
Topos_{n,1}
$$
that identifies $Topos_{(m,1)}$ equivalently with the full subcategory of $m$-localic $(n,1)$-toposes.
=--
(This is [[Higher Topos Theory|6.4.5.7]] in view of the following remarks.)
## Examples
### $(2,1)$-Toposes
If $E$ is a [[(2,1)-topos]] in which every object is *covered* by a 0-[[truncated]] object, then $E$ is equivalent to the category of (2,1)-sheaves on a 1-site (rather than merely a (2,1)-site, as is the case for general (2,1)-topoi), and is thus canonically *associated* to a 1-topos, namely the category of 1-sheaves on that same 1-site. And in fact, $E$ can be recovered from this 1-topos as the category of (2,1)-sheaves for its canonical topology.
See [[michaelshulman:truncated 2-topos]] for more.
## Related concepts
[[!include flavors of higher toposes -- list]]
## References
Section 6.4 of
* [[Jacob Lurie]], _[[Higher Topos Theory]]_
[[!redirects (n,1)-toposes]]
[[!redirects (n,1)-topoi]]
|
(n,n)-category | https://ncatlab.org/nlab/source/%28n%2Cn%29-category | Following the pattern of the notion of [[(n,r)-category]], an _$(n,n)$-category_ is a [[higher category theory|higher category]] with non-trivial cells of at most dimension $n$ and none of them guaranteed to be reversible.
So this is what is usually simply called an [[n-category]].
Note that it is possible to go on to an $(n,n+1)$-category, or $(n+1)$-[[n-poset|poset]]. You can either consider than the $n$-cells are ordered, or else consider that there are irreversible $(n+1)$-cells which are indistinguishable. (Reversible indistinguishable $(n+1)$-cells are all identities and so might as well not exist.) |
(n,r)-category | https://ncatlab.org/nlab/source/%28n%2Cr%29-category |
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### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
An $(n,r)$-category is a [[higher category theory|higher category]] such that, essentially:
* all [[k-morphisms]] for $k \gt n$ are trivial.
* all [[k-morphisms]] for $k \gt r$ are reversible.
Put another way: given a sequence of (higher) categories $C_0, C_1, ..., C_n$ in which each $C_{i+1}$ is of the form $Hom(A, B)$ for some $0$-cells $A$ and $B$ from $C_i$, let us say that $C_n$ is a depth-$n$ Hom-category of $C_0$. (We can also cleanly extend this notion to depth-$\infty$ Hom-categories, by taking the position that there are none). An $(n, r)$-category, then, is one in which every depth-$r$ Hom-category is an $\infty$-groupoid, and, furthermore, every depth-$(n+2)$ Hom-category is a [[point]]. (The appearance of $n+2$ here rather than $n$ allows us to make sense of this definition even when $n$ is as low as $-2$, and suggests that perhaps, had history gone differently, the conventions would be to number these differently.)
So $(n,r)$-categories are a generalisation of both $n$-[[n-category|categories]] and $n$-[[n-groupoid|groupoids]], covering all of the ground in between (and a bit beyond). As $n$ increases, there are many more possibilities, until there are infinitely many kinds of $(\infty,r)$-[[(infinity,n)-category|categories]].
## Definition
{#Definition}
Given a notion of [[infinity-category|$\infty$-category]] (as weak or strict as you like), then an **$(n,r)$-category** can be defined to be an $\infty$-category such that
* any $j$-morphism is an [[equivalence]], for $j \gt r$;
* any two [[parallel morphisms|parallel]] $j$-morphisms are equivalent, for $j \gt n$.
As explained below, we may assume that $n \geq -2$ and $0 \leq r \leq \max(0, n + 1)$.
For finite $r$, we can also define this inductively in terms of [[(β,r)-categories]] as follows:
+-- {: .num_defn}
###### Definition
For $-2 \leq n \leq \infty$, an **[[(n,0)-category]]** is an [[β-groupoid]] that is [[n-truncated]]: an [[n-groupoid]].
For $0 \lt r \lt \infty$, an **(n,r)-category** is an [[(β,r)-category]] $C$ such that for all [[object]]s $X,Y \in C$ the $(\infty,r-1)$-categorical [[hom-object]] $C(X,Y)$ is an $(n-1,r-1)$-category.
=--
(Even for $r = \infty$, this definition makes sense, taking $\infty - 1$ to be $\infty$, as long as we know that an $(-1,\infty)$-category is the same thing as a $(-1,0)$-category. But this may be overkill.)
You can also start with a notion of $n$-[[n-poset|poset]], then define an $(n,r)$-category to be an $(n+1)$-poset such that any $j$-morphism is an [[equivalence]] for $j \gt r$. Or, for $r \leq n$, you can start with a notion of $n$-[[n-category|category]], then define an $(n,r)$-category to be an $n$-category such that any $j$-morphism in an [[equivalence]] for $j \gt r$.
To interpret the first definition above correctly for low values of $j$, we must assume that all objects ($0$-morphisms) in a given $\infty$-category are parallel, which leads us to speak of the two $(-1)$-morphisms that serve as their common source and target and to accept any object as an equivalence between these. In particular, any $j$-morphism is an equivalence for $j \lt 1$, so if $r = 0$, then the condition is satisfied for any smaller value of $r$. Thus, we assume that $r \geq 0$.
To say that parallel $(-1)$-morphisms must be equivalent is meaningful; it requires that there be an object. One can continue to $(-2)$-morphisms and so on, but there is nothing to vary about these; so we assume that $n \geq -2$. In other words, a $(-2)$-[[(-2)-category|category]] will automatically be an $n$-category for any smaller value of $n$.
If any two parallel $j$-morphisms are equivalent, then any $j$-morphism between equivalent $(j-1)$-morphisms is an equivalence (being parallel to an identity for $j \gt 0$ and automatically for $j \lt 1$). Accordingly, any $(n,r)$-category for $r \gt n + 1$ is also an $(n,n+1)$-category. Thus, we assume that $r \leq n + 1$ (except when $n = -2$, where it would conflict with the convention $r \geq 0$ and so we simply take $r = 0$.)
## Homotopy-theoretic relation {#HomtopyTheory}
From the point of view of [[homotopy theory]], the notion of $(n,r)$-categories may be understood as a combination of the notion of [[homotopy n-type]] and that of [[directed space]].
Recall that an [[(β,0)-category]] is an [[β-groupoid]]. In light of the [[homotopy hypothesis]] -- that identifies $\infty$-groupoids with (nice) [[topological spaces]] and [[n-groupoids]] with [[homotopy n-types]] -- and in view of the notion of [[directed space]], the following terminology is suggestive:
+-- {: .standout}
An $(n,r)$-category is an $r$-directed homotopy $n$-type.
=--
Here we read
* _$0$-directed_ as _undirected_
and
* _$1$-directed_ as _directed_ .
Then, indeed, we have for instance that
* a [[(1,0)-category]] is an undirected 1-type: a 1-[[groupoid]],
* a [[(n,r)-category|(2,0)-category]] is an undirected 2-type: a [[2-groupoid]],
* etc.
* a [[(n,n)-category|(1,1)-category]] is **directed 1-type** : a [[category]],
* an [[(n,n)-category]] is an $n$-directed $n$-type: an [[n-category]],
* etc.
* an [[(β,0)-category]] is an undirected space: an [[β-groupoid]],
* an [[(β,1)-category]] is a [[directed space]]: a [[quasi-category]],
* an [[(β,n)-category]] is an $n$-directed space
* etc.
## Special cases
An [[(n,n)-category]] is simply an $n$-[[n-category|category]]. An $(n,n+1)$-category is an $(n+1)$-[[n-poset|poset]]. Note that an $\infty$-category and an $\infty$-poset are the same thing. An $(n,0)$-category is an $n$-[[n-groupoid|groupoid]]. Even though they have no special name, $(n,1)$-[[(n,1)-category|categories]] are widely studied.
For low values of $n$, many of these notions coincide. For instance, a $0$-[[0-groupoid|groupoid]] is the same as a $0$-[[0-category|category]], namely a [[set]]. And $(-1)$-[[(-1)-groupoid|groupoid]], $(-1)$-[[(-1)-category|category]], and $0$-[[0-poset|poset]] all mean the same thing (namely, a [[truth value]]) while $(-2)$-[[(-2)-groupoid|groupoid]], $(-2)$-[[(-2)-category|category]], and $(-1)$-[[(-1)-poset|poset]] likewise all mean the same thing (namely, the [[point]]).
Of particular importance is the case where $n = \infty$. See
* [[(β,n)-category]] .
### Topos cases
An analogous systematics exists for $(n,r)$-categories that in additions have the property of being a [[topos]] or [[higher topos theory|higher topos]].
* a [[(0,1)-topos]] is a [[Heyting algebra]]
* in particular, a $(0,1)$-[[Grothendieck topos]] is a [[locale]]
* a $(1,1)$-topos is a [[topos]]
* an [[(β,1)-topos]] is what [[Higher Topos Theory]] calls an $\infty$-topos
## The periodic table
There is a [[periodic table]] of $(n,r)$-categories:
<table>
<tr><th markdown="1" style="white-space: nowrap;">$n$→<br/>$r$↓</th>
<th markdown="1">$-2$</th>
<th markdown="1">$-1$</th>
<th markdown="1">$0$</th>
<th markdown="1">$1$</th>
<th markdown="1">$2$</th>
<th markdown="1">...</th>
<th markdown="1">$\infty$</th></tr>
<tr><th markdown="1">$0$</th>
<td>[[point]]</td>
<td>[[truth value]]</td>
<td>[[set]]</td>
<td>[[groupoid]]</td>
<td>[[2-groupoid]]</td>
<td>...</td>
<td>[[β-groupoid]]</td></tr>
<tr><th markdown="1">$1$</th>
<td>"</td>
<td>"</td>
<td>[[partial order|poset]]</td>
<td>[[category]]</td>
<td>[[(2,1)-category]]</td>
<td>...</td>
<td>[[(β,1)-category]]</td></tr>
<tr><th markdown="1">$2$</th>
<td>"</td>
<td>"</td>
<td>"</td>
<td>[[2-poset]]</td>
<td>[[2-category]]</td>
<td>...</td>
<td>[[(β,2)-category]]</td></tr>
<tr><th markdown="1">$3$</th>
<td>"</td>
<td>"</td>
<td>"</td>
<td>"</td>
<td>[[3-poset]]</td>
<td>...</td>
<td>[[(β,3)-category]]</td></tr>
<tr><th markdown="1">⋮</th>
<td>⋮</td>
<td>⋮</td>
<td>⋮</td>
<td>⋮</td>
<td>⋮</td>
<td>⋱</td>
<td>⋮</td></tr>
<tr><th markdown="1">$\infty$</th>
<td>point</td>
<td>truth value</td>
<td>poset</td>
<td>2-poset</td>
<td>3-poset</td>
<td>...</td>
<td>[[(β,β)-category]]/[[β-poset]]</td></tr>
</table>
## Models for weak (n,r)-categories
There are various [[model category]] models for collections of $(n,r)$-categories.
* The standard [[model structure on simplicial sets]] models [[β-groupoid|(β,0)-categories]].
* The Joyal-[[model structure on simplicial sets]] models [[(β,1)-category|(β,1)-categories]].
* The [[Charles Rezk]]-model structure for [[Theta space]]s models general $(n,r)$-categories.
## Related concepts
* [[directed (n,r)-graph]]
* [[(n,r)-site]]
* [[0-category]], [[(0,1)-category]]
* [[category]]
* [[2-category]]
* [[3-category]]
* [[n-category]]
* [[(β,0)-category]]
* [[(n,1)-category]]
* [[(β,1)-category]]
* [[(β,2)-category]]
* [[(β,n)-category]]
* **(n,r)-category**
* [[(n Γ k)-category]], [[n-fold category]]
## References
* [[John Baez]], [[Michael Shulman]], _Lectures on n-Categories and Cohomology_, ([arXiv:math/0608420](https://arxiv.org/abs/math/0608420)), in particular pp. 34-36.
[[!redirects (n,r) categories]]
[[!redirects (n,r) category]]
[[!redirects (n,r)-categories]]
[[!redirects (n,r)-category]]
[[!redirects (n, r)-category]]
[[!redirects (n, r)-categories]]
|
(n,r)-congruence | https://ncatlab.org/nlab/source/%28n%2Cr%29-congruence |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Internal $(\infty,1)$-Categories
+--{: .hide}
[[!include internal infinity-categories contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In general, the idea is that an $k$-congruence in an [[n-category]] $K$, where $k\le n$, is an "[[internal category|internal]] $(k-1)$-category" in $K$.
Here we consider the case $m\le 2$, although we allow $n$ and $m$ to be of the form $(r,s)$; see _[[(n,r)-category]]_.
## Definition
+--{: .num_defn}
###### Definition
Let $D$ be a [[2-congruence]] in a [[2-category]] $K$.
* $D$ is a **(2,1)-congruence** if it is an internal groupoid, i.e. there is a map $D_1\to D_1$ providing "inverses".
* It is a **(1,2)-congruence** if $D_1\to D_0\times D_0$ is ff.
* It is a **1-congruence** if it is both a (2,1)-congruence and a (1,2)-congruence.
* it is a **(0,1)-congruence** if $D_1\to D_0\times D_0$ is an equivalence.
=--
Note that in a 1-category,
* a 2-congruence is just an internal category (a 1-category),
* a (2,1)-congruence is an internal groupoid (a (1,0)-category),
* a (1,2)-congruence is an internal poset (a (0,1)-category), and
* a 1-congruence is an internal equivalence relation (a 0-category).
Of course, a (0,1)-congruence in any 2-category is completely determined by any object $D_0$.
+--{: .num_theorem}
###### Theorem
Let $q:X\to Y$ be a morphism in $K$. If $Y$ is $n$-truncated for $n\ge -1$, then $ker(q)$ is an $(n+1)$-congruence. This means that:
1. If $Y$ is [[groupoidal object|groupoidal]], then $ker(q)$ is a (2,1)-congruence.
1. If $Y$ is [[posetal object|posetal]], then $ker(q)$ is a (1,2)-congruence.
1. If $Y$ is [[discrete object|discrete]], then $ker(q)$ is a 1-congruence.
1. If $Y$ is [[nLab:subterminal object|subterminal]], then $ker(q)$ is a (0,1)-congruence.
In all these cases the converse is true if $K$ is [[regular 2-category|regular]] and $q$ is [[nLab:eso morphism|eso]].
=--
+--{: .proof}
###### Proof
The forward directions are fairly obvious; it is the converses which take work. Suppose first that $ker(q)$ is a (2,1)-congruence, and let $\alpha: f \to g: X \rightrightarrows Y$ be any 2-cell. Pulling back the eso $q$ along $f$ and $g$ gives $P_1\to T$ and $P_2\to T$; let $r:P \to T$ be the pullback $P_1\times_X P_2$. Since $K$ is regular, $r$ is eso. By definition of kernels, the 2-cell $\alpha r$ corresponds to a map $P\to (q/q)$. But $(q/q)\rightrightarrows C$ is a (2,1)-congruence, so composing this map with the "inverse" map $(q/q)\to(q/q)$ gives another map $P\to (q/q)$, and thereby another 2-cell $f r \to g r$ which is inverse to $\alpha r$. Finally, since $r$ is eso, precomposing with it reflects invertibility, so $\alpha$ must also be invertible. Thus $Y$ is groupoidal.
Now suppose that $ker(q)$ is a (1,2)-congruence, and let $\alpha,\beta: f\to g: T\to Y$ be two parallel 2-cells. With notation as in the previous paragraph, the 2-cells $\alpha r$ and $\beta r$ correspond to morphisms $P\rightrightarrows (q/q)$ which become isomorphic in $X$. But since $(q/q)\rightrightarrows X$ is a (1,2)-congruence, this implies that the two maps $P\rightrightarrows (q/q)$ are isomorphic, and hence $\alpha r = \beta r$. And since $r$ is eso, precomposing with it is faithful, so $\alpha=\beta$; thus $Y$ is posetal.
The discrete case follows by combining the posetal and groupoidal cases, so it remains to show that if $ker(q)$ is a (0,1)-congruence then $Y$ is subterminal. We know it is discrete, so it suffices to show that given two $f,g:T \rightrightarrows Y$ we have a 2-cell $f\to g$. Continuing with the same notation, and letting $h,k:P\to X$ be the induced maps with $q h \cong f r$ and $q k \cong g r$, we have $(h,k):P\to X\times X = (q/q)$, and therefore the 2-cell defining the fork $(q/q) \;\rightrightarrows\;X \overset{q}{\to} Y$ gives us a 2-cell $q h \to q k$ and therefore $f r \to g r$. Now $r$ is the quotient of its kernel, so for this 2-cell to induce a 2-cell $f\to g$ it suffices for it to be an action 2-cell for the actions of $ker(r)$ on $f r$ and $g r$; but this is automatic since we know $Y$ to be posetal. Thus we have a 2-cell $f\to g$ as desired, so $Y$ is subterminal.
=--
## Related concepts
* [[congruence]] [[2-congruence]]
* [[internal category]], [[internal category in an (infinity,1)-category]]
## References
* [[Mike Shulman]], _[[michaelshulman:n-congruence]]_
[[!redirects (n,r)-congruences]]
|
(n,r)-site | https://ncatlab.org/nlab/source/%28n%2Cr%29-site |
An [[(n,r)-site]] is an [[(n,r)-category]] equipped with a notion of [[coverage]].
## Examples
* [[(0,1)-site]]
* [[(1,1)-site]]
* [[(2,1)-site]], [[(2,2)-site]]
* [[(β,1)-site]]
[[!redirects (n,r)-site]]
|
(n-connected, n-truncated) factorization system | https://ncatlab.org/nlab/source/%28n-connected%2C+n-truncated%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
The $n$-connected/$n$-truncated factorization system is an [[orthogonal factorization system in an (β,1)-category]], specifically in an [[(β,1)-topos]], that generalizes the _relative [[Postnikov systems]]_ of [[βGrpd]]: it factors any morphism through its _[[n-image|(n+2)-image]]_ by an _[[n-epimorphism|(n+2)-epimorphism]]_ followed by an _[[n-monomorphism|(n+2)-monomorphism]]_.
As $n$ ranges through $(-1), 0, 1, 2, 3, \cdots$ these factorization systems form an [[k-ary factorization system|β-ary factorization system]].
## Definitions
+-- {: .num_prop}
###### Proposition
Let $\mathbf{H}$ be an [[(β,1)-topos]]. For all $(-2) \leq n \lt \infty$ the [[class]] of [[n-truncated morphisms in an infinity,1-category|n-truncated morphisms]] in $\mathbf{H}$ forms the right class in a [[orthogonal factorization system in an (β,1)-category]]. The left class is that of [[n-truncated morphisms in an infinity,1-category|n-connected morphisms]] in $\mathbf{H}$.
=--
This appears as a remark in [[Higher Topos Theory|HTT, Example 5.2.8.16]]. A construction of the factorization in terms of a [[model category]] [[presentable (β,1)-category|presentation]] is in ([Rezk, prop. 8.5](#Rezk)).
+-- {: .num_remark}
###### Remark
For $n = -1$ this says that [[effective epimorphisms in an (β,1)-category]] have the [[left lifting property]] against [[monomorphisms in an (β,1)-category]]. Therefore one may say that the effective epimorphisms in an $(\infty,1)$-topos are the [[strong epimorphisms]].
=--
## Properties
### Stability
\begin{prop}\label{nConnectedTruncatedIsStable}
For all $n$, the $n$-connected/$n$-truncated factorization system is [[stable factorization system|stable]]: the class of [[n-connected]] morphisms is preserved under [[(β,1)-pullback]].
\end{prop}
This appears as ([Lurie, prop. 6.5.1.16(6)](#Lurie)).
It follows that:
\begin{prop}\label{HomotopyPullbackPreservesNImageFactorization}
For all $n$, [[n-images]] are preserved by [[(β,1)-pullback]]
\end{prop}
\begin{proof}
Let $X \xrightarrow{f} Y$ with $n$-image $im_n(f)$.
By the [[pasting law]] its [[(infinity,1)-pullback|$\infty$-pullback]] along any $g \,\colon\, X' \xrightarrow{\;} X$ may be decomposed as two consecutive $\infty$-pullbacks:
\begin{tikzcd}
g^\ast X
\ar[r]
\ar[d]
\ar[dr, phantom, "{\mbox{\tiny(pb)}}"]
\ar[
dd,
bend right=50,
"{g^\ast f}"{swap}
]
&
X
\ar[d, "{\mbox{\tiny $n$-cnct}}"{description}]
\ar[
dd,
bend left=50,
"{f}"
]
\\
\mathrm{im}_n(g^\ast f)
\ar[r]
\ar[d]
\ar[dr, phantom, "{\mbox{\tiny(pb)}}"]
&
\mathrm{im}_n(f)
\ar[d, "{\mbox{\tiny $n$-trnct}}"{description}]
\\
Y'
\ar[
r,
"g"{swap}
]
&
Y
\end{tikzcd}
Here the pullback of the $n$-truncated map in again $n$-truncated since the right class of any orthogonal factorization system is stable under pullback. The analogous statement holds also for the $n$-connected map by Prop. \ref{nConnectedTruncatedIsStable}. Therefore the pullback of $im_n(f)$ along $g$ is indeed $im_n(g^\ast f)$, as shown.
\end{proof}
## Examples
### The case $n = -2$
A [[(-2)-truncated]] [[morphism]] is precisely an [[equivalence in an (β,1)-category]] (see there or [[Higher Topos Theory|HTT, example 5.5.6.13]]).
Moreover, every morphism is [[(-2)-connected]].
Therefore for $n = -2$ the $n$-connected/$n$-truncated factorization system says (only) that equivalences have inverses, unique up to coherent homotopy.
### The case $n = -1$
A [[(-1)-truncated]] morphism is precisely a [[monomorphism in an (β,1)-category|full and faithful morphism]].
A [[(-1)-connected]] morphism is one whose [[homotopy fiber]]s are [[inhabited]].
In [[βGrpd]] a morphism between _[[0-truncated]]_ [[object]]s ([[set]]s)
* is full and faithful precisely if it is an [[injection]];
* has non-empty fibers precisely if it is an [[epimorphism]].
Therefore between 0-truncated objects the (-1)-connected/(-1)-truncated factorization system is the [[epi/mono factorization system]] and hence [[image]] factorization.
Generally, the (-1)-connected/(-1)-truncated factorization is through the **$\infty$-categorical [[1-image]]**, the _[[homotopy image]]_ (see there for more details).
### The case $n = 0$
Let $X,Y$ be two [[groupoid]]s ([[homotopy n-type|homotopy 1-types]]) in [[βGrpd]].
A morphism $X \to Y$ is [[0-truncated]] precisely if it is a [[faithful functor]].
A morphism $X \to Y$ is [[0-connected]] precisely if it is a [[full functor]] and an [[essentially surjective functor]]: a [[essentially surjective and full functor]]
Therefore on homotopy 1-types the 0-connected/0-truncated factorization system is the [[(eso+full, faithful) factorization system]].
## References
The general abstract statement is in
* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_
A [[model category]]-theoretic discussion is in section 8 of
* {#Rezk} [[Charles Rezk]], _Toposes and homotopy toposes_ ([pdf](http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf))
Discussion in [[homotopy type theory]] is in
* [[Univalent Foundations Project]], section 7.6 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_
[[!redirects n-connected/n-truncated factorization system]]
[[!redirects (n-connected, n-truncated) factorization system]]
[[!redirects (n-epi, n-mono) factorization system]]
[[!redirects (n-epi, n-mono) factorization systems]]
|
(p,q)-string | https://ncatlab.org/nlab/source/%28p%2Cq%29-string |
> under construction
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Higher spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[type II string theory]]: a [[bound state]] of $p$-[[F1-branes]] and $q$ [[D1-branes]].
Under [[S-duality]] the [[modular group]] $SL_2(\mathbb{Z})$ acts on these $(p,q)$-tuples and mixes the $F1$- and the $D1$-brane states.
[[!include F-branes -- table]]
## Related concepts
* [[S-duality]]
* [[F-theory]]
[[!include brane bound states -- table]]
## References
* [[Igor Bandos]], _Superembedding Approach and S-Duality. A unified description of superstring and super-D1-brane_, Nucl.Phys.B599:197-227,2001 ([arXiv:hep-th/0008249](http://arxiv.org/abs/hep-th/0008249))
[[!redirects (p,q)-strings]]
|
(p,q)5-brane | https://ncatlab.org/nlab/source/%28p%2Cq%295-brane |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Higher spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[type IIB string theory]] there are [[bound states]] of [[D5-branes]] with [[NS5-branes]]. The bound state of $p \in \mathbb{Z}$ D5-branes with $q \in\mathbb{Z}$ NS5-branes is then called a _$(p,q)$-fivebrane_ or similar.
As $p$ and $q$ varies the species of $(p,q)$5-branes form the [[lattice in a vector space|lattice]] $\mathbb{Z}^2$ and are naturally [[action|acted]] on by the [[S-duality]] group [[SL(2,Z)]].
## Properties
### $(p,q)$-Brane webs
Label local [[coordinate functions]] $x^a$ on 10d [[Minkowski spacetime]] $\mathbb{R}^{9,1}$ by $0123455'6789$ and write $v_a \coloneqq \partial_{x^a}$ for the corresponding [[vector field]]
Consider a $(1,0)$5-brane (a [[D5-brane]]) along the [[multivector field]]
$v_0 v_1 v_2 v_3 v_4 v_5$ and a $(0,1)$5-brane (an [[NS-brane]]) along $v_0 v_1 v_2 v_3 v_4 v_{5'}$
| $a =$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $5'$ | $6$ | $7$ | $8$ | $9$ |
|-------|-----|-----|-----|-----|-----|-----|------|-----|-----|-----|-----|
| D5 | --- | --- | --- | --- | --- | --- | | | | | |
| NS5 | --- | --- | --- | --- | --- | | --- | | | | |
Charge conservation implies that at the [[brane intersection]] of the two a $(1,1)$5-brane emerges stretched diagonally along $v_5 + v_{g'}$, i.e. along the [[multivector field]] $v_0 v_1 v_2 v_3 v_4 (v_5 + v_{5'})$
[Aharony-Hanany 97, Sec. 3](#AharonyHanany97)
<img src="https://ncatlab.org/nlab/files/D5NS5Junction.jpg" width="500"/>
The [[worldvolume]]-[[quantum field theory]] at the [[brane intersection]] point is a [[geometric engineering of quantum field theory|geometric engineering]] of [[D=5 N=1 SYM]] .
The [[T-duality|T-dual]] perspective are [[D4/NS5-brane webs]] ([Witten 97](#Witten97)).
The result of connecting several such [[brane intersections]] are called _$(p,q)$-5brane webs_ ([Aharony-Hanany-Krol 97](#AharonyHananyKrol97)).
## Related entries
* [[(p,q)-string]]
* [[D=5 N=1 SYM]]
[[!include brane bound states -- table]]
[[!include F-branes -- table]]
## References
The original articles are:
* {#AharonyHanany97} [[Ofer Aharony]], [[Amihay Hanany]], _Branes, Superpotentials and Superconformal Fixed Points_, Nucl. Phys. B504:239-271, 1997 ([arXiv:hep-th/9704170](https://arxiv.org/abs/hep-th/9704170))
* {#AharonyHananyKrol97} [[Ofer Aharony]], [[Amihay Hanany]], [[Barak Kol]], _Webs of $(p,q)$ 5-branes, Five Dimensional Field Theories and Grid Diagrams_, JHEP 9801:002,1998 ([arXiv:hep-th/9710116](http://arxiv.org/abs/hep-th/9710116))
* [[Oren Bergman]], Gabi Zafrir, _Lifting 4d dualities to 5d_, JHEP04 (2015) 141 ([arXiv:1410.2806](https://arxiv.org/abs/1410.2806))
The [[T-duality|T-dual]] perspective are [[D4/NS5-brane webs]]:
* {#Witten97} [[Edward Witten]], _Solutions Of Four-Dimensional Field Theories Via M Theory_, Nucl. Phys. B500:3-42, 1997 ([arXiv:hep-th/9703166](https://arxiv.org/abs/hep-th/9703166))
Further intersection with [[orientifolds]]:
* [[Amihay Hanany]], [[Alberto Zaffaroni]], _Issues on Orientifolds: On the brane construction of gauge theories with $SO(2n)$ global symmetry_, JHEP 9907 (1999) 009 ([arXiv:hep-th/9903242](https://arxiv.org/abs/hep-th/9903242))
* Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Masato Taki, Futoshi Yagi, _More on 5d descriptions of 6d SCFTs_, JHEP10 (2016) 126 ([arXiv:1512.08239](https://arxiv.org/abs/1512.08239))
* Gabi Zafrir, _Brane webs in the presence of an $O5^-$-plane and 4d class S theories of type D_, JHEP07 (2016) 035 ([arXiv:1602.00130](https://arxiv.org/abs/1602.00130))
* [[Taro Kimura]], Rui-Dong Zhu, Section 2 and 3 of _Web Construction of ABCDEFG and Affine Quiver Gauge Theories_ ([arXiv:1907.02382](https://arxiv.org/abs/1907.02382))
On $(p,q)$-5-branes as [[defect branes]]:
* Tetsuji Kimura, _Defect $(p,q)$ Five-branes_, Nucl.Phys. B893 (2015) 1-20 ([arXiv:1410.8403](https://arxiv.org/abs/1410.8403), [doi:10.1016/j.nuclphysb.2015.01.023](https://doi.org/10.1016/j.nuclphysb.2015.01.023))
[[!redirects (p,q)5-branes]]
[[!redirects (p,q)5-brane web]]
[[!redirects (p,q)5-brane webs]]
[[!redirects D4/NS5-brane webs]]
|
(sub)object classifier in an (infinity,1)-topos | https://ncatlab.org/nlab/source/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos |
> This page is about object classifier objects in [[(β,1)-toposes]]. For the unrelated notion of the [[classifying topos]] of the [[theory of objects]] see at _[[classifying topos for the theory of objects]]_.
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Universes
+-- {: .hide}
[[!include universe - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A crucial ingredient in a [[topos]] is a [[subobject classifier]]. From the point of view of [[homotopy theory]], that this has to do with [[subobjects]] turns out to be a coincidence of low dimensions: subobjects are [[(-1)-truncated]] morphisms.
As discussed also at [[stuff, structure, property]], the classifying objects in [[higher topos theory]] classify more general morphisms.
When one passes all the way to [[β-toposes]], there should be objects that classify _all_ morphisms, subject to some bound on size. This is made precise in the context of [[(β,1)-topos theory]].
One way to characterize an [[(β,1)-topos]] is as
* a [[presentable (β,1)-category]]
* with [[universal colimits]]
* such that for all sufficiently large [[regular cardinal]]s $\kappa$ there is a **classifying object** for the class of all $\kappa$-compact morphisms in $X$.
This statement is originally due to [[Charles Rezk]]. It is reproduced as ([[Higher Topos Theory|Lurie HTT, theorem 6.1.6.8]]).
In terms of [[homotopy type theory]] these object classifiers are _[[types of types]]_. See there for more details and see at _[[relation between category theory and type theory]]_.
+-- {: .num_remark #ReflectonOnCharacterizationByObjectClassifier}
###### Remark
An [[object classifier]] is a (small) _self-reflection_ of the $(\infty, 1)$-topos inside itself ([[type of types]], internal [[universe]]). It possesses an [[category object in an (infinity,1)-category|internal (β,1)-topos ]] structure. See also ([[Science of Logic#WesenAlsReflexionInIhmSelbst|WdL, book 2, section 1]]).
=--
## Definition
{#Definition}
Let $\mathcal{C}$ be a [[(β,1)-category]], and let $S$ be a class of [[1-morphisms]] of $\mathcal{C}$ which is stable under [[(β,1)-pullback]]. Then an **$S$-classifier** is a [[terminal object in an (β,1)-category|terminal object]] in the sub-[[arrow category|category of arrows]] $\mathcal{C}^{\Delta^1}$ of $S$ whose morphisms are [[(β,1)-pullback]] squares in $\mathcal{C}$.
Explicitly, an $S$-classifier consists of
* a morphism $\widehat {S Type} \longrightarrow S Type$ in $S$
* such that for each $X \stackrel{p}{\to} B$ in $S$, there exists an essentially unique [[(β,1)-pullback]] square of the form
$$
\array{
X &\longrightarrow& \widehat {S Type}
\\
\downarrow^{\mathrlap{p}} && \downarrow
\\
B &\stackrel{'X'}{\longrightarrow}& S Type
}
\,.
$$
in $\mathcal{C}$
Here $'X'$ may be called the _name_, or the _[[classifying morphism]]_, or the _[[modulating morphism]]_ or the _internal reflection_ of $X$ over $B$,
For example,
* When $S$ is the class of **all monomorphisms** in $C$, an $S$-classifier is called a **subobject classifier**. For instance, every topos has a subobject classifier.
* When $S$ is the class of **all faithful morphisms** in $C$, an $S$-classifier is called a **discrete object classifier**. For instance, every (2,1)-topos has a discrete object classifier.
* When $S$ is the class of **all morphisms** in $C$, an $S$-classifier is called an **object classifier**. However, due to size issues, interesting categories tend not to have such objects, which is one reason to be interested in the next example:
* When $S$ is the class of **all relatively $\kappa$-compact morphisms** (for some regular cardinal $\kappa$--see below for the definition), an $S$-classifier is called a **$\kappa$-compact-object classifier**.
**Note on terminology:** In all cases, the "things" classified by an "(adjectives) object classifier" are _arrows_ -- this is no different from the most famous case of _[[subobject classifiers]]_, which classify _monos_. For each object $X$, a _subobject classifier_ classifies the _subobjects of $X$_. For each object $X$, an _object classifier_ classifies the _objects over $X$_.
So with that $\kappa$ fixed, we may write
$$
\array{
\widehat Type
\\
\downarrow
\\
Type
}
$$
for such a "universal bundle of $\kappa$-small objects". Intuitively this is easy to describe: a point in $Type$ corresponds to a $\kappa$-small object, hence is the "name" $'X'$ or "code for" a $\kappa$-small object, and the [[fiber]] in $\widehat Type$ over that point is the very object $X$ itself.
If one gives the projection of the universal object bundle $\widehat Type \to Type$ a name, such as $El$, and writes $El^{-1}(-)$ for its preimages then $X \simeq El^{-1}('X')$. This is, with the ${(-)}^{-1}$-suppressed, the notation used at _[Type universes a la Tarski](type+of+types#TarskiStyle)_.
## Details
### n-truncated object classifier
+-- {: .num_defn}
###### Definition
Let $C$ be an [[(β,1)-category]] and $S \in C_1$ a class of [[morphisms]] that is stable under [[(β,1)-pullback]] in $C$.
Let $Cod_C$ be the [[codomain fibration]] of $X$, i.e. the [[(β,1)-category of (β,1)-functors]]
$$
Cod_C := Func(\Delta[1], C)
$$
equipped with the [[Cartesian fibration]] $Cod_C \to C$ induced from the endpoint inclusion $\Delta[0] \to \Delta[1]$.
Write
* $Cod_C^S$ for the full [[sub-(β,1)-category]] of $Cod_C$ on the object in $S$;
* $Cod_C^{(S)}$ the non-full subcategory whose objects are the elements of $S$, and whose morphisms are squares that are pullback diagrams.
Then evaluation at $\Delta[0] \to \Delta[1]$ yields
* a [[Cartesian fibration]] $Cod_C^S \to C$;
* a [[right fibration]] $Cod_C^{(S)} \to C$.
We say a morphism $f :x \to y$ in $C$ _classifies_ $S$ -- or simply that $y$ classifies $S$ -- if it is the [[terminal object]] in $Cod_C^{(S)}$.
=--
This is [[Higher Topos Theory|HTT, notation 6.1.3.4]] and [[Higher Topos Theory|HTT, def. 6.1.6.1 ]].
#### Subobject classifier {#DetailsSubObjClassf}
+-- {: .num_defn}
###### Definition
A **subobject classifier** for $C$ is an object that classifies the class $S$ of [[monomorphism in an (β,1)-category|monomorphisms]]/[[(-1)-truncated]] morphisms in $C$.
=--
This is ([[Higher Topos Theory|HTT, def. 6.1.6.1 ]]).
+-- {: .num_example}
###### Example
The $(\infty,1)$-category [[βGrpd]] has a a subobject classifier: the [[0-groupoid]]/[[set]] $\{\emptyset,*\}$ with two elements (the two [[(-1)-truncated]] $\infty$-groupoids).
=--
+-- {: .num_prop}
###### Proposition
Every [[(β,1)-topos]] has a [[subobject classifier]].
=--
This appears as ([[Higher Topos Theory|HTT, prop. 6.1.6.3]]) and the remark below that.
#### Discrete object classifier
+-- {: .num_defn}
###### Definition
A **discrete object classifier** for $C$ is an object that classifies the class $S$ of [[faithful morphism in an (β,1)-category|faithful morphism]]/[[(0)-truncated]] morphisms in $C$.
=--
+-- {: .num_example}
###### Example
The $(\infty,1)$-category [[βGrpd]] has a a discrete object classifier: the [[1-groupoid]]/[[groupoid]] [[Set]] whose elements are sets ([[0-truncated]] $\infty$-groupoids).
=--
+-- {: .num_prop}
###### Proposition
Every [[(β,1)-topos]] has a [[discrete object classifier]].
=--
### Object classifier {#DetailsObjClassf}
**Remark/Warning.** The point of having [[subobjects]] and hence [[monomorphisms]] classified by an object in an ordinary [[topos]] may be thought of as being solely due to the fact that in a 1-[[topos]], any object necessarily classifies a _[[poset]]_ i.e. a [[(0,1)-category]] of morphisms, and the point of subobjects/monomorphisms of a given object is that they do not have [[automorphisms]].
In an $(\infty,1)$-topos we thus expect an object that classifies _all_ morphisms, in that the assignment
$$
c \mapsto Core(C_{/c})
$$
of an object $c\in C$ to the [[core]] of its [[over quasi-category|over (β,1)-category]] yields a [[(β,1)-functor]] $C^{op} \to \infty Grpd$ that is [[representable functor|representable]].
Indeed, this is _essentially_ the case -- up to size issues, that the following definitions take care of.
+-- {: .num_defn #RelativelyKappaCompact}
###### Definition
For $\kappa$ some [[cardinal]], say a morphism $f : x \to y$ in $C$ is **[[relatively k-compact morphism in an (infinity,1)-category|relatively k-compact]]** if for all [[(β,1)-pullbacks]] along $h : y' \to y$ to $\kappa$-[[compact object in an (β,1)-category|compact object]]s, $y'$, the pulled back object $h^* x'$ is itself a $\kappa$-compact object.
=--
+-- {: .num_theorem}
###### Theorem
A [[presentable (β,1)-category]] $C$ is an [[(β,1)-topos]] precisely if
1. it has [[universal colimits]];
1. for sufficiently large regular [[cardinal]]s $\kappa$, $C$ has a classifying object for relatively $\kappa$-compact morphisms.
=--
This is due to [[Charles Rezk]]. The statement appears as [[Higher Topos Theory|HTT, theorem 6.1.6.8]].
The proof essentially consists of showing that by the [[adjoint functor theorem]] (specifically, the [[representable functor theorem]]), the existence of object classifiers is equivalent to [[continuous functor|continuity]] of the [[core]] [[self-indexing]] $C^{op} \to \infty Gpd$ defined by $x\mapsto Core(C/x)$. In the presence of universal colimits, this latter condition is equivalent to all colimits being [[van Kampen colimits]], which in turn yields the connection to the Giraud-type exactness properties.
## Examples
### Object classifier in $\infty Grpd$
{#ObjectClassifierInInfinityGroupoid}
We discuss that the $\kappa$-small object classifier in the $(\infty,1)$-topos
[[βGrpd]] of [[β-groupoids]] is itself the [[core]] of the [[(β,1)-category]] $\infty Grpd_\kappa$ of $\kappa$-small $\infty$-groupoids. Observing that the connected components of this are the [[delooping]] $B Aut(F)$ of the [[automorphism β-group]] of a given [[homotopy type]] $[F]$, and using that [[βGrpd]] is [[presentable (β,1)-category|presented]] by [[Top]] $\simeq$ [[sSet]] (see also at _[[homotopy hypothesis]]_) this recovers classical theorems about the classification of [[fibrations]] in simplicial sets/topological spaces by a [[universal Kan fibration]], as listed in the [References](http://ncatlab.org/nlab/show/associated+infinity-bundle#References) at _[[associated β-bundle]]_.
+-- {: .num_prop }
###### Proposition
The $\kappa$-compact object classifier in [[βGrpd]]
is
$$
Type_\kappa := Core(\infty Grpd_\kappa)
\,,
$$
the [[core]] of the [[full sub-(β,1)-category]] of [[βGrpd]] on the $\kappa$-[[small β-groupoids]].
The corresponding [[universal bundle]] is presented by the map of [[simplicial sets]]
$$
\widehat Type_\kappa \to Type_\kappa
$$
which is the [[pullback]] of simplicial sets
$$
\array{
\widehat Type_\kappa &\to& Z_{\infty Grpd}
\\
\downarrow && \downarrow
\\
Type_\kappa &\to& \infty Grpd
}
$$
of the [[universal right fibration]] along the defining inclusion of (the [[Kan complex]] presenting) $Type_\kappa$.
=--
+-- {: .num_lemma #RelativelyCompactInInfinityGroupods}
###### Lemma
In [[βGrpd]] the relatively $\kappa$-compact morphisms, $X \to Y$, def. \ref{RelativelyKappaCompact}, are precisely those all whose [[homotopy fibers]]
$$
X_{y} := X \times_{Y} \{y\}
$$
over all [[objects]] $y \in Y$ are $\kappa$-[[small infinity-groupoids]].
=--
+-- {: .proof}
###### Proof
We may write $Y$ as an [[(β,1)-colimit]] over itself (see there)
$$
Y \simeq {\lim_{\to}}_{y \in Y} \{y\}
$$
and then use the fact that [[βGrpd]] -- being an [[(β,1)-topos]] -- has
[[universal colimits]], to obtain the [[(β,1)-pullback]] diagram
$$
\array{
{\lim_{\to}}_{y \in Y} X_y
&\stackrel{\simeq}{\to}
& X
\\
\downarrow && \downarrow
\\
{\lim_{\to}}_{y \in Y} \{y\}
&\stackrel{\simeq}{\to}&
Y
}
$$
exhibiting $X$ as an $(\infty,1)$-colimit of $\kappa$-small objects
over $Y$. By stability of $\kappa$-compact objects under
$\kappa$-small colimits (see [here](http://ncatlab.org/nlab/show/compact+object+in+an+%28infinity%2C1%29-category#StabilityUnderColimits))
it follows that $X$ is $\kappa$-compact if $Y$ is.
=--
+-- {: .proof}
###### Proof of the proposition
Since [[right fibrations]] are stable under pullback (see [here](http://ncatlab.org/nlab/show/right%2Fleft+Kan+fibration#PreservationByPullback)), this is still a right fibration. Since, up to equivalence, every morphism into a [[Kan complex]] is a
right fibration (see [here](http://ncatlab.org/nlab/show/right%2Fleft+Kan+fibration#OverKanComplex)), and since every
morphism out of a Kan complex into $\infty Grpd_\kappa$ factors
through the core $Type_\kappa$ it follows that
$Type_\kappa$ classifies all morphisms $X \to Y$ in [[βGrpd]] whose [[homotopy fibers]]
$$
X_y \simeq X \times_Y \{y\}
$$
are $\kappa$-[[compact object in an (β,1)-category|compact]].
The claim then follows with lemma \ref{RelativelyCompactInInfinityGroupods}.
=--
### Object classifier in presheaf $(\infty,1)$-toposes
Let $C$ be an [[(β,1)-category]] and $\mathbf{H} = PSh_{\infty}(C)$ the [[(β,1)-category of (β,1)-presheaves]] over $C$.
By the [[(β,1)-Yoneda lemma]],
the $\kappa$-compact object classifier here should be
the presheaf which assigns to $U \in C$ the
$\infty$-groupoid of relatively $\kappa$-compact
morphisms $X \to U$ in $PSh_\infty(C)$.
## Related concepts
* [[univalent foundations for mathematics]]
* [[type of propositions]], [[subobject classifier]], [[partial map classifier]]
* [[type of sets]], [[category of sets]], [[discrete object classifier]]
* [[type of types]], [[univalence]]
* [[classifying morphism]]
* [[Awodey's conjecture]]
## References
The general notion is due to:
* [[Jacob Lurie]], section 6.1.6 of _[[Higher Topos Theory]]_ (2009)
The object classifier in the archetypical special case of the $\infty$-topos [[βGrpd]] of [[infinity-groupoids|$\infty$-groupoids]], seen in the [[classical model structure on simplicial sets]], is discussed in
* {#KapulkinLumsdaineVoevodsky12} [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], [[Vladimir Voevodsky]], *Univalence in simplicial sets* [[arXiv:1203.2553](http://arxiv.org/abs/1203.2553)]
* {#KapulkinLumsdaine21} [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], *The Simplicial Model of Univalent Foundations (after Voevodsky)*, Journal of the European Mathematical Society **23** (2021) 2071β2126 $[$[arXiv:1211.2851](https://arxiv.org/abs/1211.2851), [doi:10.4171/jems/1050](https://doi.org/10.4171/jems/1050)$]$
as [[categorical semantics]] for [[univalence|univalent]] [[type universes]] in [[homotopy type theory]].
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+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Idea
In as far as an [[algebraic theory]] or [[Lawvere theory]] is nothing but a [[small category]] with finite [[product]]s and an [[algebra]] for the theory a product-preserving [[functor]] to [[Set]], the notion has an evident generalization to [[higher category theory]] and in particular to [[(β,1)-category]] theory.
## Definition
+-- {: .num_defn}
###### Definition
An **$(\infty,1)$-Lawvere theory** is (given by a syntactic $(\infty,1)$-category that is) an [[(β,1)-category]] $C$ with finite [[limit in a quasi-category|(β,1)-product]]s. An $(\infty,1)$-algebra for the theory is an [[(β,1)-functor]] $C \to $ [[βGrpd]] that preserves these products.
The $(\infty,1)$-category of [[β-algebras over an (β,1)-algebraic theory]] is the full [[sub-(β,1)-category]]
$$
Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C^{op})
$$
of the [[(β,1)-category of (β,1)-presheaves]] on $C^{op}$ on the product-preserving $(\infty,1)$-functors
=--
In a full $(\infty,1)$-category theoretic context this appears as [[Higher Topos Theory|HTT, def. 5.5.8.8]]. A definition in terms of [[simplicially enriched categories]] and the [[model structure on sSet-categories]] to present $(\infty,1)$-categories is in [Ros]((http://www.math.muni.cz/~rosicky/papers/infty6.pdf). The introduction of that article lists further and older occurences of this definition.
## Properties
{#Properties}
+-- {: .num_prop #PropertiesOfInfinityCategoriesOfAlgebras}
###### Proposition
Let $C$ be an [[(β,1)-category]] with finite [[limit in a quasi-category|products]]. Then
* $Alg_{(\infty,1)}(C)$ is an [[accessible (β,1)-functor|accessible]] [[localization of an (β,1)-category|localization]] of the [[(β,1)-category of (β,1)-presheaves]] $PSh_{(\infty,1)}(C^{op})$ (on the [[opposite (β,1)-category|opposite]]).
So in particular it is a [[locally presentable (β,1)-category]].
* $Alg_{(\inft)}$ is a [[compactly generated (β,1)-category]].
* The $(\infty,1)$-[[Yoneda embedding]] $j : C^{op} \to PSh_{(\infty,1)}(C^{op})$ factors through $Alg_{(\infty,1)}(C)$.
* The full subcategory $Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C)$ is stable under [[sifted colimit]]s.
=--
This is [[Higher Topos Theory|HTT, prop. 5.5.8.10]].
## Models {#Models}
There are various [[model category]] [[presentable (β,1)-category|presentations]] of $Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C^{op})$.
Recall that the [[(β,1)-category of (β,1)-presheaves]] $PSh_{(\infty,1)}(C^{op})$ itself is modeled by the [[model structure on simplicial presheaves]]
$$
PSh_{(\infty,1)}(C^{op})
\simeq
[T, sSet]^\circ
\,,
$$
where we regard $T$ as a [[Kan complex]]-[[enriched category]] and have on the right the [[sSet]]-[[enriched functor category]] with the projective or injective model structure, and $(-)^\circ$ denoting the full enriched subcategory on fibrant-cofibrant objects.
This says in particular that every weak $(\infty,1)$-functor $f : T \to \infty \mathrm{Grp}$ is equivalent to a _rectified_ on $F : T \to KanCplx$. And $f \in PSh_{(\infty,1)}(C^{op})$ belongs to $Alg_{(\infty,1)}(C)$ if $F$ preserves finite products _weakly_ in that for $\{c_i \in C\}$ a finite collection of objects, the canonical natural morphism
$$
F(c_1 \times \cdots\times \c_n)
\to
F(c_1) \times \cdots \times F(c_n)
$$
is a [[homotopy equivalence]] of [[Kan complex]]es.
If $T$ is an ordinary category with products, hence an ordinary [[Lawvere theory]], then such a functor is called a **[[homotopy T-algebra]]**. There is a model category structure on these (see there).
We now look at model category structure on _strictly_ product preserving functors $C \to sSet$, which gives an equivalent model for $Alg_{(\infty,1)}(C)$. See [[model structure on simplicial T-algebras]].
+-- {: .num_prop}
###### Proposition
Let $C$ be a [[category]] with finite [[product]]s, and let $sTAlg \subset Func(C,sSet)$ be the [[full subcategory]] of the [[functor category]] from $C$ to [[sSet]] on those functors that preserve these products.
Then $sAlg(C)$ carries the structure of a [[model category]] $sAlg(C)_{proj}$ where the weak equivalences and the fibrations are objectwise those in the standard [[model structure on simplicial sets]].
=--
This is due to ([Quillen](#Quillen)).
The inclusion $i : sAlg(C) \hookrightarrow sPSh(C^{op})_{proj}$ into the projective [[model structure on simplicial presheaves]] evidently preserves fibrations and acylclic fibrations and gives a [[Quillen adjunction]]
$$
sAlg(C)_{proj}
\stackrel{\leftarrow}{\underset{i}{\hookrightarrow}}
sPSh(C^{op})
\,.
$$
+-- {: .num_prop}
###### Proposition
The total right [[derived functor]]
$$
\mathbb{R}i : Ho(sAlg(C)_{proj}) \to Ho(sPSh(C^{op})_{proj})
$$
is a [[full and faithful functor]] and an object $F \in sPSh(C^{op})$ belongs to the [[essential image]] of $\mathbb{R}i$ precisely if it preserves products up to [[weak homotopy equivalence]].
=--
This is due to ([Bergner](#Bergner)).
It follows that the natural $(\infty,1)$-functor
$$
(sAlg(C)_{proj})^\circ \stackrel{}{\to}
PSh_{(\infty,1)}(C^{op})
$$
is an [[equivalence of quasi-categories|equivalence]].
A comprehensive statement of these facts is in [[Higher Topos Theory|HTT, section 5.5.9]].
## Examples
### Simplicial 1-algebras
For $T$ (the [[syntactic category]] of) an ordinary [[algebraic theory]] (a [[Lawvere theory]]) let $T Alg$ be the category of its ordinary algebras, the ordinary product-preserving functors $T \to Set$.
We may regard $T$ as an $(\infty,1)$-category and consider its $(\infty,1)$-algebras. By the above discussion, these are modeled by product-presering functors $T \to sSet$. But this are equivalently [[simplicial object]]s in $T$-algebras
$$
[T, sSet]_\times \simeq T Alg^{\Delta^{op}}
\,.
$$
There is a standard [[model structure on simplicial T-algebras]] and we find that simplicial $T$-1-algebras model $T$-$(\infty,1)$-algebras.
### Homotopy $T$-algebras
For $T$ an ordinary Lawvere theory, there is also a model category structure on ordinary functors $T \to sSet$ that preserve the products only up to weak equivalence. Such functors are called [[homotopy T-algebra]]s.
This model structure is equivalent to the [[model structure on simplicial T-algebras]] (see [[homotopy T-algebra]] for details) but has the advantage that it is a left [[proper model category]].
### Simplicial theories
There is a notion of _simplicial algebraic theory_ that captures some class of $(\infty,1)$-algebraic theories. For the moment see section 4 of ([Rezk](#Rezk))
### Structure-$(\infty,1)$-sheaves
A pre[[geometry (for structured (β,1)-toposes)]] is a (multi-sorted) $(\infty,1)$-algebraic theory. A _structure $(\infty,1)$-sheaf_ on an [[(β,1)-topos]] $\mathcal{X}$ in the sense of [[structured (β,1)-topos]]es is an $\infty$-algebra over this theory
$$
\mathcal{O} : \mathcal{T} \to \mathcal{X}
$$
in the $(\infty,1)$-topos $\mathcal{X}$ -- a special one satisfying extra conditions that make it indeed behave like a sheaf of _function algebras_ .
### Symmetric monoidal $(\infty,1)$-Categories and $E_\infty$-algebras {#EInfty}
There is a $(2,1)$-algebraic theory whose algebras in
[[(β,1)Cat]] are [[symmetric monoidal (β,1)-categories]]. Hence monoids in these algebras are [[E-β algebra]]s (see [[monoid in a monoidal (β,1)-category]]).
This is in ([Cranch](#Cranch)). For more details see
[[(2,1)-algebraic theory of E-infinity algebras]].
## Related concepts
* [[algebraic theory]] / [[Lawvere theory]] / [[essentially algebraic theory]]
* [[2-Lawvere theory]]
* **algebraic $(\infty,1)$-theory** / [[essentially algebraic (β,1)-theory]]
* [[simplicial Lawvere theory]]
* [[monad]] / [[(β,1)-monad]]
* [[operad]] / [[(β,1)-operad]]
## References
The model structure presentation for the $(\infty,1)$-category of $(\infty,1)$-algebras goes back all the way to
* [[Dan Quillen]], _Homotopical Algebra_ Lectures Notes in Mathematics 43, SpringerVerlag, Berlin, (1967)
{#Quillen}
A characterization of $(\infty,1)$-categories of $(\infty,1)$-algebras in terms of [[sifted colimit]]s is given in
* J. Rosicky _On homotopy varieties_ ([pdf](http://www.math.muni.cz/~rosicky/papers/infty6.pdf))
using the incarnation of $(\infty,1)$-categories as [[simplicially enriched categories]].
An $(\infty,1)$-categorical perspective on these homotopy-algebraic theories is given in
* [[Andre Joyal]], _The theory of quasi-categories and its applications_, lectures at CRM Barcelona February 2008, draft [hc2.pdf](http://www.crm.cat/HigherCategories/hc2.pdf#page=44)_
from page 44 on.
A detailed account in the context of a general theory of [[(β,1)-category of (β,1)-presheaves]] is the context of section 5.5.8 of
* [[Jacob Lurie]], _[[Higher Topos Theory]]_ .
The [[model category]] [[presentable (infinity,1)-category|presentations]] of $(\infty,1)$-algebras is studied in
* [[Charles Rezk]], _Every homotopy theory of simplicial algebras admits a proper model_ ([math/0003065](http://arxiv.org/abs/math/0003065)) ,
{#Rezk}
where it is shown that every such model is [[Quillen equivalence|Quillen equivalent]] to a left [[proper model category]]. The article uses a monadic definition of $(\infty,1)$-algebras.
A discussion of [[homotopy T-algebra]]s and their strictification is in
* [[Bernard Badzioch]], _Algebraic theories in homotopy theory_ Annals of Mathematics, 155 (2002), 895-913 ([JSTOR](http://www.jstor.org/stable/3062135))
and for multi-sorted theories in
* [[Julie Bergner]], _Rigidification of algebras over multi-sorted theories_ , Algebraic and Geometric Topoogy 7, 2007.
{#Bergner}
A discussion of [[E-β algebra]]-structures in terms of $(\infty,1)$-algebraic theories is in
* [[James Cranch]], _Algebraic Theories and $(\infty,1)$-Categories_ ([arXiv](http://arxiv.org/abs/1011.3243))
{#Cranch}
See also
* [MO discussion](http://mathoverflow.net/questions/118500/what-is-a-simplicial-commutative-ring-from-the-point-of-view-of-homotopy-theory)
Expressed as a higher form of [[Lawvere theory]] see
* [[John D. Berman]], _Higher Lawvere theories_ ([arXiv:1903.02991](https://arxiv.org/abs/1903.02991))
[[!redirects (infinity,1)-algebraic theory]]
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[[!redirects (β,1)-algebraic theories]]
[[!redirects algebraic (β,1)-theory]]
[[!redirects algebraic (β,1)-theories]]
[[!redirects infinity1-algebraic theory]]
|
(β,1)-category of (β,1)-categories > history | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-categories+%3E+history | < [[(β,1)-category of (β,1)-categories]] |
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+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
For $R \in CRing_\infty$ an [[E-β ring]], the [[(β,1)-modules]] over $R$ with [[homomorphisms]] between them form an [[(β,1)-category]], the _$(\infty,1)$-category of $(\infty,1)$-modules over $R$_.
## Properties
### Compact generation, dualizable and perfect objects
{#CompactGeneration}
+-- {: .num_prop}
###### Propositon
Let $R$ be an [[A-β ring]]. The (β,1)-category of β-modules $R Mod$ is a [[compactly generated (β,1)-category]] and the [[compact object in an (β,1)-category|compact objects]] coincide with the [[perfect modules]]
If $R$ is commutative ([[E-β ring|E-β]]) then the perfect modules (and hence the compact objects) also coincide with the [[dualizable objects]].
=--
The first statement is ([[Higher Algebra|HA, prop. 7.2.4.2]]), the second ([[Higher Algebra|HA, prop. 7.2.4.4]]). For [[chain complexes]] this also appears as ([BFN 08, lemma 3.5](#BFN08)).
### Stable Dold-Kan correspondence
{#StableDoldKan}
For $R$ an ordinary [[ring]], write $H R$ for the corresponding [[Eilenberg-MacLane spectrum]].
+-- {: .num_theorem #StableDoldKan}
###### Theorem
For $R$ any [[ring]] (or [[ringoid]], even) there is a [[Quillen equivalence]]
$$
H R Mod \simeq Ch_\bullet(R Mod)
$$
between model structure on $H R$-module spectra and the [[model structure on chain complexes]] (unbounded) of ordinary $R$-[[module]]s.
This presents a corresponding [[equivalence of (β,1)-categories]]. If $R$ is a commutative ring, then this is an equivalence of [[symmetric monoidal (β,1)-categories]].
=--
This equivalence on the level of [[homotopy categories]] is due to ([Robinson](#Robinson)). The refinement to a Quillen equivalence is ([SchwedeShipley, theorem 5.1.6](#SchwedeShipley)). See also the discussion at _[[stable model categories]]_. A direct description as an equivalence of $(\infty,1)$-categories appears as ([Lurie, theorem 7.1.2.13](#Lurie)).
+-- {: .num_remark}
###### Remark
This is a stable version of the [[Dold-Kan correspondence]].
=--
See at __[[algebra spectrum]]_ for the corresponding statement for $H R$-algebra spectra and [[dg-algebras]].
### Periodicity
For $E$ a [[periodic ring spectrum]], then $E Mod$ ought to inherit a $\mathbb{Z}/2\mathbb{Z}$-[[β-action]]. See at _[periodic ring spectrum -- Periodicity of modules](periodic+ring+spectrum#PeriodicityOnModules)_
## Related concepts
* [[Mod]]
* [[2Mod]]
* [[nMod]]
## References
Modules over algebras over an arbitrary [[(β,1)-operad]] are discussed in section 3.3 of
* [[Jacob Lurie]], _[[Higher Algebra]]_
Modules specifically over [[A-β algebras]] are discussed in section 4.2 there.
The equivalence between the [[homotopy categories]] of $H R$-module spectra and $Ch_\bullet(R Mod)$ is due to
* Alan Robinson, _The extraordinary derived category_ , Math. Z. 196 (2) (1987) 231–238.
{#Robinson}
The refinement of this statement to a [[Quillen equivalence]] is due to
* [[Stefan Schwede]], [[Brooke Shipley]], _Stable model categories are categories of modules_ , Topology 42 (2003), 103-153 ([pdf](http://www.math.uic.edu/~bshipley/classTopFinal.pdf))
{#SchwedeShipley}
Discussion in the context of [[derived algebraic geometry]] includes
* {#BFN08} [[David Ben-Zvi]], [[John Francis]], [[David Nadler]], section 3.1 of _Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry_, J. Amer. Math. Soc. 23 (2010), no. 4, 909-966 ([arXiv:0805.0157](http://arxiv.org/abs/0805.0157))
[[!redirects (β,1)-categories of (β,1)-modules]]
[[!redirects (β,1)-category of β-modules]]
[[!redirects (β,1)-categories of β-modules]]
[[!redirects (infinity,1)-category of (infinity,1)-modules]]
[[!redirects (infinity,1)-categories of (infinity,1)-modules]]
[[!redirects (β,1)-category of modules]]
[[!redirects (β,1)-categories of modules]]
[[!redirects (infinity,1)-category of modules]]
[[!redirects (infinity,1)-categories of modules]]
[[!redirects (β,1)Mod]]
|
(β,1)-comparison lemma | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29-comparison+lemma |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Category theory
+-- {: .hide}
[[!include quasi-category theory contents]]
=--
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
# The $(\infty,1)$-comparison lemma
* table of contents
{: toc}
## Idea
The **(β,1)-comparison lemma** says that, under certain conditions, a functor between [[(β,1)-sites]] induces an equivalence between the categories of [[(β,1)-sheaves]] on the sites.
In [this paper](#Hoyois2013), Lemma C.3, Hoyois proves the following comparison lemma.
+-- {: .num_theorem #Hoyois2013}
###### Lemma
Let $D$ be a [[locally small]] [[(β,1)-category]], $C$ a [[small]] (β,1)-category, and $u : C \to D$ a fully faithful functor. Let $\tau$ and $\rho$ be quasi-topologies on $C$ and $D$, respectively. Suppose that:
a. Every $\tau$-[[sieve]] is generated by a cover $\{U_i \to X\}$ such that:
1. the fiber products $U_{i_0} \times_X \cdots\times_X U_{i_n}$ exist and are preserved by $u$;
2. $\{u(U_i) \to u(X)\}$ is a $\bar\rho$-cover.
b. For every $X \in C$ and every $\rho$-sieve $R \hookrightarrow u(X)$, $u^*(R) \hookrightarrow X$ is a $\bar\tau$-sieve in $C$.
c. Every $X \in D$ admits a $\bar\rho$-cover $\{U_i \to X\}$ such that the fiber products $U_{i_0} \times_X \cdots \times_X U_{i_n}$ exist and belong to the essential image of $u$.
Then the adjunction $u^* \dashv u_*$ restricts to an equivalence of β-categories $Shv_\rho(D) \simeq Shv_\tau(C)$.
=--
Here, a _quasi-topology_ is a collection of sieves closed under pullback and $\bar\tau$ is the coarsest topology containing a quasi-topology $\tau$. The stability under pullback ensures that $Shv_\tau(C)=Shv_{\bar\tau}(C)$.
## Generalization
It seems difficult to find a useful generalization not assuming the existence of some pullbacks. For the conclusion of the [lemma](#Hoyois2013), the following conditions (b is unchanged) are both necessary and sufficient:
a. For every $\tau$-sieve $U \hookrightarrow X$, $a_\rho u_!(U \to X)$ is an equivalence.
b. For every $X \in C$ and every $\rho$-sieve $R \hookrightarrow u(X)$, $u^*(R) \hookrightarrow X$ is a $\bar\tau$-sieve in $C$.
c. For every $X \in D$, its image in $Shv_\rho(D)$ belongs to the smallest subcategory generated by the image of $C$ under colimits.
We can take these conditions a and b to define, respectively, the notions of [[cover-preserving functor]] (continuous functor) and [[comorphism of sites]] (cocontinuous functor) for (β,1)-sites.
## Related concepts
* [[comparison lemma]]
* [[dense subsite]]
## References
* [[Marc Hoyois]], _A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula_. [arXiv:1309.6147](https://arxiv.org/abs/1309.6147)
{#Hoyois2013} |
(β,1)-end | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29-end |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{: toc}
## Idea
This is the generalization of the notion of *[[ends]]* from [[category theory]] to [[(β,1)-category theory]].
## Definition
### End of a space-valued functor
Given an [[(β,1)-functor]] $F \colon J^{op} \times J \to \infty Gpd$ to [[βGroupoids]] from the [[product category|product]] of a [[locally small (β,1)-category]] $J$ with its [[opposite (β,1)-category]], we define its *end* to be the [[(β,1)-categorical hom-space]] in the [[(β,1)-category of (β,1)-functors]] from the [[hom-functor]] of $J$ to $F$:
$$
\int_J F
\coloneqq
\infty Gpd^{J^{op} \times J}
\big(
J(-,-), F(-,-)
\big)
$$
### Ends and coends as a weighted (co)limit
More generally, given an [[(β,1)-functor]] $F \colon J^{op} \times J \to C$ to [[βGroupoids]] to any [[locally small (β,1)-category]] $C$,
we define its (co)end, when it exists, as the (co)[[representing object]] of
$$
C\left(c, \int_J F \right)
\simeq
\int_J C(c, F)
\simeq
\infty Gpd^{J^{op} \times J}
\big(
J(-,-), C(c, F(-,-))
\big)
$$
$$
C\left(\int^J F, c \right) \simeq
\int_{J^{op}} C(F, c) \simeq
\infty Gpd^{J \times J^{op}}( J^{op}(-,-), C(F(-,-), c) )
$$
When $C = \infty Gpd$, this definition agrees with the previous.
### As (co)limits over the twisted arrow category
Let $F : J^{op} \times J \to C$ be a functor between (β,1)-categories.
We can express the (co)end as (co)limits over the [[twisted arrow (β,1)-category]] by
$$
\int_J F \simeq \lim \left( Tw(J) \xrightarrow{(src, tgt)} J^{op} \times J
\xrightarrow{F} C \right)
$$
$$
\int^J F \simeq \colim \left( Tw(J^{op})^{op} \xrightarrow{(src, tgt)} J^{op} \times J
\xrightarrow{F} C \right)
$$
The second of these, for example, follows from the fact the projection
$p : Tw(J^{op}) \to J \times J^{op}$ is the left fibration classified by $J^{op}(-,-)$, and so we can write the hom-functor as the left Kan extension $p_!(1) = J^{op}(-,-)$ and take an adjoint transpose.
## Properties
+-- {: .num_prop }
###### Proposition (Fubini)
Given a functor $F : J^\op \times J \times K^\op \times K \to C$, if the relevant ends/coends exist, we have natural equivalences
$$
\int_J \int_K F \simeq \int_{J \times K} F
\qquad \qquad
\int^J \int^K F \simeq \int^{J \times K} F
$$
=--
+-- {: .proof}
###### Proof
For ends, using the fact $Tw$ preserves products we can compute
$$
\begin{aligned}
\int_J \int_K F
&\simeq \lim_{j \to j' \in Tw(J)} \lim_{k \to k' \in Tw(K)} F(j,j',k,k')
\\&\simeq \lim_{(j,k) \to (j',k') \in \Tw(J \times K)} F(j,j',k,k')
\\&\simeq \int_{J \times K} F
\end{aligned}
$$
The same argument applies for coends.
=--
+-- {: .num_prop }
###### Proposition
Given functors $F,G : X \to Y$ between two (β,1)-categories, the space of natural transformations between them is given by the end
$$ Y^X(F, G) \simeq \int_{x \in X} Y(F(x), G(x)) $$
=--
This is proposition 5.1 of [Gepner-Haugseng-Nikolaus 15](#GepnerHaugsengNikolaus15).
## Related concepts
* [[twisted arrow (β,1)-category]]
* [[lax (β,1)-colimit]]
#References
* {#GepnerHaugsengNikolaus15} [[David Gepner]], [[Rune Haugseng]], [[Thomas Nikolaus]], _Lax colimits and free fibrations in $\infty$-categories_ ([arXiv:1501.02161](http://arxiv.org/abs/1501.02161))
[[!redirects (β,1)-ends]]
[[!redirects (β,1)-coend]]
[[!redirects (β,1)-coends]] |
(β,1)-limit | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29-limit |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of [[limit]] and [[colimit]] generalize from [[category theory]] to [[(β,1)-category theory]]. One model for [[(β,1)-categories]] are [[quasi-categories]]. This entry discusses limits and colimits in quasi-categories.
## Definition
+-- {: .num_defn}
###### Definition
For $K$ and $C$ two [[quasi-category|quasi-categories]] and $F : K \to
C$ an [[(β,1)-functor]] (a morphism of the underlying [[simplicial set]]s) , the **limit** over $F$ is, if it exists, the [[terminal object in a quasi-category|quasi-categorical terminal object]] in the [[over quasi-category]] $C_{/F}$:
$$
\underset{\leftarrow}{\lim} F \coloneqq TerminalObj(C_{/F})
$$
(well defined up to a contractible space of choices).
A **colimit** in a quasi-category is accordingly a limit in the [[opposite quasi-category]].
=--
+-- {: .num_remark}
###### Remark
Notice from the discussion at [[join of quasi-categories]] that there are two definitions -- denoted $\star$ and $\diamondsuit$ -- of join, which yield results that differ as simplicial sets, though are equivalent as quasi-categories.
The notation $C_{/F}$ denotes the definition of [[over quasi-category]] induced from $\star$, while the notation $C^{/F}$ denotes that induced from $\diamondsuit$. Either can be used for the computation of limits in a quasi-category, as for quasi-categorical purposes they are weakly equivalent.
So we also have
$$
\underset{\leftarrow}{\lim} F \coloneqq TerminalObj(C^{/F})
\,.
$$
See [[Higher Topos Theory|HTT, prop 4.2.1.5]].
=--
## Properties
### In terms of slice $\infty$-categories
{#InTermsOfSlices}
+-- {: .num_prop}
###### Proposition
Let $\mathcal{C}$ be a [[quasi-category]] and let
$f \colon K \to \mathcal{C}$ be a [[diagram]] with $(\infty,1)$-colimiting [[cocone]] $\tilde f \colon K \star \Delta^0 \to \mathcal{C}$. Then the induced map of [[slice (β,1)-categories|slice quasi-categories]]
$$
\mathcal{C}_{/\tilde f} \to \mathcal{C}_{f}
$$
is an [[acyclic Kan fibration]].
=--
+-- {: .num_prop #SlicingOverLimitingCone}
###### Proposition
For $F \colon \mathcal{K} \to \mathcal{C}$ a diagram in an $(\infty,1)$-category and $\underset{\leftarrow}{\lim} F$ its limit, there is a natural [[equivalence of (β,1)-categories]]
$$
\mathcal{C}_{/F} \simeq \mathcal{C}_{/\underset{\leftarrow}{\lim} F}
$$
between the [[slice (β,1)-categories]] over $F$ (the $(\infty,1)$-category of $\infty$-[[cones]] over $F$) and over just $\underset{\leftarrow}{\lim}F$.
=--
+-- {: .proof}
###### Proof
Let $\tilde F \colon \Delta^0 \star \mathcal{K} \to \mathcal{C}$
be the limiting cone. The canonical cospan of $\infty$-functors
$$
\ast \to \Delta^0 \star \mathcal{K} \leftarrow \mathcal{K}
$$
induces a span of slice $\infty$-categories
$$
\mathcal{C}_{/\underset{\leftarrow}{\lim}F}
\leftarrow
\mathcal{C}_{/\tilde F}
\rightarrow
\mathcal{C}_{/F}
\,.
$$
The right functor is an equivalence by prop. \ref{SlicingOverLimitingCone}. The left functor is induced by restriction along an op-[[final (β,1)-functor]] (by the Examples discussed there) and hence is an equivalence by the discussion at _[[slice (β,1)-category]]_ ([Lurie, prop. 4.1.1.8](#Lurie)).
=--
This appears for instance in ([Lurie, proof of prop. 1.2.13.8](#Lurie)).
### In terms of $\infty$-Hom adjunction
The definition of the limit in a quasi-category in terms of terminal objects in the corresponding [[over quasi-category]] is well adapted to the particular nature the incarnation of $(\infty,1)$-categories by quasi-categories. But more intrinsically in $(\infty,1)$-category theory, it should be true that there is an [[adjunction]] characterization of
$(\infty,1)$-limits : limit and colimit, should be (pointwise or global) [[right adjoint|right]] and [[left adjoint|left]]
[[adjoint (infinity,1)-functor]] of the constant diagram $(\infinity,1)$-functor,
$const : C \to Func(K,C)$.
$$
(colim \dashv const \dashv lim)
:
Func(K,C)
\stackrel{\overset{lim}{\to}}{\stackrel{\overset{const}{\leftarrow}}
{\underset{colim}{\to}}}
Func(*,C) \simeq C
\,.
$$
By the discussion at [[adjoint (β,1)-functor]] ([[Higher Topos Theory|HTT, prop. 5.2.2.8]]) this requires exhibiitng a morphism $\eta : Id_{Func(K,C)} \to const colim$ in $Func(Func(K,C),Func(K,C))$ such that for every $f \in Func(K,C)$ and $Y \in C$
the induced morphism
$$
Hom_{C}(colim(f),Y) \to Hom_{Func(K,C)}(const colim(f), const Y)
\stackrel{Hom(\eta, const Y)}{\to}
Hom_{Func(K,Y)}(f, const Y)
$$
is a weak equivalence in $sSet_{Quillen}$.
But first consider the following pointwise characterization.
+-- {: .num_prop}
###### Proposition
Let $C$ be a [[quasi-category]], $K$ a [[simplicial set]]. A [[co-cone]] [[diagram]]
$\bar p : K \star \Delta[0] \to C$ with cone point $X \in C$
is a [[colimit|colimiting]] [[diagram]] (an initial object in
$C_{p/}$) precisely if for every object $Y \in C$ the morphism
$$
\phi_Y : Hom_C(X,Y) \to Hom_{Func(K,C)}(p, const Y)
$$
induced by the morpism $ p \to const X$ that is encoded by $\bar p$ is an
equivalence (i.e. a [[homotopy equivalence]] of [[Kan complex]]es).
=--
+-- {: .proof}
###### Proof
This is [[Higher Topos Theory|HTT, lemma 4.2.4.3]].
The key step is to realize that $Hom_{Func(K,C)}(p, const Y)$ is given (up to equivalence) by
the [[pullback]] $C^{p/} \times_C \{Y\}$ in [[sSet]].
Here is a detailed way to see this,
using the discussion at [[hom-object in a quasi-category]].
We have that $Hom_{Func(K,C)}(p, const Y)$ is given by
$(C^K)^{p/} \times_{C^K} const Y$. We compute
$$
\begin{aligned}
((C^K)^{p/} \times_{C^K} const Y)_n
& =
Hom_{{\Delta[0]}/sSet}( \Delta[0] \diamondsuit \Delta[n] , C^K ) \times_{(C^K)_n} \{const Y\}
\\
& =
Hom_{{\Delta[0]}/sSet}( \Delta[0] \coprod_{\Delta[n]} \Delta[n] \times \Delta[1] , C^K )
\times_{(C^K)_n} \{const Y\}
\\
& =
\{p\}
\times_{Hom(\Delta[0],C^K)}
Hom(\Delta[0], C^K)
\times_{Hom(\Delta[n], C^K)}
Hom(\Delta[n] \times \Delta[1], C^K)
\times_{Hom(\Delta[n], C^K)}
\{const Y\}
\\
& =
\{p\}
\times_{Hom(K,C)}
Hom(K,C)
\times_{Hom(\Delta[n]\times K,C)}
Hom(\Delta[n]\times K \times \Delta[1], C)
\times_{Hom(\Delta[n]\times K, C)}
Hom(\Delta[n],C)
\times_{\Delta[n],C}
\{Y\}
\\
&=
\{p\}
\times_{Hom(K,C)}
Hom(K \diamondsuit \Delta[n], C)
\times_{Hom(\Delta[n],C)}
\{Y\}
\\
&=
(C^{p/}\times_C \{Y\})_n
\end{aligned}
$$
Under this identification, $\phi_Y$ is the morphism
$$
\left(
C^{X/}
\stackrel{\phi'}{\to}
C^{\bar p/}
\stackrel{\phi''}{\to}
C^{p/}
\right)
\times_C \{Y\}
\,,
$$
in [[sSet]] where $\phi'$ is a section of the map $C^{\bar p/} \to C^{X/}$,
(which one checks is an acyclic [[Kan fibration]]) obtained by choosing composites
of the co-cone components with a given morphism $X \to Y$.
The morphism $\phi''$ is a [[left fibration]] (using [[Higher Topos Theory|HTT, prop. 4.2.1.6]])
One finds that the morphism $\phi''$ is a [[left fibration]].
The strategy for the completion of the proof is: realize that the first condition
of the proposition is equivalent to $\phi''$ being an acyclic Kan fibration, and the
second statement equivalent to $\phi''_Y$ being an acyclic Kan fibration, then
show that these two conditions in turn are equivalent.
=--
### In terms of products and equalizers
A central theorem in ordinary [[category theory]] asserts that a [[category]] has [[limit]]s already if it has [[product]]s and [[equalizer]]s. The analog statement is true here:
+-- {: .num_prop}
###### Proposition
Let $\kappa$ be a [[regular cardinal]]. An [[(β,1)-category]] $C$ has all $\kappa$-small limits precisely if it has [[equalizer]]s and $\kappa$-small [[product]]s.
=--
This is [[Higher Topos Theory|HTT, prop. 4.4.3.2]].
### In terms of homotopy limits {#TermsOfHomotopy}
The notion of [[homotopy limit]], which exists for [[model categories]] and in particular for [[simplicial model categories]] and in fact in all plain [[Kan complex]]-[[enriched categories]] -- as described in more detail at [[homotopy Kan extension]] -- is supposed to be a model for $(\infty,1)$-categorical limits. In particular, under sending the Kan-complex enriched categories $C$ to quasi-categories $N(C)$ using the [[homotopy coherent nerve]] functor, homotopy limits should precisely corespond to quasi-categorical limits. That this is indeed the case is asserted by the following statements.
+-- {: .num_prop}
###### Proposition
Let $C$ and $J$ be [[Kan complex]]-[[enriched categories]] and let $F : J \to C$ be an [[sSet]]-[[enriched functor]].
Then a [[cocone]] $\{\eta_i : F(i) \to c\}_{i \in J}$ under $F$ exhibits the object $c \in C$ as a [[homotopy colimit]] (in the sense discussed in detail at [[homotopy Kan extension]]) precisely if the induced morphism of quasi-categories
$$
\bar {N(F)} : N(J)^{\triangleright} \to N(C)
$$
is a quasi-categorical colimit [[diagram]] in $N(C)$.
=--
Here $N$ is the [[homotopy coherent nerve]], $N(J)^{\triangleright}$ the [[join of quasi-categories]] with the point, $N(F)$ the image of the simplicial functor $F$ under the homotopy coherent nerve and $\bar{N(F)}$ its extension to the join determined by the cocone maps $\eta$.
+-- {: .proof}
###### Proof
This is [[Higher Topos Theory|HTT, theorem 4.2.4.1]]
A central ingredient in the proof is the fact, discused at [[(β,1)-category of (β,1)-functors]] and at [[model structure on functors]], that [[sSet]]-[[enriched functor]]s do model [[(β,1)-functor]]s, in that for $A$ a [[combinatorial simplicial model category]], $S$ a [[quasi-category]] and $\tau(S)$ the corresponding $sSet$-category under the left adjoint of the [[homotopy coherent nerve]], we have an [[equivalence of quasi-categories]]
$$
N(([C,A]_{proj})^\circ)
\simeq
Func(S, N(A^\circ))
$$
and the same is trued for $A$ itself replaced by a [[chunk of a model category|chunk]] $U \subset A$.
With this and the discussion at [[homotopy Kan extension]], we find that the cocone components $\eta$ induce for each $a \in [C,sSet]$ a [[homotopy equivalence]]
$$
C(c,a) \stackrel{}{\to}
[J^{op}, C](j F, const a)
$$
which is hence equivalently an equivalence of the corresponding [[hom-object in a quasi-category|quasi-categorical hom-objects]]. The claim follows then from the above discussion of characterization of (co)limits in terms of $\infty$-hom adjunctions.
=--
+-- {: .num_cor}
###### Corollary
The quasi-category $N(A^\circ)$ [[presentable (β,1)-category|presented]] by a [[combinatorial simplicial model category]] $A$ has all small quasi-categorical limits and colimits.
=--
+-- {: .proof}
###### Proof
This is [[Higher Topos Theory|HTT, 4.2.4.8]].
It follows from the fact that $A$ has
(pretty much by definition of [[model category]] and [[combinatorial model category]])
all [[homotopy limit]]s and [[homotopy colimit]]s (in fact all [[homotopy Kan extension]]s)
by the above proposition.
=--
Since $(\infty,1)$-categories equivalent to those of the form $N(A^\circ)$ for $A$ a [[combinatorial simplicial model category]] are precisely the [[locally presentable (β,1)-categories]], it follows from this in particular that every locally presentable $(\infty,1)$-category has all limits and colimits.
### Commutativity of limits {#CommutativityOfLimits}
The following proposition says that if for an $(\infty,1)$-functor $F : X \times Y \to C$ limits (colimits) over each of the two variables exist separately, then they commute.
+-- {: .num_prop }
###### Proposition
Let $X$ and $Y$ be [[simplicial set]]s and $C$ a [[quasi-category]]. Let $p : X^{\triangleleft} \times Y^{\triangleleft} \to C$ be a [[diagram]]. If
1. for every object $x \in X^{\triangleleft}$ (including the cone point) the restricted diagram $p_x : Y^{\triangleleft} \to C$ is a limit diagram;
1. for every object $y \in Y$ (not including the cone point) the restricted diagram $p_y : X^{\triangleleft} \to C$ is a limit diagram;
then, with $c$ denoting the cone point of $Y^{\triangleleft}$, the restricted diagram, $p_c : X^{\triangleleft} \to C$ is also a limit diagram.
=--
+-- {: .proof}
###### Proof
This is [[Higher Topos Theory|HTT, lemma 5.5.2.3]]
=--
In other words, suppose that $\lim_x F(x,y)$ exists for all $y$ and $\lim_y F(x,y)$ exists for all $x$ and also that $\lim_y (\lim_x F(x,y))$ exists, then this object is also $\lim_x \lim_y F(x,y)$.
## Examples
### $\infty$-Limits of special shape
#### Coproduct
...
#### Pullback / Pushout
See also [[(β,1)-pullback]].
The non-degenerate cells of the [[simplicial set]] $\Delta[1] \times \Delta[1]$ obtained as the [[cartesian product]] of the simplicial 1-[[simplex]] with itself look like
$$
\array{
(0,0) &\to& (1,0)
\\
\downarrow &\searrow& \downarrow
\\
(0,1) &\to& (1,1)
}
$$
A **sqare** in a [[quasi-category]] $C$ is an image of this in $C$, i.e. a morphism
$$
s : \Delta[1] \times \Delta[1] \to C
\,.
$$
The simplicial square $\Delta[1]^{\times 2}$ is [[isomorphism|isomorphic]], as a [[simplicial set]], to the [[join of simplicial sets]] of a 2-[[horn]] with the point:
$$
\Delta[1] \times \Delta[1]
\simeq
\{v\} \star \Lambda[2]_2
=
\left(
\array{
v &\to& 1
\\
\downarrow &\searrow& \downarrow
\\
0 &\to& 2
}
\right)
$$
and
$$
\Delta[1] \times \Delta[1]
\simeq
\Lambda[2]_0 \star \{v\}
=
\left(
\array{
0&\to& 1
\\
\downarrow &\searrow& \downarrow
\\
2 &\to& v
}
\right)
\,.
$$
If a square $\Delta[1] \times \Delta[1] \simeq \Lambda[2]_0 \star \{v\} \to C$ exhibits $\{v\} \to C$ as a colimit over $F : \Lambda[2]_0 \to C$, we say the colimit
$$
v := \lim_\to F := F(1) \coprod_{F(0)} F(2)
$$
is [[generalized the|the]] **pushout** of the diagram $F$.
##### Pasting law of pushouts {#PushoutPasting}
We have the following $(\infty,1)$-categorical analog of the familiar [pasting law of pushouts](http://ncatlab.org/nlab/show/pullback#Pasting) in ordinary [[category theory]]:
+-- {: .num_prop}
###### Proposition
A pasting diagram of two squares is a morphism
$$
\Delta[2] \times \Delta[1] \to C
\,.
$$
Schematically this looks like
$$
\array{
x &\to& y &\to& z
\\
\downarrow && \downarrow && \downarrow
\\
x' &\to& y' &\to& z'
}
\,.
$$
If the left square is a pushout diagram in $C$, then the right square is precisely if the outer square is.
=--
+-- {: .proof}
###### Proof
A proof appears as [[Higher Topos Theory|HTT, lemma 4.4.2.1]]
=--
#### Coequalizer
...
#### Quotients
* [[infinity-quotient]]
#### Tensoring and cotensoring with an $\infty$-groupoid {#Tensoring}
##### Recap of the 1-categorical situation
An ordinary [[category]] with [[limit]]s is canonically [[power|cotensored]] over [[Set]]:
For $S, T \in $ [[Set]] and $const_T : S \to Set$ the [[diagram]] parameterized by $S$ that is constant on $T$, we have
$$
\lim_{\leftarrow} \, const_T \simeq T^S
\,.
$$
Accordingly the cotensoring
$$
(-)^{(-)} : Set^{op} \times C \to C
$$
is defined by
$$
c^S := \lim_{\leftarrow} (S \stackrel{const_c}{\to} C)
= \prod_{S} c
\,.
$$
And by continuity of the [[hom-functor]] this implies the required natural isomorphisms
$$
Hom_C(d,c^S) = Hom_C(d, {\lim_{\leftarrow}}_S c) \simeq {\lim_{\leftarrow}}_S
Hom_C(d,c) \simeq Set(S,Hom_C(d,C))
\,.
$$
Correspondingly if $C$ has [[colimit]]s, then the [[copower|tensoring]]
$$
(-) \otimes (-) : Set \times C \to C
$$
is given by forming [[colimit]]s over constant diagrams: $S \otimes c := {\lim_{\to}}_S c$, and again by continuity of the [[hom-functor]] we have the required natural isomorphism
$$
Hom_C(S \otimes c, d) =
Hom_C({\lim_{\to}}_S c,d)
\simeq
{\lim_{\leftarrow}}_S Hom_C(c,d)
\simeq
Set(S,Hom_C(c,d))
\,.
$$
Of course all the colimits appearing here are just [[coproduct]]s and all limits just [[product]]s, but for the generalization to $(\infty,1)$-categories this is a misleading simplification, it is really the notion of limit and colimit that matters here.
##### Definition
We expect for $S, T \in $ [[βGrpd]] and for $const_T : S \to \infty Grpd$ the constant diagram, that
$$
\lim_{\leftarrow} \, const_T \simeq T^S
\,,
$$
where on the right we have the [[internal hom]] of $\infty$-groupoids, which is modeled in the [[model structure on simplicial sets]] $sSet_{Quillen}$ by the fact that this is a [[closed monoidal category|closed]] [[monoidal category]].
Correspondingly, for $C$ an $(\infty,1)$-category with colimits, it is [[copower|tensored]] over [[βGrpd]] by setting
$$
(-)\otimes (-) : \infty Grpd \times C \to C
$$
$$
S \otimes c := {\lim_{\to}}_S c
\,,
$$
where now on the right we have the $(\infty,1)$-categorical colimit over the constant diagram $const : S \to C$ of shape $S$ on $c$.
Then by the $(\infty,1)$-continuity of the hom, and using the above characterization of the [[internal hom]] in $\infty Grpd$ we have the required natural equivalence
$$
Hom_C(S \otimes c, d)
=
Hom_C({\lim_{\to}}_S c, d)
\simeq
{\lim_{\leftarrow}}_S Hom_C(c,d)
\simeq
\infty Grpd(S,Hom_C(c,d))
\,.
$$
The following proposition should assert that this is all true
+-- {: .num_prop #TensoringProposition}
###### Proposition
The $(\infty,1)$-categorical colimit ${\lim_{\to}} c$ over the diagram of shape $S \in \infty Grpd$ constant on $c \in C$ is characterized by the fact that it induces natural equivalences
$$
Hom_C({\lim_{\to}}_S c, d)
\simeq
\infty Grpd(S, Hom_C(c,d))
$$
for all $d \in C$.
=--
This is essentially [[Higher Topos Theory|HTT, corollary 4.4.4.9]].
+-- {: .num_cor #EveryInfinityGroupoidIsHomotopyColimitOfConstantFunctorOverItself}
###### Corollary
Every [[β-groupoid]] $S$ is the $(\infty,1)$-colimit in [[βGrpd]] of the constant diagram on the [[point]] over itself:
$$
S \simeq {\lim_{\to}}_S const_*
\,.
$$
=--
This justifies the following definition
+-- {: .num_defn}
###### Definition
For $C$ an $(\infty,1)$-category with colimits, the **tensoring of $C$ over $\infty Grpd$** is the $(\infty,1)$-functor
$$
(-) \otimes (-) : \infty Grpd \times C \to C
$$
given by
$$
S \otimes c = \lim_{\to} (const_c : S \to C)
\,.
$$
=--
See [[Higher Topos Theory|HTT, section 4.4.4]].
##### Models {#ModelsForTensoring}
We discuss models for $(\infty,1)$-(co)limits in terms of ordinary [[category theory]] and [[homotopy theory]].
\begin{lemma}
If $C$ is [[presentable (infinity,1)-category|presented]] by a [[simplicial model category]] $A$, in that $C \simeq A^\circ$, then the
$(\infty,1)$-tensoring and $(\infty,1)$-cotensoring of $C$ over [[βGrpd]]
is modeled by the ordinary [[copower|tensoring]] and [[power|powering]]
of $A$ over [[sSet]]:
For $\hat c \in A$ cofibant and representing an object $c \in C$ and for $S \in sSet$ any simplicial set, we have an equivalence
$$
c \otimes S \simeq \hat C \cdot S
\,.
$$
\end{lemma}
+-- {: .proof}
###### Proof
The powering in $A$ satisfies the [[natural isomorphism]]
$$
sSet(S,A(\hat c,\hat d)) \simeq A(\hat c \cdot S, \hat d)
$$
in [[sSet]].
For $\hat c$ a cofibrant and $\hat d$ a fibrant representative, we have that both sides here are [[Kan complex]]es that are equivalent to the corresponding [[derived hom space]]s in the corresponding $(\infty,1)$-category $C$, so that this translates into an equivalence
$$
Hom_C(c \cdot S, d)
\simeq
\infty Grpd(S, Hom_C(c,d))
\,.
$$
The claim then follows from the above proposition.
=--
### Limits in over-$(\infty,1)$-categories {#InOvercategories}
+-- {: .num_prop }
###### Proposition
For $C$ an $(\infty,1)$-category, $X : D \to C$ a diagram, $C/X$ the [[over-(β,1)-category]] and $F : K \to C/X$ a diagram in the over-$(\infty,1)$-category, then the [[(β,1)-limit]] $\lim_{\leftarrow} F$ in $C/X$ coincides with the $(\infty,1)$-limit $\lim_{\leftarrow} F/X$ in $C$.
=--
+-- {: .proof}
###### Proof
Modelling $C$ as a [[quasi-category]] we have that $C/X$ is given by the [[simplicial set]]
$$
C/X : [n] \mapsto Hom_X([n] \star D, C)
\,,
$$
where $\star$ denotes the [[join of simplicial sets]]. The limit $\lim_{\leftarrow} F$ is the initial object in $(C/X)/F$, which is the quasi-category given by the simplicial set
$$
(C/X)/F : [n] \mapsto Hom_{F}( [n] \star K, C/X)
\,.
$$
Since the join preserves colimits of simplicial sets in both arguments, we can apply the [[co-Yoneda lemma]] to decompose $[n] \star K = {\lim_{\underset{{[r] \to [n]\star K}}{\to}}} [r]$, use that the [[hom-functor]] sends colimits in the first argument to limits and obtain
$$
\begin{aligned}
Hom([n] \star K, C/X)
&\simeq
Hom( {\lim_{\to}}_r [r], C/X)
\\
& \simeq {\lim_{\leftarrow}}_r Hom([r], C/X)
\\
& \simeq {\lim_{\leftarrow}}_r Hom_F( [r] \star D, C )
\\
& \simeq Hom_F({\lim_{\to}}_r ([r] \star D), C )
\\
& \simeq Hom_F( ({\lim_{\to}}_r[r]) \star D, C )
\\
& \simeq Hom_F(([n] \star K) \star D, C)
\\
& \simeq Hom_F([n] \star (K \star D), C)
\end{aligned}
\,.
$$
Here $Hom_F([r]\star D,C)$ is shorthand for the hom in the (ordinary) [[under category]] $sSet^{D/}$ from the canonical inclusion $D \to [r] \star D$ to $X : D \to C$. Notice that we use the 1-categorical analog of the statement that we are proving here when computing the colimit in this under-category as just the colimit in $sSet$. We also use that the [[join of simplicial sets]], being given by [[Day convolution]] is an associative tensor product.
In conclusion we have an isomorphism of simplicial sets
$$
(C/X)/F \simeq C/(X/F)
$$
and therefore the initial objects of these quasi-categories coincide on both sides. This shows that $\lim_{\leftarrow} F$ is computed as an initial object in $C/(X/F)$.
=--
### Limits and colimits with values in $\infty Grpd$ {#WithValInooGrpd}
Limits and colimits over a [[(β,1)-functor]] with values in the [[(β,1)-category]] [[β-Grpd]] of [[β-groupoids]] may be reformulated in terms of the [[universal fibration of (β,1)-categories]], hence in terms of the [[(β,1)-Grothendieck construction]].
Let [[βGrpd]] be the [[(β,1)-category]] of [[β-groupoid]]s. Let the [[(β,1)-functor]] $Z|_{Grpd} \to \infty Grpd^{op}$ be the [[universal fibration of (infinity,1)-categories|universal β-groupoid fibration]] whose fiber over the object denoting some $\infty$-groupoid is that very $\infty$-groupoid.
Then let $X$ be any [[β-groupoid]] and
$$
F : X \to \infty Grpd
$$
an [[(β,1)-functor]]. Recall that the [[Cartesian fibration|coCartesian fibration]] $E_F \to X$ classified by $F$ is the pullback of the [[universal fibration of β-groupoids]] $Z|_{Grpd}$ along F:
$$
\array{
E_F &\to& Z|_{Grpd}
\\
\downarrow && \downarrow
\\
X &\stackrel{F}{\to}& \infty Grpd
}
$$
+-- {: .num_prop #InfinityGroupoidalCoLimitsViaIntegrationAndSlicing}
###### Proposition
Let the assumptions be as above. Then:
* The $\infty$-colimit of $F$ is equivalent to the [[(β,1)-Grothendieck construction]] $E_F$:
$$
\underset{\longrightarrow}{\lim} F \simeq E_F
$$
* The $\infty$-limit of $F$ is equivalent to the [[β-groupoid of sections]] of $E_F \to X$
$$
\underset{\longleftarrow}{\lim} \simeq \Gamma_X(E_F)
\,.
$$
=--
The statement for the colimit is corollary 3.3.4.6 in [[Higher Topos Theory|HTT]]. The statement for the limit is corollary 3.3.3.4.
+-- {: .num_remark }
###### Remark
The form of the statement in prop. \ref{InfinityGroupoidalCoLimitsViaIntegrationAndSlicing} is the special case of the general form of [[internal (co-)limits]], here internal to the [[(β,1)-topos]] [[βGrpd]] with $Core(\inftyGrpd_{small})$ its small [[object classifier]]. See at _[internal (co-)limit -- Groupoidal homotopy (co-)limits](internal+%28co-%29limit#ExamplesInfinityGroupoidal)_ for more on this.
=--
### Limits and colimits with values in $(\infty,1)$Cat
{#ColimitsInInfinityCat}
+-- {: .num_prop }
###### Proposition
For $F : D \to $ [[(β,1)Cat]] an [[(β,1)-functor]], its $\infty$-colimit is given by forming the [[(β,1)-Grothendieck construction]] $\int F$ of $F$ and then inverting the [[Cartesian morphism]]s.
Formally this means, with respect to the [[model structure for Cartesian fibrations]] that there is a [[natural isomorphism]]
$$
\lim_\to F \simeq (\int F)^\sharp
$$
in the [[homotopy category]] of the presentation of $(\infty,1)$-category by [[marked simplicial sets]].
=--
This is [[Higher Topos Theory|HTT, corollary 3.3.4.3]].
For the special case that $F$ takes values in ordinary categories see also at [[2-limit]] the section [2-limits in Cat](http://ncatlab.org/nlab/show/2-limit#2ColimitsInCat).
### Limits in $\infty$-functor categories
For $C$ an ordinary [[category]] that admits small [[limit]]s and [[colimit]]s, and for $K$ a [[small category]], the [[functor category]] $Func(D,C)$ has all small limits and colimits, and these are computed objectwise. See [[limits and colimits by example]]. The analogous statement is true for an [[(β,1)-category of (β,1)-functors]].
+-- {: .num_prop}
###### Proposition
Let $K$ and $C$ be [[quasi-categories]], such that $C$ has all [[limit in a quasi-category|colimits]] indexed by $K$.
Let $D$ be a small quasi-category. Then
* The [[(β,1)-category of (β,1)-functors]] $Func(D,C)$ has all $K$-indexed colimits;
* A morphism $K^\triangleright \to Func(D,C)$ is a colimiting cocone precisely if for each object $d \in D$ the induced morphism $K^\triangleright \to C$ is a colimiting cocone.
=--
+-- {: .proof}
###### Proof
This is [[Higher Topos Theory|HTT, corollary 5.1.2.3]]
=--
## Related concepts
* [[limit]], [[internal limit]]
* [[2-limit]]
* **$(\infty,1)$-limit
* [[finite (β,1)-limit]], [[relative (β,1)-limit]]
* [[homotopy limit]]
* [[lax (β,1)-colimit]]
## References
### General
The definition of limit in quasi-categories is due to
* [[AndrΓ© Joyal]], _Quasi-categories and Kan complexes_ Journal of Pure and Applied Algebra 175 (2002), 207-222.
A brief survey is on page 159 of
* [[AndrΓ© Joyal]], _The theory of quasicategories and its applications_ lectures at [Simplicial Methods in Higher Categories](http://www.crm.es/HigherCategories/), ([pdf](http://www.crm.cat/HigherCategories/hc2.pdf))
A detailed account is in [definition 1.2.13.4, p. 48](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=48) in
* [[Jacob Lurie]], _[[Higher Topos Theory]]_
{#Lurie}
* [[Denis-Charles Cisinski]], _Do homotopy limits compute limits in the associated quasicategory in the non-combinatorial model category case?_, [MO/176111/176142](http://mathoverflow.net/questions/176111/do-homotopy-limits-compute-limits-in-the-associated-quasicategory-in-the-non-com/176142#176142).
A discussion of [[weighted limit|weighted]] $(\infty,1)$-limits is in
* Martina Rovelli, _Weighted limits in an (β,1)-category_, 2019, [arxiv:1902.00805](https://arxiv.org/abs/1902.00805)
A discussion of free colimit completion constructions is in
* [[Charles Rezk]], _Free colimit completion in β-categories_ ([arXiv:2210.08582](https://arxiv.org/abs/2210.08582))
Discussion [[category internal to an (infinity,1)-topos|internal to]] any [[(β,1)-topos]]:
* [[Louis Martini]], [[Sebastian Wolf]], *Limits and colimits in internal higher category theory* [[arXiv:2111.14495](https://arxiv.org/abs/2111.14495)]
### In homotopy type theory
A formalization of some aspects of $(\infty,1)$-limits in terms of [[homotopy type theory]] is [[Coq]]-coded in
* [[Guillaume Brunerie]], _[HoTT/Coq/Limits](https://github.com/guillaumebrunerie/HoTT/tree/master/Coq/Limits)_
See also
* [[Egbert Rijke]], _Homotopy Colimits and a Descent Theorem_, March 14, 2013 ([video](http://video.ias.edu/univalent-1213-0314-EgbertRijke))
[[!redirects limit in a quasi-category]]
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[[!redirects (β,1)-coproduct]]
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[[!redirects (β,1)-pushout]]
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[[!redirects (β,1)-copower]]
[[!redirects infinity1-limit]]
[[!redirects infinity1-colimit]]
|
(β,1)-local geometric morphism | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29-local+geometric+morphism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The analog in the context of [[(β,1)-topos]] theory of a [[local geometric morphism]] in [[topos theory]].
## Definition
+-- {: .num_defn }
###### Definition
A **local (β,1)-geometric morphism** $f : \mathbf{H} \to \mathbf{S}$ between [[(β,1)-topos]]es $\mathbf{H},\mathbf{S}$ is
* an [[(β,1)-geometric morphism]]
$$
(f^* \dashv f_*)
:
\mathbf{H} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}
{\to}} \mathbf{S}
$$
* such that
1. a further [[right adjoint]]
$f^! : \mathbf{S} \to \mathbf{H}$
to the [[direct image]] functor exists:
$$
(f^* \dashv f_* \dashv f^!) :
\mathbf{H}
\stackrel{\overset{f^*}{\leftarrow}}
{\stackrel{\underset{f_*}{\to}}{\underset{f^!}{\leftarrow}}}
\mathbf{S}
$$
1. and $f$ is a [[β-connected (β,1)-geometric morphism]].
If $f : \mathbf{H} \to \mathbf{S}$ is the [[global section]] [[(β,1)-geometric morphism]] in the [[over-(β,1)-category]] [[Topos]]$/\mathbf{S}$, then we say that $\mathbf{H}$ is a **local $(\infty,1)$-topos** over $\mathbf{S}$.
=--
+-- {: .num_remark }
###### Remark
If $\mathbf{S} = $ [[βGrpd]] then the extra condition that $f$ is [[β-connected (β,1)-geometric morphism]] is automatic (see [Properties -- over βGrpd](#OverInfGrpd)).
=--
## Properties {#Properties}
### Over $\infty Grpd$ {#OverInfGrpd}
+-- {: .num_prop }
###### Proposition
Every local $(\infty,1)$-topos over [[βGrpd]] has [[homotopy dimension]] $\leq 0$.
=--
See [[homotopy dimension]] for details.
+-- {: .num_prop }
###### Proposition
If an [[(β,1)-geometric morphism]] $f : \mathbf{H} \to $ [[βGrpd]] has an extra [[right adjoint]] $f^!$ to its [[direct image]], then $\mathbf{H}$ is an [[β-connected (β,1)-topos]].
=--
+-- {: .proof}
###### Proof
By the general properties of [[adjoint (β,1)-functor]]s it is sufficient to show that $f_! f^* \simeq Id$. To see this, we use that every [[β-groupoid]] $S \in $ [[βGrpd]] is the [[(β,1)-colimit]] (as discussed there) over itself of the [[(β,1)-functor]] constant on the point: $S \simeq {\lim_\to}_{S} *$.
The [[left adjoint]] $f^*$ preserves all [[(β,1)-colimit]]s, but if $f_*$ has a right adjoint, then it does, too, so that for all $S$ we have
$$
f_* f^* {\lim_\to}_S * \simeq {\lim_\to}_S f_* f^* *
\,.
$$
Now $f_*$, being a [[right adjoint]] preserves the [[terminal object in an (β,1)-category|terminal object]] and so does $f^*$ by definition of [[(β,1)-geometric morphism]]. Therefore
$$
\cdots \simeq {\lim_\to}_S * \simeq S
\,.
$$
=--
### Concrete objects
Every local $(\infty,1)$-geometric morphism induces a notion of [[concrete (β,1)-sheaves]]. See there for more (also see [[cohesive (β,1)-topos]]).
## Examples
### Local over-$(\infty,1)$-toposes
+-- {: .num_prop }
###### Proposition
Let $\mathbf{H}$ be any [[(β,1)-topos]] (over [[βGrpd]]) and let $X \in \mathbf{H}$ be an [[object]] that is [[small-projective]]. Then the [[over-(β,1)-topos]] $\mathbf{H}/X$ is local.
=--
+-- {: .proof}
###### Proof
We check that the [[global section]] [[(β,1)-geometric morphism]] $\Gamma : \mathbf{H}/X \to $ [[βGrpd]] preserves [[(β,1)-colimit]]s.
The functor $\Gamma$ is given by the [[hom-functor]] out of the [[terminal object]] of $\mathcal{H}/X$, this is $(X \stackrel{Id}{\to} X)$:
$$
\Gamma : (A \stackrel{f}{\to} X) \mapsto Hom_{\mathbf{H}/X}(Id_X, f)
\,.
$$
The [[derived hom-space|hom-β-groupoid]]s in the [[over-(β,1)-category]] are (as discussed there) [[homotopy fiber]]s of the hom-sapces in $\mathbf{H}$: we have an [[(β,1)-pullback]] diagram
$$
\array{
\mathbf{H}/X(Id_X, (A \to X)) &\to&
\mathbf{H}(X,A)
\\
\downarrow && \downarrow^{f_*}
\\
* &\stackrel{Id_X}{\to}& \mathbf{H}(X,X)
}
\,.
$$
Overserve that [[(β,1)-colimit]]s in the [[over-(β,1)-category]] $\mathbf{H}/X$ are computed in $\mathbf{H}/X$.
$$
{\lim_{\to}}_i (A_i \stackrel{f_i}{\to} X)
\simeq
({\lim_\to}_i A_i) \to X
\,.
$$
If $X$ is [[small-projective]] then by definition we have
$$
\mathbf{H}(X, {{\lim}_\to}_i A_i)
\simeq
{\lim_\to}_i \mathbf{H}(X, A_i)
\,,
$$
Inserting all this into the above $(\infty,1)$-pullback gives the $(\infty,1)$-pullback
$$
\array{
\mathbf{H}/X(Id_X, {\lim_\to}_i (A_i \to X)) &\to&
{\lim_\to}_i \mathbf{H}(X, A_i)
\\
\downarrow && \downarrow^{f_*}
\\
* &\stackrel{Id_X}{\to}& \mathbf{H}(X,X)
}
\,.
$$
By [[universal colimits]] in the [[(β,1)-topos]] [[βGrpd]], this [[(β,1)-pullback]] of an [[(β,1)-colimit]] is the $(\infty,1)$-colimit of the separate pullbacks, so that
$$
\Gamma({\lim_\to}_i (A_i \to X)))
\simeq
\mathbf{H}/X(Id_X, {\lim_\to}_i (A_i \to X))
\simeq
{\lim_\to}_i \mathbf{H}/X(Id_X,(A_i \to X))
\simeq
{\lim_\to}_i \Gamma(A_i \to X)
\,.
$$
So $\Gamma$ does commute with colimits if $X$ is [[small-projective]]. Since all [[(β,1)-topos]]es are [[locally presentable (β,1)-categories]] it follows by the [[adjoint (β,1)-functor]] that $\Gamma$ has a [[right adjoint|right]] [[adjoint (β,1)-functor]].
=--
## Related concpepts
* [[locally connected topos]] / [[locally β-connected (β,1)-topos]]
* [[connected topos]] / [[β-connected (β,1)-topos]]
* [[strongly connected topos]] / [[strongly β-connected (β,1)-topos]]
* [[totally connected topos]] / [[totally β-connected (β,1)-topos]]
* [[local topos]] / **local (β,1)-topos**.
* [[cohesive topos]] / [[cohesive (β,1)-topos]]
and
* [[locally connected site]], [[locally β-connected site]]
* [[connected site]]
* [[local site]]
* [[cohesive site]], [[(β,1)-cohesive site]]
## References
The 1-categorical notion is discussed in
* [[Peter Johnstone]], [[Ieke Moerdijk]], _Local maps of toposes_ Proc. London Math. Soc. (1989) s3-58 (2): 281-305. ([pdf](http://plms.oxfordjournals.org/content/s3-58/2/281.full.pdf+html))
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|
(β,1)-logic | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29-logic |
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###Context###
#### Type theory
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[[!include type theory - contents]]
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#### Foundations
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[[!include foundations - contents]]
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#Contents#
* table of contents
{:toc}
## Idea
As a [[category]] has [[internal logic]], so a [[(β,1)-category]] has internal _$(\infty,1)$-logic_
## Examples
See
* [[internal logic of an (β,1)-topos]]
## Related concepts
* [[type theory]], [[logic]]
* [[2-type theory]], [[2-logic]]
* [[(β,1)-type theory]], **(β,1)-logic**
|
(β,1)Operad | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29Operad | +-- {: .rightHandSide}
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### Context
#### [[categories of categories - contents|categories of categories]]
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[[!include categories of categories - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[(β,2)-category]] of [[(β,1)-operads]]. The [[operad|operadic]] generalization of [[(β,1)Cat]].
## Properties
[[!include table - models for (infinity,1)-operads]]
## Related concepts
* [[Cat]], [[Operad]]
* [[2Cat]]
* [[(β,1)Cat]], [[(β,1)Operad]]
* [[(β,n)Cat]]
category: category
[[!redirects (infinity,1)Operad]] |
(β,2)-category of cobordisms | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C2%29-category+of+cobordisms |
## Idea
An [[(β,n)-category of cobordisms]] for $n = 2$.
## Related concepts
* [[2d TQFT]], [[TCFT]]
* [[Calabi-Yau object]]
[[!redirects (infinity,2)-category of cobordisms]] |
(β,2)-category theory | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C2%29-category+theory |
## Idea
The theory of [[(β,2)-categories]]...
## Related concepts
[[!include table of category theories]] |
(β,2)-functor | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C2%29-functor |
## Idea
[[(infinity,n)-functor]] for $n = 2$
## Related entries
* [[2d TQFT]], [[TCFT]]
[[!redirects (β,2)-functors]]
[[!redirects (infinity,2)-functor]]
[[!redirects (infinity,2)-functors]]
|
(β,n)-category theory | https://ncatlab.org/nlab/source/%28%E2%88%9E%2Cn%29-category+theory |
## Idea
The theory of [[(β,n)-categories]]
## Related concepts
[[!include table of category theories]]
|
0-bundle | https://ncatlab.org/nlab/source/0-bundle |
## Idea
In the context of [[principal infinity-bundles]] a 0-bundle is equivalently a [[function]].
## Examples
* [[prequantum 0-bundle]].
[[!redirects 0-bundles]]
|
0-category | https://ncatlab.org/nlab/source/0-category |
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### Context ###
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definition
A __$0$-category__ (or __$(0,0)$-category__) is, up to [[equivalence of categories|equivalence]], the same as a [[set]] (or [[class]]).
+-- {: .num_remark}
###### Remark
Although this terminology may seem strange at first, it simply follows the logic of $n$-[[n-category|categories]] (and $(n,r)$-[[(n,r)-category|categories]]). To understand these, it is very helpful to use [[negative thinking]] to see sets as the beginning of a sequence of concepts: sets, [[category|categories]], [[2-category|2-categories]], [[3-category|3-categories]], etc. Doing so reveals patterns such as the [[periodic table]]; it also sheds light on the theory of [[homotopy groups]] and [[n-stuff]].
For example, there should be a $1$-category of $0$-categories; this is the [[Set|category of sets]]. Then a [[enriched category|category enriched]] over this is a $1$-category (more precisely, a [[locally small category]]). Furthermore, an enriched groupoid is a [[groupoid]] (or $1$-groupoid), so a $0$-category is the same as a [[0-groupoid]].
To some extent, one can continue to define a [[(-1)-category]] to be a [[truth value]] and a [[(-2)-category]] to be a triviality (that is, there is exactly one). These don\'t fit the pattern perfectly; but the concepts of [[(-1)-groupoid]] and [[(-2)-groupoid]] for them do work perfectly, as does the concept of [[0-poset]] for a truth value.
=--
+-- {: .num_remark}
###### Remark
Interpreted literally, $0$-category or $(0, 0)$-category would be an $\infty$-category such that every $j$-cell for $j \gt 0$ is an equivalence, and any two such $j$-cells that are parallel are equivalent. The picture that apparently emerges from this description might suggest a [[symmetric proset]], a set equipped with an [[equivalence relation]], or something even more complicated than that. One *could* thus say that a $0$-category is a symmetric proset, when considered just up to [[isomorphism of categories|isomorphism]]. But it is more appropriate in higher category theory to consider these things up to equivalence rather than up to isomorphism; when we do this, a $0$-category is equivalent to a set again.
=--
## Related concepts
* **0-category**, [[(0,1)-category]]
* [[category]]
* [[2-category]]
* [[3-category]]
* [[n-category]]
* [[(β,0)-category]]
* [[(β,1)-category]]
* [[(β,2)-category]]
* [[(β,n)-category]]
* [[(n,r)-category]]
[[!redirects (0,0)-category]]
[[!redirects 0-categories]]
[[!redirects (0,0)-categories]] |
0-dimensional TQFT | https://ncatlab.org/nlab/source/0-dimensional+TQFT | #Contents#
* automatic table of contents goes here
{:toc}
## Idea
A _0-dimensional TQFT_ is a [[TQFT]] regarded in the sense of [[FQFT]] as a [[representation]] of the [[category]] of 0-dimensional [[cobordism]]s.
This degenerate case turns out to exhibit a nontrivial amount of interesting information, in particular if regarded in the context of _super_ QFT.
## Definition
### 0-Dimensional Cobordisms
The [[category]] $Cob_0$ of 0-dimensional [[cobordism]]s is the [[symmetric monoidal category]] $Cob_0$ having the $-1$-dimensional [[manifold]] $\emptyset$ as the only [[object]] and [[isomorphism]] classes of compact $0$-dimensional manifolds as morphisms. Clearly $Cob_0$ is equivalent to $\mathbf{B}\mathbb{N}$
### 0-Dimensional TQFT
A **0-dimensional [[TQFT]]** (with values in $\mathbb{Z}$-[[module]]s) is a [[monoidal functor]]
$$
Z\colon Cob_0\to \mathbb {Z} Mod
\,.
$$
By definition of [[monoidal functor]], one has $Z({\emptyset})=\mathbb{Z}$ and so $Z$ is completely (and freely) determined by the assignment $Z(\{pt\}\in End_\mathbb{Z}(\mathbb{Z})=\mathbb{Z}$. In other words, the space of 0-dimensional TQFTs is $\mathbb{Z}$.
### Over a manifold
One can consider TQFTs with a target manifold $X$: all bordisms are required to have a map to $X$.
In dimension $0$, morphisms in $Cob_0(X)$ are the topological [[monoid]] $\bigcup_{n\geq 1} Sym^n(X)$. In particular, continuous tensor functors from $Cob_0(X)$ to $\mathbb{Z}$-modules are naturally identified with degree 0 [[integral cohomology]] $H^0(X;\mathbb{Z})$.
### Extended version
The picture becomes more interesting if one goes from topological field theory to [[extended topological quantum field theory]]. Indeed, from this point of view, to the $-1$-dimensional vacuum is assigned the symmetric monoidal [[0-category]] $\mathbb{Z}$, and consequently, the infinity-version of the space of all $0$-dimensional TQFTs is the [[Eilenberg-Mac Lane spectrum]]. It follows that the space of extended $0$-dimensional TQFTs with target $X$ (taking values in $\mathbb{Z}$-modules) is the graded [[integral cohomology]] ring $H^*(X;\mathbb{Z})$.
### Super version
From the [[differential geometry]] point of view, a relation between de Rham cohomology of a smooth manifold $X$ and $0$-dimensional functorial field theories arises if one moves from topological field theory to $(0|1)$-supersymmetric field theory, see [[Axiomatic field theories and their motivation from topology]].
> It would be interesting to describe a direct connection between the extended and the susy theory; it should parallel the usual Cech-de Rham argument
|
0-functor | https://ncatlab.org/nlab/source/0-functor |
As a $0$-[[0-category|category]] is simply a [[set]], so a __$0$-functor__ is simply a [[function]]. See also $n$-[[n-functor|functor]].
[[!redirects 0-functor]]
[[!redirects 0-functors]]
|
0-groupoid | https://ncatlab.org/nlab/source/0-groupoid |
#Contents#
* table of contents
{:toc}
## Idea
A _0-groupoid_ or **[[homotopy n-type|0-type]]** is a [[set]]. This terminology may seem strange at first, but it is very helpful to see sets as the beginning of a sequence of concepts: sets, [[groupoid|groupoids]], [[2-groupoid|2-groupoids]], [[3-groupoid|3-groupoids]], etc. Doing so reveals patterns such as the [[periodic table]]. (It also sheds light on the theory of [[homotopy group]]s and [[n-stuff]].)
For example, there should be a $1$-groupoid of $0$-groupoids; this is the [[underlying groupoid]] of the [[Set|category of sets]]. Then a [[enriched category|groupoid enriched]] over this is a $1$-groupoid (more precisely, a [[locally small category|locally small groupoid]]). Furthermore, an enriched category is a [[category]] (or $1$-category), so a $0$-groupoid is the same as a [[0-category]].
One can continue to define a [[(β1)-groupoid]] to be a [[truth value]] and a [[(β2)-groupoid]] to be a triviality (that is, there is exactly one).
## Related concepts
[[!include homotopy n-types - table]]
[[!redirects 0-groupoids]]
[[!redirects 0-type]]
[[!redirects 0-types]]
|
0-morphism | https://ncatlab.org/nlab/source/0-morphism |
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###Context###
#### Higher category theory
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[[!include higher category theory - contents]]
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#Contents#
* table of contents
{:toc}
## Definition
Following the patterns of [[k-morphism]]s in a [[category]] or [[higher category]] it makes sense to say
**$0$-morphism** for an _[[object]]_ of that category.
## Related concepts
* [[object]]
* [[morphism]]
* [[2-morphism]]
* [[k-morphism]]
[[!redirects 0-morphisms]] |
0-object > history | https://ncatlab.org/nlab/source/0-object+%3E+history | < [[0-object]]
[[!redirects 0-object -- history]] |
0-poset | https://ncatlab.org/nlab/source/0-poset | A __0-poset__ is a [[truth value]]. Compare the concept of $1$[[1-poset|poset]] (a [[partial order|poset]]) and $(-1)$-[[(-1)-poset|poset]] (which is trivial); compare also with $(-1)$-[[(-1)-category|category]] and $0$-[[0-groupoid|groupoid]], which mean the same thing for different reasons.
The point of 0-posets is that they complete some patterns in the [[periodic table]] of $n$-categories, in particular the progression of $n$-[[n-poset|posets]].
For example, there should be a $0$-[[0-category|category]] of $0$-posets; a $0$-category is simply a [[set]], and this set is the set of truth values, classically
$$
(-1)Pos := \{\bot, \top\}
\,.
$$
Actually, we should expect the $0$-category of $0$-posets to be a $1$-poset; this is simply a poset, and indeed truth values do form a poset (where $\bot \leq \top$).
If we equip the category of $0$-posets with its [[cartesian monoidal category|monoidal cartesian structure]] (which is [[logical conjunction|conjunction]], the logical AND operation), then an $\infty$-[[enriched category|category enriched]] over this should be a $1$-poset; and indeed it is (up to [[equivalence of categories]]) a poset (although up to isomorphism only, a category enriched over truth values under conjunction is actually a set equipped with a [[preorder]]).
See [[(β1)-category]] for references on this sort of [[negative thinking]].
|
0-sphere | https://ncatlab.org/nlab/source/0-sphere |
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###Context###
#### Spheres
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[[!include spheres -- contents]]
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#Contents#
* table of contents
{:toc}
## Idea
The _0-sphere_ $S^0$ is the [[n-sphere]] for $n = 0$: The [[disjoint union]] of two [[points]],
$$
S^0 \;\;\simeq\;\; \ast \sqcup \ast
\,.
$$
This definition (even if possibly not be familiar from standard texts of [[point-set topology]]) is certainly justified by the fact that the 2-element set is clearly:
* the [[unit sphere]] inside the [[real line]] $\mathbb{R}^1$,
and as such part of the general pattern that the [[n-sphere]] is the [[unit sphere]] inside $\mathbb{R}^{n+1}$;
* the [[topological space]]/[[homotopy type]] whose [[suspension]] is equivalently the [[1-sphere]] (the [[circle]]):
$$
S^1 \,\simeq\, \mathrm{S} S^0
\,,
$$
thus being the base case of the [[induction|inductive]] definition of the [[n-spheres|$n$-spheres]]:
$$
S^{n+1}
\;\simeq\;
\mathrm{S} S^{n}
\;\simeq\;
\mathrm{S} \mathrm{S} S^{n-1}
\;\simeq\;
\mathrm{S}^{n+1} S^0
\,.
$$
In fact, with [[suspension]] $\mathrm{S}X$ understood as the [[homotopy pushout]] of the [[terminal object|terminal]] [[map]] $X \to \ast$ along itself, one has also
$$
S^0 \;\coloneqq\; \mathrm{S} S^{-1}
$$
for $S^{-1} \,\coloneqq\, \varnothing$ the [[empty set]].
These spheres of degenerate dimensions play an imprtant role in [[homotopy theory]], see for instance the [[cofibrantly generated model category|generating cofibrations]] in the [[classical model structure on topological spaces]] ([here](classical+model+structure+on+topological+spaces#TopologicalGeneratingCofibrations)).
Of course, the underlying object of $S^0$, simple as it is, plays various other roles, too:
* in [[topos theory]], the two element set plays the role of the [[subobject classifier]] among [[Sets]];
* in [[formal logic]] this is also known as the [[classical mathematics|classical]] *[[boolean domain]]*: the [[set]] of [[classical mathematics|classical]] [[truth values]];
and in [[data type|data]]-[[type theory]] one also speaks of the [[type]] $Bit$ of [[bits]];
* that same type regarded in [[homotopy type theory]] again plays the role of the $0$-sphere in the sense of *[[sphere types]]* -- see also at *[boolean domain -- In HoTT](boolean+domain#InHomotopyTypeTheory)*;
* in [[category theory]] the [[disjoint union]] of two copies of the [[terminal object]] (be it the 2-element [[set]] or the corresponding [[discrete category]], etc.) is often denoted "$\mathbf{2}$"
(cf. e.g. at *[[Stone duality]]*).
## Related concepts
* [[1-sphere]]
* [[2-sphere]]
* [[4-sphere]]
* [[7-sphere]]
[[!redirects 0-spheres]]
[[!redirects (-1)-sphere]]
[[!redirects (-1)-spheres]]
|
1-category | https://ncatlab.org/nlab/source/1-category |
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### Context
#### Category theory
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[[!include category theory - contents]]
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#### Higher category theory
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[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In the context of [[higher category theory]] one sometimes needs, for emphasis, to say _1-category_ for _[[category]]_.
## Details
Fix a meaning of $\infty$-[[infinity-category|category]], however weak or strict you wish. Then a __$1$-category__ is an $\infty$-category such that every [[2-morphism]] is an [[equivalence]] and all [[parallel pairs]] of [[k-morphism|j-morphisms]] are equivalent for $j \geq 2$. Thus, up to equivalence, there is no point in mentioning anything beyond $1$-morphisms, except whether two given parallel $1$-morphisms are equivalent.
If you rephrase equivalence of $1$-morphisms as [[equality]], which gives the same result up to [[equivalence of categories|equivalence]], then all that is left in this definition is a [[category]]. Thus one may also say that a __$1$-category__ is simply a category.
The point of all this is simply to fill in the general concept of $n$-[[n-category|category]]; nobody thinks of $1$-categories as a concept in their own right except simply as categories.
## Related concepts
The notions of $1$-[[1-groupoid|groupoid]] and $1$-[[1-poset|poset]] are defined on the same basis.
[[!redirects 1-categories]]
[[!redirects 1βcategories]]
|
1-category equipped with relations | https://ncatlab.org/nlab/source/1-category+equipped+with+relations | #Contents#
* automatic table of contents goes here
{:toc}
## Idea ##
A [[2-category equipped with proarrows]] is a 2-category together with a 2-category of "proarrows" which are intended to generalize the arrows of $K$ in the same way that [[profunctors]] generalize the [[functors]] in [[Cat]]. Since profunctors are a [[categorification]] of [[relations]], it is natural to think of decategorifying such equipments to give a structure on a 1-category that equips it with "relations". We call this structure a *1-category equipped with relations*.
## Definition ##
### (1,2)-categories equipped with proarrows
Recall that a *2-category equipped with proarrows* (aka "proarrow equipment" or "equipment") can be defined as a certain sort of [[double category]], with $\mathcal{V}(\underline{K}) = K$. If, in such a double category, any two squares with the same boundary are equal, we say that it is is a **(1,2)-category equipped with proarrows**, or a **(1,2)-category proarrow equipment**. This is equivalent to requiring that the 2-category of proarrows (and hence also the underlying 2-category of arrows) is [[locally posetal 2-category|locally posetal]], i.e. a (1,2)-category.
For example, if $V$ is any [[quantale]], then $V Cat$ is naturally a (1,2)-category equipped with proarrows. In particular, taking $V=\mathbb{2}$, we have a (1,2)-category proarrow equipment whose objects are [[preorders]].
### 1-categories equipped with relations
A **1-category equipped with relations** is a (1,2)-category equipped with proarrows, regarded as a double category $\underline{K}$, together with an [[involution]] $\underline{K}^{h op} \cong \underline{K}$ which is (isomorphic to) the identity on objects and (vertical) arrows. Here $\underline{K}^{h op}$ denotes the horizontal opposite of a double category obtained by reversing the horizontal (pro-)arrows but not the vertical ones. We also call this structure a **relation equipment** or a **1-category proarrow equipment**.
In particular, the definition implies that we have an involution $K \cong K^{co}$ which is the identity on objects and arrows, which for a (1,2)-category means that $K$ is actually (equivalent to) a 1-category. Note though that the 2-category of proarrows (which we now call "relations") is still (like [[Rel]]) a (1,2)-category, not necessarily a 1-category.
For example, for any quantale $V$, the sub-2-category of $V Cat$ consisting of the *symmetric* $V$-categories (those where $A(x,y) = A(y,x)$) is a 1-category equipped with relations. In particular, for $V=\mathbb{2}$, we have the relation equipment $\underline{Rel}$ of sets, functions, and binary relations.
In general, we can think of a relation equipment as generalizing some of the properties of $\underline{Rel}$. For instance, internal relations in any [[regular category]] also form a relation equipment.
## Cartesian 1-categories equipped with relations ##
It is proven in
* Carboni, Kelly, Wood, "A 2-categorical approach to change of base and geometric morphisms, I" ([PDF](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1991__32_1/CTGDC_1991__32_1_47_0/CTGDC_1991__32_1_47_0.pdf))
that a (1,2)-category is a [[cartesian bicategory]] precisely when it is a [[cartesian object]] in a suitable 2-category of proarrow equipments (where we make a bicategory $M$ into an equipment by taking the proarrows to be those of $M$ and the arrows to be the "maps" in $M$, i.e. the morphisms having right adjoints). Here is a rough sketch of the argument, using the double-category description of equipments.
+-- {: .un_theorem}
###### Theorem
Let $\underline{K}$ be a 1-category equipped with relations, which is a [[cartesian object]] in the 2-category of relation equipments (that is, it is a **cartesian relation equipment**). Then $\mathcal{H}(\underline{K})$ is a cartesian bicategory.
=--
+-- {: .proof}
###### Proof
That $\underline{K}$ is a cartesian object means, in particular, that it is a [[pseudomonoid]] in the 2-category of equipments. By lifting the coherence data from arrows to representable proarrows, it follows that $\mathcal{H}(\underline{K})$ is a monoidal 2-category. Being a cartesian object also gives a cartesian product on objects and proarrows, with diagonals $\Delta\colon X\to X\times X$, and lifting these arrows to representable proarrows $\Delta_\bullet$ and $\Delta^\bullet$ gives each object a commutative monoid and comonoid structure. Now for any proarrow $\phi\colon X\to Y$, the square
$$\array{X & \overset{\phi}{\to} & Y\\
^\Delta \downarrow & \Downarrow & \downarrow^\Delta\\
X\times X& \underset{\phi\times \phi}{\to} & Y\times Y}$$
in $\underline{K}$ induces 2-cells, i.e. inequalities, $\Delta_\bullet \phi \le (\phi\times\phi)\Delta_\bullet$ and $\phi \Delta^\bullet \le \Delta^\bullet(\phi\times\phi)$.
=--
A [[bicategory of relations]] is a (1,2)-category which is a cartesian bicategory, and which also satisfies some additional conditions. We can also construct this structure starting from a relation equipment.
+-- {: .un_theorem}
###### Theorem
Let $\underline{K}$ be a relation equipment satisfying the hypotheses of the previous theorem, and suppose in addition that every proarrow $\phi\colon x\nrightarrow y$ in $\underline{K}$ can be written as $f_\bullet g^\bullet$ for some (vertical) arrows $f$ and $g$. (That is, "tabulations" in a certain sense exist.) Then $\mathcal{H}(\underline{K})$ is a [[bicategory of relations]].
=--
+-- {: .proof}
###### Sketch of Proof
We first verify the axiom $\Delta^\bullet \Delta_\bullet = 1$. Since $\Delta^\bullet \Delta_\bullet$ is the restriction of $1_{X\times X}$ along $\Delta$ on both sides, it suffices to show that
$$\array{X & \overset{1_X}{\to} & X\\
^\Delta\downarrow &\Downarrow& \downarrow^\Delta\\
X\times X& \underset{1_{X\times X}}{\to} & X\times X}$$
is a cartesian 2-cell in $\underline{K}$. But if we have any other square
$$\array{A & \overset{\phi}{\to} & B\\
^{(f,g)}\downarrow &\Downarrow& \downarrow^{(h,k)}\\
X\times X& \underset{1_{X\times X}}{\to} & X\times X}$$
then $(f,g)$ factoring through $\Delta$ means that $f=g$, and likewise $h=k$. Composing the given square with the projection
$$\array{X\times X & \overset{1_{X\times X}}{\to} & X\times X\\
\downarrow &\Downarrow & \downarrow\\
X& \underset{1_X}{\to} & X}$$
(which comes from being a cartesian object in $Equipments$), we obtain a square
$$\array{A & \overset{\phi}{\to} & B \\
^f\downarrow &\Downarrow & \downarrow^g\\
X& \underset{1_X}{\to} & X}$$
which factors the given square through the putative cartesian one. The factorization is unique since all 2-cells are unique.
We now verify the Frobenius axiom $\Delta^\bullet \Delta_\bullet = (1\times \Delta_\bullet)(\Delta^\bullet \times 1)$. Since $\Delta$ is associative, we have a square
$$\array{X & \overset{1_X}{\to} & X\\
^\Delta\downarrow && \downarrow^\Delta\\
X\times X & \Downarrow & X\times X\\
^{1\times \Delta}\downarrow && \downarrow^{\Delta\times 1}\\
X\times X\times X & \underset{1_{X\times X\times X}}{\to} & X\times X\times X}$$
and therefore a square
$$\array{X\times X & \overset{\Delta^\bullet \Delta_\bullet}{\to} & X\times X\\
^{1\times \Delta}\downarrow && \downarrow^{\Delta\times 1}\\
X\times X\times X & \underset{1_{X\times X\times X}}{\to} & X\times X\times X}$$
and it suffices to show that this is a cartesian 2-cell. So suppose given a square
$$\array{A & \overset{\phi}{\to} & B\\
^{(f,g,g)}\downarrow & \Downarrow & \downarrow^{(h,h,k)}\\
X\times X\times X & \underset{1_{X\times X\times X}}{\to} & X\times X\times X.}$$
The fact that $g$ and $h$ appear twice is equivalent to saying that the left and right boundaries of this square factor through $1\times\Delta$ and $\Delta\times 1$, respectively. Now by assumption, $\phi = u_\bullet v^\bullet$ for some $u\colon C\to B$ and $v\colon C\to A$. Thus our square is equivalent to one
$$\array{C & \overset{1_C}{\to} & C\\
^{(f v,g v,g v)}\downarrow & \Downarrow & \downarrow^{(h u,h u,k u)}\\
X\times X\times X & \underset{1_{X\times X\times X}}{\to} & X\times X\times X.}$$
But this is just a 2-cell in the vertical category $K$, which is a 1-category; hence we have $(f v,g v, g v) = (h u, h u, k u)$ and thus $f v = h u = g v = k u$. Calling their common value $m$, we thus have a composite square
$$\array{C & = & C & = & C\\
^{(m,m)}\downarrow && \downarrow^{m} && \downarrow^{(m,m)}\\
X\times X & \underset{\Delta^\bullet }{\to} & X & \underset{\Delta_\bullet}{\to} & X\times X}$$
(since $\Delta m = (m,m)$) which gives us the desired factorization. The other Frobenius axiom is, of course, dual.
=--
+-- {: .un_corollary}
###### Corollary
If $\underline{K}$ is a relation equipment satisfying the hypotheses of the theorem, then $\mathcal{H}(\underline{K})$ is an [[allegory]].
=--
+-- {: .proof}
###### Proof
It is shown [here](http://rfcwalters.blogspot.com/2009/10/categorical-algebras-of-relations.html) that any bicategory of relations is an allegory.
=--
## See also ##
Other attempted axiomatizations of the same idea "something that acts like the category of relations in a regular category" include:
* [[allegory]]
* [[bicategory of relations]]
## References
A comparison of "regular proarrow equipments" with "regular [[fibrations]] of [[subobjects]]" is in
* [[Finn Lawler]], *Fibrations of predicates and bicategories of relations*, [arXiv](http://arxiv.org/abs/1502.08017)
A study of equipments of relations from a double-categorical viewpoint, and a characterization of those that arise from some [[factorization system]], is in
* [[Michael Lambert]], *Double categories of relations*, [arxiv](https://arxiv.org/abs/2107.07621) 2021
[[!redirects (1,2)-category equipment]]
[[!redirects (1,2)-category equipped with proarrows]]
[[!redirects 1-category equipped with proarrows]]
[[!redirects relation equipment]]
[[!redirects relation equipments]]
[[!redirects 1-category relation equipment]]
[[!redirects 1-category equipment]]
[[!redirects 1-category equipments]]
[[!redirects 1-category proarrow equipment]]
[[!redirects (1,2)-category proarrow equipment]] |
1-dimensional Chern-Simons theory | https://ncatlab.org/nlab/source/1-dimensional+Chern-Simons+theory | +-- {: .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]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
### Idea
By the general mechanism of [[schreiber:β-Chern-Simons theory]], every [[invariant polynomial]] of total degree 2 induces a 1-dimensional Chern-Simons-like theory.
## Examples
### For the first Chern class
By the general mechanism of [[schreiber:β-Chern-Simons theory]] there is a Chern-Simons action functional associated to the first [[Chern class]], or rather to the corresponding [[invariant polynomial]], which is simply the [[trace]] map on the [[unitary Lie algebra]]
$$
tr : \mathfrak{u}(n) \to \mathbb{R}
\,.
$$
This yields an [[action functional]] for a 1-dimensional [[QFT]] as follows:
The [[configuration space]] over a 1-dimensional $\Sigma$ is the [[groupoid of Lie algebra valued 1-forms]] $\Omega^1(\Sigma, \mathfrak{u})$. After identifying $\Sigma \subset \mathbb{R}$ this may be identified with the space of $\mathfrak{u}(n)$-valued functions.
The action functional is simply the [[trace]] operation
$$
S_{CS}(\phi) = \int_\Sigma tr(\phi)
\,.
$$
Degenerate as this situation is, it can be useful to regard the [[trace]] as a Chern-Simons action functional.
* Arguments for a role in large $N$ gauge theory are in ([Nair 06](#Nair)).
* The _[[spectral action]]_ is of this form.
### For a group character, on a coadjoint orbit
For $G$ a suitable [[Lie group]] (compact, semi-simple and simply connected) the [[Wilson loops]] of $G$-[[principal connections]] are equivalently the [[partition functions]] of a 1-dimensional Chern-Simons theory.
This appears famously in the formulation of [[Chern-Simons theory]] [with Wilson lines](Chern-Simons+theory#WithWilsonLineObservables). More detailes are at _[[orbit method]]_.
### For a symplectic Lie 0-algebroid
A [[symplectic manifold]] regarded as a [[symplectic Lie n-algebroid]] with $n = 0$ induces a 1d Chern-Simons theory whose [[Chern-Simons form]] is a Liouville form of the symplectic form.
This case is discussed in ...
## Related concepts
* [[higher dimensional Chern-Simons theory]]
* **1d Chern-Simons theory**
* [[2d Chern-Simons theory]]
* [[3d Chern-Simons theory]]
* [[4d Chern-Simons theory]]
* [[5d Chern-Simons theory]]
* [[6d Chern-Simons theory]]
* [[7d Chern-Simons theory]]
* [[AKSZ sigma-models]]
* [[string field theory]]
* [[infinite-dimensional Chern-Simons theory]]
## References
### For the first Chern class
A discussion of 1d CS theory in the context of large $N$-gauge theory is in
* V.P. Nair, _The Matrix Chern-Simons One-form as a Universal Chern-Simons Theory_ Nucl.Phys.B750:289-320,2006 ([arXiv:hep-th/0605007](http://arxiv.org/abs/hep-th/0605007))
{#Nair}
An exposition of this theory formulated via an [[extended Lagrangian]] in [[higher geometric quantization]] is in section 1 of
* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:A higher stacky perspective on Chern-Simons theory]]_
Further discussion is in section 5.7 of
* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_
### For a symplectic Lie 0-algebroid
A 1d Chern-Simons theory with target a [[cotangent bundle]] is discussed in
* [[Ryan Grady]], [[Owen Gwilliam]], _One-dimensional Chern-Simons theory and the $\hat{A}$ genus_ ([arXiv:1110.3533](http://arxiv.org/abs/1110.3533))
{#GradyGwilliam}
[[!redirects 1-dimensional Chern-Simons theories]]
[[!redirects 1d Chern-Simons theory]]
[[!redirects 1d Chern-Simons theories]]
|
1-functor | https://ncatlab.org/nlab/source/1-functor |
As a $1$-[[1-category|category]] is simply a [[category]], so a __$1$-functor__ is simply a [[functor]]. See also $n$-[[n-functor|functor]].
[[!redirects 1-functor]]
[[!redirects 1-functors]]
|
1-groupoid | https://ncatlab.org/nlab/source/1-groupoid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
By _1-groupoid_ one means -- for emphasis -- a _[[groupoid]]_ regarded as an _[[β-groupoid]]_.
A quick way to make this precise is to says that a 1-groupoid is a [[Kan complex]] which is [[equivalence|equivalent]] ([[homotopy equivalence|homotopy equivalent]]) to the [[nerve]] of a groupoid: a [[simplicial coskeleton|2-coskeletal]] Kan complex. More abstractly this is a [[truncated object of an (β,1)-category|1-truncated]] [[β-groupoid]].
More generally and more vaguely: Fix a meaning of $\infty$-[[infinity-groupoid|groupoid]], however weak or strict you wish. Then a __$1$-groupoid__ is an $\infty$-groupoid such that all [[parallel pairs]] of $j$-morphisms are [[equivalence|equivalent]] for $j \geq 2$. Thus, up to equivalence, there is no point in mentioning anything beyond $1$-morphisms, except whether two given parallel $1$-morphisms are equivalent. If you rephrase equivalence of $1$-morphisms as [[equality]], which gives the same result up to [[equivalence of categories|equivalence]], then all that is left in this definition is a [[groupoid]]. Thus one may also say that a __$1$-groupoid__ is simply a groupoid.
The point of all this is simply to fill in the general concept of $n$-[[n-groupoid|groupoid]]; nobody thinks of $1$-groupoids as a concept in their own right except simply as groupoids. Compare $1$-[[1-category|category]] and $1$-[[1-poset|poset]], which are defined on the same basis.
## Related concepts
[[!include homotopy n-types - table]]
[[!redirects 1-groupoids]]
|
1-morphism | https://ncatlab.org/nlab/source/1-morphism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
* [[object]]
* [[morphism]] / **1-morphism**
[[horizontal morphism]], [[vertical morphism]]
* [[2-morphism]]
* [[3-morphism]]
* [[k-morphism]]
***
A **1-morphism** is a [[morphism]] in an [[n-category]]. One says 1-morphism for emphasis as a special case of [[k-morphism]].
[[!redirects 1-morphism]]
[[!redirects 1-morphisms]]
|
1-poset | https://ncatlab.org/nlab/source/1-poset | Fix a meaning of $\infty$-[[infinity-category|category]], however weak or strict you wish. Then a __$1$-poset__ is an $\infty$-category such that every $2$-morphism is an [[equivalence]] and all [[parallel pair]]s of $j$-morphisms are equivalent for $j \geq 1$. Thus, up to equivalence, there is no point in mentioning anything beyond $1$-morphisms, not even whether two given parallel $1$-morphisms are equivalent. Up to [[equivalence of categories|equivalence]], therefore, all that is left in this definition is a [[partial order|poset]]. Thus one may also say that a __$1$-poset__ is simply a poset.
The point of all this is simply to fill in the general concept of $n$-[[n-poset|poset]]; nobody thinks of $1$-posets as a concept in their own right except simply as posets. Compare $1$-[[1-category|category]] and $1$-[[1-groupoid|groupoid]], which are defined on the same basis. |
1-topos | https://ncatlab.org/nlab/source/1-topos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A __$1$-topos__, or __$(1,1)$-topos__, is simply a [[topos]] in the usual sense of the word. The prefix $1$- may be added when also discussing [[higher category theory|higher categorical]] types of topoi in [[higher topos theory]] such as [[2-topos]], $(\infty,1)$-[[(infinity,1)-topoi|topos]], or even $\infty$-[[infinity-topos|topos]]. Compare that a [[(0,1)-topos]] is a [[Heyting algebra]].
Similarly, a __Grothendieck $1$-topos__, or __Grothendieck $(1,1)$-topos__, is simply a [[Grothendieck topos]]. Compare that a [[Grothendieck (0,1)-topos]] is a [[frame]] (or [[locale]]).
Note that a 1-topos is not exactly a particular sort of [[2-topos]] or $\infty$-topos, just as a [[Heyting algebra]] is not a particular sort of 1-topos. The (1,2)-category of locales (i.e. (0,1)-topoi) embeds fully in the 2-category of Grothendieck 1-topoi by taking sheaves, but a locale is not identical to its topos of sheaves (and in fact no nontrivial 1-topos can be a poset), in that the following [[diagram]] of [[functors]] can *not* be filled by a [[natural isomorphism]]:
\begin{tikzcd}
\mathrm{Top}_{(0,1)}
\ar[rr, "\mathrm{Sh}(-)"]
\ar[d, "{\mathrm{forget}}"{pos=.9,yshift=5pt,sloped}]
\ar[drr, phantom, "{\#}"]
&&
\mathrm{Top}_{(1,1)}
\ar[d, "{\mathrm{forget}}"{pos=.9,yshift=5pt,sloped}]
\\
\mathrm{Cat}_{(0,1)}
\ar[rr, hook]
&&
\mathrm{Cat}_{(1,1)}
\end{tikzcd}
Likewise, one expects every Grothendieck 1-topos to give rise to a 2-topos or $\infty$-topos of [[stacks]], hopefully producing a full embedding of some sort.
## Related concepts
[[!include flavors of higher toposes -- list]]
[[!redirects (1,1)-topos]]
[[!redirects Grothendeick 1-topos]]
[[!redirects Grothendeick (1,1)-topos]]
[[!redirects 1-toposes]]
[[!redirects 1-topoi]] |
10d supergravity | https://ncatlab.org/nlab/source/10d+supergravity |
[[supergravity]] in [[dimension]] 10.
* [[type II supergravity]], [[double field theory]]
* [[heterotic supergravity]] |
10j symbol | https://ncatlab.org/nlab/source/10j+symbol |
In the context of fixed [[triangulations]] of spacetime arising in [[spin foam models]], the [[10j symbol]] is a spin network that arises in the partition function for the [[Barrett-Crane model]] of [[Riemannian quantum gravity]]. [[Greg Egan]], [[J. Daniel Christensen]], and [[John Baez]] formulated an algorithm in C++ to describe these symbols that arise in this discrete [[spin foam model]].
##Definition
A [[10j symbol]] is a spin(4) [[spin network]] containing five vertices and ten edges, where the edges are labels by [[spins]]. This symbol can be understood as a function which receives ten spins as input and produces a complex number as output, this process underpins the calculation of the partition function for the [[Barrett-Crane model]], which depends on the contraction of tensors at the vertices.
##References
* J. Daniel Christensen and Greg Egan, _An Efficient Algorithm for the Riemannian 10j Symbols._ [arxiv:gr-qc/0110045v3](http://arxiv.org/abs/gr-qc/0110045v3)
* Baez, John C., J. Daniel Christensen, and Greg Egan. _Asymptotics of 10j Symbols_ Classical and Quantum Gravity 19, no. 24 (December 21, 2002): 6489β6513. [https://doi.org/10.1088/0264-9381/19/24/315.](https://arxiv.org/abs/gr-qc/0208010) |
11-dimensional Chern-Simons theory | https://ncatlab.org/nlab/source/11-dimensional+Chern-Simons+theory | +-- {: .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]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The most basic version of [[higher dimensional Chern-Simons theory]] over a ([[compact topological space|compact]]) [[smooth manifold]] $X$ of [[dimension]] 11 has as [[field (physics)|fields]] [[cocycles]] $\hat D \colon X \to \mathbf{B}^5 U(1)_{conn}$ in degree-5 [[ordinary cohomology]] and whose [[action functional]] is given by the [[fiber integration in ordinary differential cohomology]] of the [[cup product in ordinary differential cohomology]] of such a field with itself:
$$
\hat D \mapsto \exp\left(\tfrac{i}{\hbar}\int_X \hat D \cup \hat D\right)
\,.
$$
This is the direct generalization of $U(1)$-[[3d Chern-Simons theory]] of the abelian [[7d Chern-Simons theory]], and all three are related by [[holography]] to the [[self-dual higher gauge field]] in dimension 2,6, and 10, respectively.
However, for applications in [[string theory]] more refined versions of these theories matter. For instance in 7d the full [[6d (2,0)-superconformal QFT]] contains not just a single abelian [[higher self-dual gauge field]] and accordingly the corresponding [[7d Chern-Simons theory]] is richer, namely is, by [AdS7/CFT6](#AdS-CFT#AdS7CFT6), the [[KK-compactification]] of [[11-dimensional supergravity]] on $S^4$. Similarly, in 10-dimensions the [[RR-field]] of [[type II superstring theory]] is a [[higher self-dual gauge field]] whose quantization law is of the form that makes it qualify ([Moore-Witten 99](#MooreWitten99)) as the holographic boundary theory of an 11d Chern-Simons theory. However, as a configuration of the [[RR-field]] is a [[cocycle]] in [[twisted differential K-theory]], so there should be an 11d Chern-Simons theory given ([Belov-Moore 06](#BelovMoore06)) by the [[fiber integration in differential cohomology]] of the [[cup product in differential cohomology]] in K-theory.
## Related concepts
* [[11d supergravity]]
* [[higher dimensional Chern-Simons theory]]
* [[1d Chern-Simons theory]]
* [[2d Chern-Simons theory]]
* [[3d Chern-Simons theory]]
* [[4d Chern-Simons theory]]
* [[5d Chern-Simons theory]]
* [[6d Chern-Simons theory]]
* [[7d Chern-Simons theory]]
## References
The [[self-dual higher gauge field]] nature (see there for more) in terms of a [[quadratic form]] on [[differential K-theory]] is discussed originally around
* {#MooreWitten99} [[Gregory Moore]], [[Edward Witten]], _Self-Duality, Ramond-Ramond Fields, and K-Theory_, JHEP 0005:032 (2000) [[arXiv:hep-th/9912279](http://arxiv.org/abs/hep-th/9912279), [doi:10.1088/1126-6708/2000/05/032](https://doi.org/10.1088/1126-6708/2000/05/032)]
and ([Freed 00](#Freed00)) for [[type I superstring theory]], and for [[type II superstring theory]] in
* {#Wittem} [[Edward Witten]], _Duality Relations Among Topological Effects In String Theory_, JHEP 0005:031,2000 ([arXiv:hep-th/9912086](http://arxiv.org/abs/hep-th/9912086))
* {#FreedHopkins00} [[Daniel Freed]], [[Michael Hopkins]], _On Ramond-Ramond fields and K-theory_, JHEP 0005 (2000) 044 ([arXiv:hep-th/0002027](http://arxiv.org/abs/hep-th/0002027))
* {#FMW00} D. Diaconescu, [[Gregory Moore]], [[Edward Witten]], _$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory_, Adv.Theor.Math.Phys.6:1031-1134,2003 ([arXiv:hep-th/0005090](http://arxiv.org/abs/hep-th/0005090)), summarised in _A Derivation of K-Theory from M-Theory_ ([arXiv:hep-th/0005091](http://arxiv.org/abs/hep-th/0005091))
with more refined discussion in [[twisted differential K-theory|twisted differential]] [[KR-theory]] in
* {#DFM09} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf))
See at _[[orientifold]]_ for more on this. The relation to [[11d Chern-Simons theory]] is made manifest in
* {#BelovMooreII} Dmitriy Belov, [[Greg Moore]], _Type II Actions from 11-Dimensional Chern-Simons Theories_ ([arXiv:hep-th/0611020](http://arxiv.org/abs/hep-th/0611020))
Review is in
* {#Szabo12} [[Richard Szabo]], section 3.6 and 4.6 of _Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology_ ([arXiv:1209.2530](http://arxiv.org/abs/1209.2530))
[[!redirects 11d Chern-Simons theory]] |
11d supergravity Lie 3-algebra | https://ncatlab.org/nlab/source/11d+supergravity+Lie+3-algebra | [[!redirects supergravity Lie 3-algebra]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
The _supergravity Lie 3-algebra_ $\mathfrak{sugra}(10,1)$ or [[M2-brane]] extension $\mathfrak{m}2\mathfrak{brane}$ is a [[super L-β algebra]] that is a shifted [[β-Lie algebra cohomology|extension]]
$$
0 \to b^2 \mathbb{R} \to
\mathfrak{sugra}(10,1) \to
\mathfrak{siso}(10,1)
\to
0
$$
of the [[super Poincare Lie algebra]] $\mathfrak{siso}(10,1)$ in 10+1 dimensions induced by the exceptional degree 4-[[Lie algebra cohomology|super Lie algebra cocycle]]
$$
\mu_4 = \bar \psi \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b
\;\;
\in CE(\mathfrak{siso}(10,1))
\,.
$$
This is the same mechanism by which the [[String Lie 2-algebra]] is a shifted central extension of $\mathfrak{so}(n)$.
## Properties
### The Chevalley-Eilenberg algebra
+-- {: .num_prop #TheCEAlgebra}
###### Proposition
The [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{sugra}(10,1))$ is generated on
* elements $\{e^a\}$ and $\{\omega^{ a b}\}$ of degree $(1,even)$
* a single element $c$ of degree $(3,even)$
* and elements $\{\psi^\alpha\}$ of degree $(1,odd)$
with the differential defined by
$$
d_{CE} \, \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}
$$
$$
d_{CE} \, e^{a } = \omega^a{}_b \wedge e^b + \frac{i}{2}\bar \psi \Gamma^a \psi
$$
$$
d_{CE} \, \psi = \frac{1}{4} \omega^{ a b} \Gamma_{a b} \psi
$$
$$
d_{CE} \, c = \frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b
\,.
$$
=--
> (fill in details)
### Hidden super Lie 1-algebra
At the end of ([D'Auria-Fre 82](#DAuriaFr82)) the authors ask for a super Lie 1-algebra $\mathfrak{g}$, equipped with a degree-3 element $A$ in its [[Chevalley-Eilenberg algebra]], and equipped with a homomorphism $p\colon \mathfrak{g}\longrightarrow \mathbb{R}^{10,1\vert \mathbf{32}}$ such that the pullback of the 4-cocycle $\mu_4$ along $p$ is trivialized by $A$:
$$
p^\ast \mu_4 = d_{CE}A
\,.
$$
In the [[model structure on L-infinity algebras|homotopy theory of L-infinity algebra]] this means that
$$
\array{
\mathfrak{g} &\longrightarrow& \ast
\\
{}^{\mathllap{p}}
\downarrow
&\swArrow_{\mathrlap{A}}&
\downarrow
\\
\mathbb{R}^{10,1\vert \mathbf{32}}
&\underset{\mu_4}{\longrightarrow}&
\mathbf{B}^3 \mathbb{R}
}
\,.
$$
Compare this to the characterization of $\mathfrak{sugra}(10,1)$ as the [[homotopy fiber]] of $\mu_4$, hence as the _universal_ solution to this situation
$$
\array{
\mathfrak{sugra}(10,1) &\longrightarrow& \ast
\\
\downarrow
&\swArrow_{}&
\downarrow
\\
\mathbb{R}^{10,1\vert \mathbf{32}}
&\underset{\mu_4}{\longrightarrow}&
\mathbf{B}^3 \mathbb{R}
}
\,.
$$
In any case, in ([D'Auria-Fre 82](#DAuriaFr82)) possible choices for $p \colon \mathfrak{g} \to \mathbb{R}^{10,1\vert\mathbf{32}}$ are found.
Curiously, the bosonic [[body]] of $\mathfrak{g}$ is such that when adapted to a compactification to 4d, then it is the [[exceptional tangent bundle]] on which the [[U-duality]] group [[E7]] has a canonical action.
In ([BAIPV 04](#BAIPV04)) these solutions are shown to extend to a 1-parameter family of solutions. Further comments are in ([Andrianopoli-D'Auria-Ravera 16](#AndrianopoliDAuriaRavera16)).
### Relation to M5-brane action functional
The supergravity Lie 3-algebra carries a 7-cocycle (the one that induces the [[supergravity Lie 6-algebra]]-extension of it). The corresponding WZW term is that of the [[M5-brane]] in its [[Green-Schwarz action functional]]-like formulation.
[[!include brane scan]]
### Relation to the 11-dimensional polyvector super Poincaré-algebra
{#Polyvector}
#### Via derivations
+-- {: .num_prop }
###### Proposition
Let $\mathfrak{der}(\mathfrak{sugra}(10,1))$ be the [[automorphism β-Lie algebra]] of $\mathfrak{sugra}(10,1)$. This is a [[dg-Lie algebra]]. Write
$\mathfrak{der}(\mathfrak{sugra}(10,1))_0$ for the ordinary [[Lie algebra]] in degree 0.
This is [[isomorphic]] to the polyvector-extension of the [[super PoincarΓ© Lie algebra]] (see there) in $d = 10+1$ -- the
"[[M-theory super Lie algebra]]" -- with "2-brane central charge": the Lie algebra spanned by generators
$\{P_a, Q_\alpha, J_{a b}, Z^{a b}\}$ and graded Lie brackets those of the [[super PoincarΓ© Lie algebra]] as well as
$$
[Q_\alpha, Q_\beta] = i (C \Gamma^a)_{\alpha \beta} P_a + (C \Gamma_{a b})Z^{a b}
$$
$$
[Q_\alpha, Z^{a b}] = 2 i (C \Gamma^{[a})_{\alpha \beta}Q^{b]\beta}
$$
etc.
=--
This observation appears implicitly in ([Castellani 05, section 3.1](#Castellani05)), see ([FSS 13](#FSS13)).
+-- {: .proof}
###### Proof
With the presentation of the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{sugra}(10,1))$ as in prop. \ref{TheCEAlgebra} above, the generators are identified with [[derivation]]s on $CE(\mathfrak{sugra}(10,1))$ as
$$
P_a = [d_{CE}, \frac{\partial}{\partial e^a} ]
$$
and
$$
Q_\alpha = [d_{CE}, \frac{\partial}{\partial \psi^\alpha} ]
$$
and
$$
J_{a b} = [d_{CE}, \frac{\partial}{\partial \omega^{a b}} ]
$$
and
$$
Z^{a b} = [d_{CE}, e^a \wedge e^b \wedge \frac{\partial}{\partial c}]
$$
etc. With this it is straightforward to compute the commutators.
Notably the last term in
$$
[Q_\alpha, Q_\beta] = i (C \Gamma^a)_{\alpha \beta} P_a + (C \Gamma_{a b})Z^{a b}
$$
arises from the contraction of the 4-cocycle $\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b$ with $\frac{\partial}{\partial \psi^\alpha}\wedge \frac{\partial}{\partial \psi^\beta}$.
=--
#### Via the Heisenberg Lie 3-algebras
(...)
## Applications
The field configurations of 11-dimensional [[supergravity]] may be identified with [[β-Lie algebra-valued forms]] with values in $\mathfrak{sugra}(10,1)$.
See [[D'Auria-Fre formulation of supergravity]].
## Related concepts
[[supergravity Lie 6-algebra]] $\to$ **supergravity Lie 3-algebra** $\to$ [[super PoincarΓ© Lie algebra]]
* [[4d supergravity Lie 2-algebra]]
* [[extended supersymmetry]]
* [[type II supergravity Lie 2-algebra]]
* [[type II supersymmetry algebra]]
* [[M-theory supersymmetry algebra]]
## References
The [[Chevalley-Eilenberg algebra]] of $\mathfrak{sugra}(10,1)$ first appears in (3.15) of
* {#DAuriaFr82} [[Riccardo D'Auria]], [[Pietro FrΓ©]], _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982)
and later in the textbook
* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro FrΓ©]], _[[Supergravity and Superstrings - A Geometric Perspective]]_
The manifest interpretation of this as a [[Lie 3-algebra]] and the supergravity field content as [[β-Lie algebra valued forms]] with values in this is mentioned in
* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:L-β algebra connections]]_
* {#FSS13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_
A systematic study of the super-[[Lie algebra cohomology]] involved is in
* [[John Baez]], [[John Huerta]], _Division algebras and supersymmetry I_ ([arXiv:0909.0551](http://arxiv.org/abs/0909.0551))
* [[John Baez]], [[John Huerta]], _Division algebras and supersymmetry II_ ([arXiv:1003.34360](http://arxiv.org/abs/1003.3436))
See also [[division algebra and supersymmetry]].
Further discussion of its "hidden" super Lie algebra includes
* {#BAIPV04} [[Igor Bandos]], [[JosΓ© de AzcΓ‘rraga]], [[JosΓ© M. Izquierdo]], Moises Picon, Oscar Varela, _On the underlying gauge group structure of $D=11$ supergravity_, Phys. Lett. B **596** (2004) 145-155 [[arXiv;hep-th/0406020](http://arxiv.org/abs/hep-th/0406020)]
* [[Igor Bandos]], [[Jose de Azcarraga]], Moises Picon, Oscar Varela, _On the formulation of $D=11$ supergravity and the composite nature of its three-from field_, Annals Phys. **317** (2005) 238-279 [[arXiv:hep-th/0409100](https://arxiv.org/abs/hep-th/0409100)]
* {#AndrianopoliDAuriaRavera16} [[Laura Andrianopoli]], [[Riccardo D'Auria]], [[Lucrezia Ravera]], _Hidden Gauge Structure of Supersymmetric Free Differential Algebras_ [[arXiv:1606.07328](https://arxiv.org/abs/1606.07328)]
Further review:
* [[Lucrezia Ravera]], *On the hidden symmetries of $D=11$ supergravity* [[arXiv:2112.00445](https://arxiv.org/abs/2112.00445)]
The computation of the automorphism Lie algebra of $\mathfrak{sugra}(10,1)$ is in
* {#Castellani05} [[Leonardo Castellani]], _Lie derivatives along antisymmetric tensors and the M-theory superalgebra_, J. Phys. Math. **3** (2011) 1-7 [[arXiv:hep-th/0508213](http://arxiv.org/abs/hep-th/0508213)]
A similar argument with more explicit use of the Lie 3-algebra as underlying the [[Green-Schwarz action functional|Green-Schwarz-like action functional]] for the [[M5-brane]] is in
* [[Dmitri Sorokin]], [[Paul Townsend]], _M-theory superalgebra from the M-5-brane_, Phys. Lett. B **412** (1997) 265-273 [[arXiv:hep-th/9708003](http://arxiv.org/abs/hep-th/9708003)]
[[!redirects m2brane]]
|
120-cell | https://ncatlab.org/nlab/source/120-cell |
#Contents#
* table of contents
{:toc}
## Idea
...one of the [[regular polytopes]] in [[dimension]] 4...
...hence a higher dimensional analogs of the [[Platonic solids]]...
\begin{centre}
\begin{imagefromfile}
"file_name": "Chilton120Cell.jpg",
"width": 360
\end{imagefromfile}
\end{centre}
> graphics grabbed from [Stillwell 01](#Stillwell01)
## Related concepts
* [[24-cell]]
* [[600-cell]]
## References
* {#Stillwell01} John Stillwell, _The story of the 120-Cell_, Notices of the American Mathematical Society 2001 ([pdf](https://www.ams.org/notices/200101/fea-stillwell.pdf))
See also
* Wikipedia, _[120-cell](https://en.wikipedia.org/wiki/120-cell)_ |
13-sphere | https://ncatlab.org/nlab/source/13-sphere |
+-- {: .rightHandSide}
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###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[n-sphere]] of [[dimension]] $n = 13$.
## Properties
### AKM-Theorem
In higher-dimensional analogy to the [[AKM-theorem]] we have:
+-- {: .num_prop}
###### Proposition
The [[13-sphere]] is the [[quotient space]] of the (right-)[[octonionic projective plane]] by the left multiplication [[action]] by [[Sp(1)]]:
$$
\mathbb{O}P^2 / \mathrm{Sp}(1)
\simeq
S^{13}
$$
=--
([Atiyah-Berndt 02](#AtiyahBerndt02))
## References
* {#AtiyahBerndt02} [[Michael Atiyah]], [[JΓΌrgen Berndt]], in: Surv. Differ. Geom. VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck (International Press, Somerville, MA, 2003) 1-27 ([arXiv:math/0206135](https://arxiv.org/abs/math/0206135), [doi:10.4310/SDG.2003.v8.n1.a1](https://dx.doi.org/10.4310/SDG.2003.v8.n1.a1))
|
15-sphere | https://ncatlab.org/nlab/source/15-sphere |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[sphere]] of [[dimension]] 15.
## Properties
### Octonionic Hopf fibration
{#OctonionicHopfFibration}
The 15-sphere participates in the [[octonionic Hopf fibration]], the analog of the complex [[Hopf fibration]] with the field of [[complex numbers]] replaced by the division ring of [[octonions]] $\mathbb{O}$.
$$
\array{
S^7 &\hookrightarrow& S^15
\\
&& \downarrow^\mathrlap{p}
\\
&& S^8
}
$$
Here the idea is that $S^{15}$ can be construed as $\{(x, y) \in \mathbb{O}^2: {|x|}^2 + {|y|}^2 = 1\}$, with $p$ mapping $(x, y)$ to $x/y$ as an element in the [[projective line]] $\mathbb{P}^1(\mathbb{O}) \cong S^8$, with each [[fiber]] a [[torsor]] parametrized by octonionic [[scalars]] $\lambda$ of unit [[norm]] (so $\lambda \in S^7$).
### Other properties
* $S^{15}$ is the only sphere that admits three homogeneous Einstein metrics.
* It is the only sphere that appears as a regular orbit in three cohomogeneity one actions on projective spaces, namely of $SU(8)$, $Sp(4)$ and $Spin(9)$ on $\mathbb{C}P^8$, $\mathbb{H}P^4$ and $\mathbb{O}P^2$, respectively ([OPPV, p. 1](#OPPV))
## References
* {#OPPV} Liviu Ornea, Maurizio Parton, Paolo Piccinni, Victor Vuletescu, _Spin(9) geometry of the octonionic Hopf fibration_, ([arXiv:1208.0899](http://arxiv.org/abs/1208.0899)) |
1d Dijkgraaf-Witten theory | https://ncatlab.org/nlab/source/1d+Dijkgraaf-Witten+theory |
+-- {: .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]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
## Idea
The 1-dimensional analog of 3d [[Dijkgraaf-Witten theory]].
For the moment a detailed discussion is at _[[local prequantum field theory]]_ in the section _[1d Dijkgraaf-Witten theory](prequantum+field+theory#1dDWTheory)_.
[[!redirects 1-dimensional Dijkgraaf-Witten theory]]
|
1d WZW model | https://ncatlab.org/nlab/source/1d+WZW+model |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Wess-Zumino-Witten theory
+--{: .hide}
[[!include infinity-Wess-Zumino-Witten theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In the context of [[infinity-Wess-Zumino-Witten theory - contents|higher dimensional WZW models]] the following 1-dimensional [[sigma-models]] are seen to be examples:
* the free [[non-relativistic particle]];
* the massive [[Green-Schwarz action functional|Green-Schwarz]] [[superparticle]] (the [[D0-brane|D0]] [[super p-brane]]).
([Azcarraga-Izqierdo 95, section 8.3 and 8.7](#AzcarragaIzqierdo)) .
## Examples
### Free massive non-relativistic particle
Write
$$
H \coloneqq G/R
$$
for the [[coset]] obtained as the [[quotient]] of the [[Galilei group]] in some [[dimension]] $d$ by the [[rotation group|group of]] [[rotations]]. This $H$ has a canonical global [[coordinate chart]] $(t,x, \dot x)$. We may regard it as the first order [[jet bundle]] to the [[bundle]] $\mathbb{R}^d \times \mathbb{R} \to \mathbb{R}$ whose [[sections]] are [[trajectories]] in [[Cartesian space]] $\mathbb{R}^d$ (the [[field bundle]] for the 1d [[sigma-model]] with [[target space]] $\mathbb{R}^d$).
Among the $H$-[[left invariant differential 2-forms]] on $H$ is
$$
\omega_m \coloneqq m (d_{dR} x - \dot x d_{dR} t) \wedge d_{dR} \dot x
$$
for some $m \in \mathbb{R}$ (where a contraction of vectors is understood).
This is a representative of a degree-2 [[cocycle]] in the [[Lie algebra cohomology]] of $Lie(H)$. We may regard this as the [[curvature]] of a [[connection on a bundle|connection]] [[differential 1-form|1-form]]
$$
A \coloneqq m \dot x (d_{dR} x - \frac{1}{2} \dot x d_{dR} t)
\,.
$$
Hence the value of the [[action functional]] of the corresponding 1d pure (topological) WZW model on a field configuration is
$$
m \int_{\Sigma_1} \dot x (d_{dR} x - \frac{1}{2} \dot x d_{dR} t)
=
m \int_{\Sigma_1} \frac{\partial L}{ \partial \dot x}(d x - \dot x)+ L d t
\,,
$$
where $L(x, \dot x) d t = \frac{1}{2}m \dot x^2 d t$ is the [[Lagrangian]] of the the free [[non-relativistic particle]] of [[mass]] $m$.
Evaluated on [[jet prolongations]] of [[sections]] of the [[field bundle]], for which the relation $d x = \dot x d t$ holds, then the first term of this expression vanishes and so the resulting WZW-type [[action functional]] is that of the free [[non-relativistic particle]].
See ([Azcarraga-Izqierdo, section 8.3](#AzcarragaIzqierdo)) for a useful account.
## References
* {#AzcarragaIzqierdo} [[JosΓ© de AzcΓ‘rraga]], J. Izqierdo, sections 8.3 and 8.7 of _[[Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics]]_, Cambridge monographs of mathematical physics, (1995)
[[!redirects 1d WZW models]] |
1lab | https://ncatlab.org/nlab/source/1lab | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
\tableofcontents
## Idea
The **1lab** is an online reference maintained by [[AmΓ©lia Liao]] primarily for [[category theory]] done in [[univalent type theory|univalent]] [[cubical type theory]] implemented in [[cubical Agda]].
Web:
* [1lab website](https://1lab.dev/)
* [Github page for the 1lab](https://github.com/plt-amy/1lab)
The name "1lab" is apparently a reference to (the name of) the "[[nLab|$n$Lab]]", with the specification $n = 1$ meant to highlight this focus on [[univalent category|type-theoretic]] [[1-categories]] (as opposed to [[higher category theory|higher categories]], whose type-theoretic formulation remains underdeveloped, generally).
## Related entries
[[!include proof assistants and formalization projects -- list]]
|
1T relation | https://ncatlab.org/nlab/source/1T+relation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[knot theory]], by the _1-term relations_, or _1T relations_ for short, one means the [[relation]] on the [[linear span]] of the [[set]] of [[chord diagrams]] which relates every chord diagram with an "isolated chord" (one not intersecting any other chord in the diagram) to [[zero]].
The [[quotient space]] of the [[linear span]] of [[chord diagrams]] by the 1T and by the [[4T relations]] is the [[domain]] for _[[unframed weight systems]]_ on chord diagrams. If one omits the 1T relation and imposes only the [[4T relation]], one speaks of _[[framed weight systems]]_.
## Related concepts
[[!include chord diagrams and weight systems -- table]]
## References
Original articles
* {#BarNatan95} [[Dror Bar-Natan]], _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (<a href="https://doi.org/10.1016/0040-9383(95)93237-2">doi:10.1016/0040-9383(95)93237-2</a>, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))
* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))
(for [[horizontal chord diagrams]])
Textbook accounts
* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Section 4 of: _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv/1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))
* [[David Jackson]], [[Iain Moffat]], Section 11 of: _An Introduction to Quantum and Vassiliev Knot Invariants_, Springer 2019 ([doi:10.1007/978-3-030-05213-3](https://link.springer.com/book/10.1007/978-3-030-05213-3))
Lecture notes:
* {#BarNatanStoimenow97} [[Dror Bar-Natan]], Alexander Stoimenow, _The Fundamental Theorem of Vassiliev Invariants_ ([arXiv:q-alg/9702009](https://arxiv.org/abs/q-alg/9702009))
[[!redirects 1T relations]]
[[!redirects 1T-relation]]
[[!redirects 1T-relations]]
[[!redirects 1-term relation]]
[[!redirects 1-term relations]]
|
2-adjunction | https://ncatlab.org/nlab/source/2-adjunction | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A **2-adjunction** is a common name for various kinds of adjunctions in [[2-category theory]]; not only adjunctions between [[2-categories]] themselves, but more generally adjunctions within an arbitrary [[3-category]]. Since there are various different levels of strictness at which one works in 2-category theory, there are various different kinds of 2-adjunction, including:
* A [[strict 2-adjunction]], involving [[strict 2-categories]], [[strict 2-functors]], [[strict 2-natural transformations]], and on-the-nose [[triangle identities]]. This can be generalized to an arbitrary [[strict 3-category]].
* A [[biadjunction]], involving [[weak 2-categories]] ([[bicategories]]), weak 2-functors ([[pseudofunctors]]), [[pseudonatural transformations]], and triangle identities up to isomorphism. This can be generalized to an arbitrary [[tricategory]].
* The word [[pseudoadjunction]] is sometimes used interchangeably with "biadjunction", but often it refers specifically to a notion definable internal to a [[Gray-category]]. When specialized to the canonical Gray-category $Gray$, this produces a notion involving strict 2-categories and strict 2-functors, but *pseudo* natural transformations, and triangle identities up to isomorphism.
* A [[lax 2-adjunction]] involves triangle identities only up to noninvertible transformation, and perhaps [[lax 2-functors]] and/or [[lax 2-natural transformations]] as well. If only the triangle identities are lax, this can be defined at any of the above levels of strictness and internalized in any of the above ways; but if the functors or transformations are lax, then it doesn't fit very easily into a well-known abstract 3-categorical structure, since there is no 3-category (as usually understood) including lax functors or transformations.
There are also more specialized kinds of 2-adjunction, such as
* An [[idempotent 2-adjunction]] is a categorification of an [[idempotent adjunction]].
* A [[lax-idempotent 2-adjunction]] is a weaker sort of categorification, relevant to [[lax-idempotent 2-monads]].
## Related concepts
* [[adjoint functor]], [[adjoint triple]], [[adjoint quadruple]]
* [[proadjoint]], [[Hopf adjunction]]
* **2-adjunction**
[[biadjunction]], [[pseudoadjunction]], [[lax 2-adjunction]]
[[strict adjoint 2-functor]]
[[lax-idempotent 2-adjunction]]
* [[adjoint (infinity,1)-functor]]
* [[(β,n)-category with adjoints]]
[[!redirects 2-adjointness]]
[[!redirects 2-adjunctions]]
[[!redirects adjoint 2-functor]]
[[!redirects 2-adjoint]]
[[!redirects 2-adjoints]]
[[!redirects left 2-adjoint]]
[[!redirects left 2-adjoints]]
[[!redirects right 2-adjoint]]
[[!redirects right 2-adjoints]]
|
2-algebraic geometry | https://ncatlab.org/nlab/source/2-algebraic+geometry |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Where in [[algebraic geometry]] one considers [[spaces]] which are [[Isbell duality|formally dual]] to [[commutative rings]], in "2-algebraic geometry" one considers spaces formally dual to _[[2-rigs]]_, namely to certain [[tensor categories]] or more generally to [[tensor (β,1)-categories]].
Curiously, the more general idea of regarding certain [[abelian categories]] and certain linear [[A-β categories]] ([[stable (β,1)-categories]]) as analogs of categories of [[quasicoherent sheaves]] on some space (i.e. ignoring the monoidal structure) is much older, this is the program of [[derived noncommutative geometry]], see also at [derived algebraic geometry -- Relation to noncommutative geometry](derived+algebraic+geometry#RelationToDerivedNoncommutativeGeometry).
The systematic use of the tensor product structure here goes back to ([Balmer 02](#Balmer02)) and the concept of the [[spectrum of a tensor triangulated category]]. In ([Lurie](#Lurie)) the concept of "2-[[affine scheme]]" is studied more systematically. Then ([CJF 11](#CJF11)) introduce the terminology of "2-algebraic geometry".
## Related concepts
* [[spectrum of a tensor triangulated category]]
* [[spectrum of a symmetric monoidal (β,1)-category]]
## References
* {#Balmer02} [[Paul Balmer]], _Presheaves of triangulated categories and reconstruction of schemes_, Mathematische Annalen __324__:3 (2002), 557-580 [dvi](http://www.math.ucla.edu/~balmer/Pubfile/Reconstr.dvi), [pdf](http://www.math.ucla.edu/~balmer/Pubfile/Reconstr.pdf) [ps](http://www.math.ucla.edu/~balmer/Pubfile/Reconstr.ps); _The spectrum of prime ideals in tensor triangulated categories_, J. Reine Angew. Math. __588__:149–168, 2005 [dvi](http://www.math.ucla.edu/~balmer/Pubfile/Spectrum.dvi) [pdf](http://www.math.ucla.edu/~balmer/Pubfile/Spectrum.pdf) [ps](http://www.math.ucla.edu/~balmer/Pubfile/Spectrum.ps); _Spectra, spectra, spectra - Tensor triangular spectra versus Zariski spectra of endomorphism rings_, Alg. and Geom. Topology __10__:3 (2010) 1521-1563 [dvi](http://www.math.ucla.edu/~balmer/Pubfile/SSS.dvi) [pdf](http://www.math.ucla.edu/~balmer/Pubfile/SSS.pdf) [ps](http://www.math.ucla.edu/~balmer/Pubfile/SSS.ps)
* {#Lurie} [[Jacob Lurie]], _[[Tannaka duality for geometric stacks]]_.
* {#CJF11} [[Alexandru Chirvasitu]], [[Theo Johnson-Freyd]], _The fundamental pro-groupoid of an affine 2-scheme_, Applied Categorical Structures, Vol 21, Issue 5 (2013), pp. 469β522. ([DOI](http://dx.doi.org/10.1007/s10485-011-9275-y), [arXiv:1105.3104](http://arxiv.org/abs/1105.3104))
* {#Brandenburg14} [[Martin Brandenburg]], _Tensor categorical foundations of algebraic geometry_ ([arXiv:1410.1716](http://arxiv.org/abs/1410.1716))
|
2-categorical limit > history | https://ncatlab.org/nlab/source/2-categorical+limit+%3E+history | < [[2-categorical limit]]
[[!redirects 2-categorical limit -- history]] |
2-category | https://ncatlab.org/nlab/source/2-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Idea
The notion of a _2-category_ generalizes that of [[category]]: a 2-category is a [[higher category theory|higher category]], where on top of the objects and morphisms, there are also 2-morphisms. In old texts, strict 2-categories are occasionally called _hypercategories_.
A 2-category consists of
* [[objects]];
* 1-[[morphisms]] between objects;
* [[2-morphisms]] between morphisms.
The morphisms can be [[composition|composed]] along the objects, while the 2-morphisms can be composed in two different directions: along objects -- called [[horizontal composition]] -- and along morphisms -- called [[vertical composition]]. The composition of morphisms is allowed to be associative only up to [[coherent]] [[associator]] 2-morphisms.
2-categories are also a [[horizontal categorification]] of [[monoidal categories]]: they are like monoidal categories with many objects.
2-categories provide the context for discussing
* [[adjunction]]s;
* [[monad]]s.
The concept of 2-category generalizes further in [[higher category theory]] to [[n-categories]], which have [[k-morphism]]s for all $k\le n$.
2-categories form a [[3-category]], [[2Cat]].
## Definitions
### Strict 2-categories
The easiest definition of 2-category is that it is a category [[enriched category|enriched]] over the [[cartesian monoidal category]] [[Cat]]. Thus it has a collection of objects,. and for each pair of objects a category $hom(x,y)$. The objects of these hom-categories are the morphisms, and the morphisms of these hom-categories are the 2-morphisms. This produces the classical notion of [[strict 2-category]].
### General 2-categories
{#Weak}
For some purposes, strict 2-categories are too strict: one would like to allow composition of morphisms to be associative and unital only up to coherent invertible 2-morphisms. A direct generalization of the above "enriched" definition produces the classical notion of [[bicategory]].
One can also obtain notions of 2-category by specialization from the case of higher categories. Specifically, if we fix a meaning of $\infty$-[[infinity-category|category]], however weak or strict we wish, then we can define a __$2$-category__ to be an $\infty$-category such that every 3-morphism is an [[equivalence]], and all parallel pairs of $j$-morphisms are equivalent for $j \geq 3$. It follows that, up to equivalence, there is no point in mentioning anything beyond $2$-morphisms, except whether two given parallel $2$-morphisms are equivalent. In some models of $\infty$-categories, it is possible to make this precise by demanding that all parallel pairs of $j$-morphisms are actually *equal* for $j\geq 3$, producing a simpler notion of 2-category in which we can speak about [[equality]] of 2-morphisms instead of equivalence. (This is the case for both strict $2$-categories and bicategories.)
All of the above definitions produce "equivalent" theories of 2-category, although in some cases (such as the fact that every bicategory is equivalent to a strict 2-category) this requires some work to prove. On the nLab, we often use the word "2-category" in the general sense of referring to whatever model one may prefer, but usually one in which composition is weak; a [[bicategory]] is an adequate definition. One should beware, however, that in the literature it is common for "2-category" to refer only to *strict* 2-categories.
A 2-category in which all 1-morphisms and 2-morphisms are invertible is a [[2-groupoid]].
## Examples
* The archetypical 2-category is [[Cat]], the 2-category whose
* objects are [[categories]];
* morphisms are [[functor]]s;
* 2-morphisms are [[natural transformation]];
horizontal composition of 2-morphisms is the [[Godement product]].
This happens to be a [[strict 2-category]].
* More generally, for $V$ any enriching category (such as a Benabou [[cosmos]]), there is a 2-category $V Cat$ whose
* objects are $V$-[[enriched categories]];
* morphisms are $V$-enriched functors; and
* 2-morphisms are $V$-natural transformations.
* On the other hand, for any such $V$ we also have a [[bicategory]] $V$-[[Prof]] whose
* objects are $V$-[[enriched categories]];
* morphisms are $V$-[[profunctor]]s; and
* 2-morphisms are natural transformations between these.
* If $C$ is a category with [[pullbacks]], then there is a bicategory [[Span]]$(C)$ whose
* objects are the objects of $C$;
* morphisms are [[spans]] in $C$; and
* 2-morphisms are morphisms of spans.
* Every [[monoidal category]] $C$ may be thought of as a [[bicategory]] $\mathbf{B}C$ (its [[delooping]]). This has
* a single object $\bullet$;
* morphisms are the objects of $C$: $(\mathbf{B}C)_1 = C_0$;
* 2-morphisms are the morphisms of $C$ : $(\mathbf{B}C)_2 = C_1$;
[[horizontal composition]] in $\mathbf{B}C$ is the [[tensor product]] in $C$ and [[vertical composition]] in $\mathbf{B}C$ is composition in $C$.
Conversely, every 2-category with a single object comes from a monoidal category this way, so the concepts are effectively equivalent. (Precisely: the 2-category of _pointed_ 2-categories with a single object is equivalent to that of monoidal categories). For more on this relation see [[delooping hypothesis]], [[k-tuply monoidal n-category]], and [[periodic table]].
* Every [[2-groupoid]] is a 2-category. For instance
* for $A$ any [[abelian group]], the double [[delooping]] $\mathbf{B}^2 A$ is the strict 2-category with a single object, a single 1-morphisms, set of 2-moprhisms being $A$ and both horizontal composition as well as [[vertical composition]] being the product in $A$.
* for $G$ any [[2-group]], its single [[delooping]] is a 2-groupoid with a single object.
* Every [[topological space]] has a [[path 2-groupoid]].
* Every [[(β,2)-category]] has a **homotopy 2-category**, obtained by dividing out all 3-morphisms and higher.
## Properties
### Double nerve
An ordinary [[category]] has a [[nerve]] which is a [[simplicial set]]. For 2-categories one may consider their [[double nerve]] which is a [[bisimplicial set]].
There is also a 2-nerve. ([LackPaoli](#LackPaoli))
(...)
### Model category structure {#ModelCategoryStructure}
There is a [[model category]] structure on 2-categories -- sometimes known as the [[folk model structure]] -- that models the [[(2,1)-category]] underlying [[2Cat]] ([Lack](#LackFolkModel)).
For strict 2-categories this is the restriction of the corresponding [[folk model structure]] on [[strict omega-categories]].
* The weak equivalences are the [[2-functor]]s that are equivalences of 2-categories.
* The acyclic fibrations are the [[k-surjective functor]]s for all $k$.
#### Free resolutions {#FreeResolution}
**Theorem** A strict 2-category $C$ is cofibrant precisely if the underlying 1-category $C_1$ is a [[free category]].
This is theorem 4.8 in ([LackStrict](#LackStrict)). This is a special case of the more general statement that free strict $\omega$-categories are given by [[computad]]s.
**Example (free resolution of a 1-category).** Let $C$ be an ordinary category (a 1-category) regarded as a strict 2-category. Then the cofibrant resolution $\hat C \stackrel{\simeq}{\to} C$ is the strict 2-category given as follows:
* the objects of $\hat C$ are those of $C$;
* the morphisms of $\hat C$ are finite sequences of composable morphisms of $C$, and composition is concatenation of such sequences
(hence $(\hat C)_1$ is the [[free category]] on the [[quiver]] underlying $C$);
* the 2-morphisms of $\hat C$ are _generated_ from 2-morphisms $c_{f,g}$ of the form
$$
\array{
&& y
\\
& {}^{\mathllap{f}}\nearrow &\Downarrow^{c_{f,g}}& \searrow^{\mathrlap{g}}
\\
x && \underset{g \circ_C f }{\to} && z
}
$$
and their formal inverses
$$
\array{
&& y
\\
& {}^{\mathllap{f}}\nearrow &\Uparrow^{c_{f,g}^{-1}}& \searrow^{\mathrlap{g}}
\\
x && \underset{g \circ_C f }{\to} && z
}
$$
for all composable $f,g \in Mor(C)$ with composite (in $C$!) $g \circ_C f$;
subject to the _relation_ that for all composable triples $f,g,h \in Mor(C)$
the following equation of 2-morphisms holds
$$
\array{
y &\to& &\stackrel{g}{\to}& &\to& && z
\\
\uparrow &\seArrow^{c_{f,g}}& && & \nearrow & && \downarrow
\\
{}^{\mathllap{f}}\uparrow && & \nearrow & &&&& \downarrow^{\mathrlap{h}}
\\
\uparrow & \nearrow & && &\Downarrow^{c_{h,(g\circ_C f)}}& && \downarrow
\\
x &\to& &\underset{h \circ (g \circ_C f)}{\to}& &\to& &\to& w
}
\;\;\;
=
\;\;\;
\array{
y &\to& &\stackrel{g}{\to}& &\to& && z
\\
\uparrow &\searrow& && & & &\swArrow_{c_{g,h}}& \downarrow
\\
{}^{\mathllap{f}}\uparrow && & \searrow & &&&& \downarrow^{\mathrlap{h}}
\\
\uparrow &\Downarrow_{c_{f,(g \circ_C h)}}& && &\searrow& && \downarrow
\\
x &\to& &\underset{( h \circ_C g) \circ f}{\to}& &\to& &\to& w
}
$$
**Observation** Let $D$ be any strict 2-catgeory. Then a [[pseudofunctor]] $C \to D$ is the same as a strict 2-functor $\hat C \to D$.
## 2-categorical concepts
**constructions**
* [[opposite 2-category]]
**extra properties**
* [[regular 2-category]]
* [[exact 2-category]]
* [[coherent 2-category]]
* [[extensive 2-category]]
* [[2-pretopos]]
* [[2-topos]]
**types of morphisms**
* [[subcategory]]
* [[faithful morphism]]
* [[fully faithful morphism]]
* [[conservative morphism]]
* [[pseudomonic morphism]]
* [[discrete morphism]]
**specific versions**
* globular [[strict 2-category]]
* [[bicategory]]
**limit notions**
* [[2-limit]]
* [[strict 2-limit]]
* [[flexible limit]]
* [[PIE limit]]
**model structures**
* [[canonical model structure]]
* [[2-trivial model structure]]
## Related concepts
* [[0-category]], [[(0,1)-category]]
* [[category]]
* **2-category**
[[equivalence in a 2-category]]
[[localization of a 2-category]]
[[2-type theory]], [[directed type theory]]
* [[3-category]]
* [[n-category]]
* [[(β,0)-category]]
* [[(β,1)-category]]
* [[(β,2)-category]]
* [[(β,n)-category]]
* [[(n,r)-category]]
* [[double category]]
## References
{#References}
Despite its being frequently attributed to Ehresmann, the notion of [[strict 2-categories]] is due to:
* {#BΓ©nabou65} [[Jean BΓ©nabou]], Example (5) of: *CatΓ©gories relatives*, C. R. Acad. Sci. Paris **260** (1965) 3824-3827 [[gallica](https://gallica.bnf.fr/ark:/12148/bpt6k4019v/f37.item)]
> (conceived as [[Cat]]-[[enriched categories]] and called *2-categories*)
* {#Maranda65} [[Jean-Marie Maranda]], Def. 1 in: *Formal categories*, Canadian Journal of Mathematics **17** (1965) 758-801 [[doi:10.4153/CJM-1965-076-0](https://doi.org/10.4153/CJM-1965-076-0), [pdf](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A7C463460EB8CAC64C2CA340F870CF80/S0008414X00039729a.pdf/formal-categories.pdf)]
> (conceived as [[Cat]]-[[enriched categories]] and called *categories of the second type*)
both apparently following or inspired by the earlier definition of *[[double categories]]* due to
* [[Charles Ehresmann]], *CatΓ©gories double et catΓ©gories structurΓ©es*, C.R. Acad. Paris 256 (1963) 1198-1201 [[[Ehresmann-CategoriesDoubles.pdf:file]], [gallica](https://gallica.bnf.fr/ark:/12148/bpt6k3208j/f1246)]
An early definition also appears in the following, where it is mistakenly attributed to [[Charles Ehresmann]]:
* {#EK65} [[Samuel Eilenberg]], [[G. Max Kelly]], *Closed Categories*, p. 425 in: [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. RΓΆhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]
> (expressed entirely in components, under the name *hypercategories*)
The fundamental structure of the [[2-category of categories]] (namely the [[vertical composition]], [[horizontal composition]] and the [[whiskering]] of [[natural transformations]]) was first described in:
* {#Godement58} [[Roger Godement]], Appendix (pp. 269) of: *Topologie algΓ©brique et theorie des faisceaux*, ActualitΓ©s Sci. Ind. **1252**, Hermann, Paris (1958) [[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), [[Godement-TopologieAlgebrique.pdf:file]]]
Early discussion of the general notion of bicategories:
* {#BΓ©nabou1967} [[Jean BΓ©nabou]], *Introduction to Bicategories*, Lecture Notes in Mathematics **47** Springer (1967) 1-77 [[doi:10.1007/BFb0074299](http://dx.doi.org/10.1007/BFb0074299)]
Exposition and review:
* {#KellyStreet74} [[Max Kelly]], [[Ross Street]], *Review of the elements of 2-categories*, Sydney Category Seminar 1972/1973, in [[G. Max Kelly]] (ed.) Lecture Notes in Mathematics **420**, Springer (1974) [[doi:10.1007/BFb0063101](https://doi.org/10.1007/BFb0063101)]
* {#Street96} [[Ross Street]], *Categorical Structures*, in Handbook of Algebra Vol. 1 (ed. M. Hazewinkel), Elsevier Science, Amsterdam (1996) [<a href="https://doi.org/10.1016/S1570-7954(96)80019-2">doi:10.1016/S1570-7954(96)80019-2</a>, [pdf](http://maths.mq.edu.au/~street/45.pdf), [[Street-CategoricalStructures.pdf:file]], [ISBN:978-0-444-82212-3](https://shop.elsevier.com/books/handbook-of-algebra/hazewinkel/978-0-444-82212-3)]
* [[Ross Street]], _Encyclopedia article on 2-categories and bicategories_ ([pdf](http://www.maths.mq.edu.au/~street/Encyclopedia.pdf))
* [[Tom Leinster]], _Basic bicategories_ ([arXiv:9810017](http://arxiv.org/abs/math/9810017))
* {#Lack10} [[Steve Lack]], _A 2-categories companion_, In: Baez J., May J. (eds.) *[[Towards Higher Categories]]*. The IMA Volumes in Mathematics and its Applications, vol 152. Springer 2010 ([arXiv:math.CT/0702535](http://arxiv.org/abs/math.CT/0702535), [doi:10.1007/978-1-4419-1524-5_4](https://doi.org/10.1007/978-1-4419-1524-5_4))
> (including discussion of ([[strict 2-limits|strict]]) [[2-limits]])
* [[John Power]], _2-Categories_, BRICS Notes Series 1998 ([pdf](http://www.brics.dk/NS/98/7/BRICS-NS-98-7.pdf))
Comprehensive textbook accounts:
* [[Ofer Gabber]], [[Lorenzo Ramero]], Chapter 2 of: *Foundations for almost ring theory* ([arXiv:math/0409584](https://arxiv.org/abs/math/0409584))
* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], *2-Dimensional Categories*, Oxford University Press (2021) [[arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378)]
On [[coherence theorems]]:
* {#Power89} [[A. John Power]], _A general coherence result._ J. Pure Appl. Algebra 57 (1989), no. 2, 165–173. [doi:10.1016/0022-4049(89)90113-8](http://dx.doi.org/10.1016/0022-4049%2889%2990113-8) [MR0985657](http://www.ams.org/mathscinet-getitem?mr=985657)
Relation between bicategories and Tamsamani weak 2-categories:
* {#LackPaoli} [[Steve Lack]], [[Simona Paoli]], _2-nerves for bicategories_ ([arXiv](http://arxiv.org/abs/math/0607271))
* [[Simona Paoli]], _From Tamsamani weak 2-categories to bicategories_ ([arXiv](http://www.maths.mq.edu.au/~simonap/Bicategories_Rev_4.pdf))
There is a [[model category]] structure on 2-categories -- the [[canonical model structure]] -- that models the [[(2,1)-category]] underlying [[2Cat]]:
* {#LackStrict} [[Steve Lack]], _A Quillen Model Structure for 2-Categories_, K-Theory 26: 171–205, 2002. ([website](http://www.maths.usyd.edu.au/u/stevel/papers/qmc2cat.html))
* {#LackFolkModel} [[Steve Lack]], _A Quillen Model Structure for Biategories_, K-Theory 33: 185-197, 2004. ([website](http://www.maths.usyd.edu.au/u/stevel/papers/qmcbicat.html))
Discussion of weak 2-categories in the style of [[A-infinity categories]] is (using [[dendroidal sets]] to model the higher [[operads]]) in
* Andor Lucacs, _Dendroidal weak 2-categories_ ([arXiv:1304.4278](http://de.arxiv.org/abs/1304.4278))
* Jonathan Chiche, _La théorie de l'homotopie des 2-catégories_, thesis, [arXiv](http://arxiv.org/abs/1411.6936).
[[!redirects 2-category]]
[[!redirects 2-categories]]
[[!redirects weak 2-category]]
[[!redirects weak 2-categories]]
[[!redirects hypercategory]]
[[!redirects hypercategories]]
[[!redirects 2-categorical]]
|
2-category equipped with proarrows | https://ncatlab.org/nlab/source/2-category+equipped+with+proarrows |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### 2-Category theory
+-- {: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
# Contents
* automatic table of contents goes here
{: toc}
## Idea
A **2-category equipped with proarrows**, also called a *proarrow equipment* or simply an *equipment*, is a [[2-category]] together with extra "proarrows" which enable one to reproduce more category theory inside it than is possible in a general 2-category. The ur-example is [[Cat]] equipped with [[profunctors]].
See also [[framed bicategory]].
## Motivation
Just as ordinary [[category theory]] provides a framework in which one can do "formal mathematics," one of the (many) purposes of [[higher category theory]] is to provide a framework in which one can do [[formal category theory]]. In particular, many concepts in ordinary category theory can be interpreted internally in a [[2-category]], in a way which specializes to the original concept in [[Cat]]. Examples of such concepts include [[adjunctions]], [[monads]], [[Grothendieck fibrations]], [[Kan extensions]], and [[fully faithful morphisms]].
However, these internalizations are not always "correct" in every 2-category. For instance, in the 2-category $V Cat$ of [[enriched categories|categories enriched]] in a [[monoidal category]] $V$, internal adjunctions and monads give the correct notion of $V$-adjunction and $V$-monad, but internal fully-faithfulness for a $V$-functor is weaker than the "correct" notion of $V$-fully-faithfulness. More technically, in many cases the naive notion of Kan extension is not sufficient and one needs "pointwise" Kan extensions; with some more work these can also be defined in a 2-category, but the resulting notion (though correct in $Cat$) is not always correct in $V Cat$.
Furthermore, some concepts of category theory are difficult to interpret at all in a 2-category. The notion of [[limit]] in ordinary category theory can be interpreted in a 2-category by identifying the limit of a functor $D\to C$ as its Kan extension along the unique functor $D\to *$. However, in enriched category theory the more important notion is that of [[weighted limit]], and there is no straightforward way to interpret this in $V Cat$.
What is missing is that the 2-category $V Cat$ doesn't natively supply any information about the $V$-valued hom-functors in a 2-category. (In $Cat$ these hom-functors can be recovered by looking at [[comma categories]], which can be interpreted internally as [[comma objects]]---in some sense this is what all the above internalizations are secretly doing.) Thus, all of these problems can be remedied by equipping a 2-category with extra structure which describes these hom-functors, or more generally describes a notion of [[profunctor]].
There are several not-quite-equivalent ways to describe this extra structure. One due to Street and Walters, called a [[Yoneda structure]], involves assigning to each object $A$ a "presheaf object" $P A$ and a "Yoneda arrow" $A\to P A$; a profunctor $A\to B$ is then identified with an arrow $B \to P A$. The notion of *proarrow equipment*, due to Wood, instead postulates an additional bicategory of "proarrows" and specifies their relationship to the ordinary arrows. One can then define fully faithful morphisms, pointwise Kan extensions, weighted limits, etc. relative to this structure, in a way which specializes to the correct notions in $V Cat$.
## Definitions {#Definitions}
There are several equivalent ways to define proarrow equipments on a 2-category.
### Definition as a 2-functor {#DefinitionAsA2Functor}
Let $K$ be a [[2-category]]. The following structure is said to **equip $K$ with proarrows**.
* A 2-category $M$, whose arrows are called *proarrows*.
* A functor $K\to M$ which is [[bijective-on-objects functor|bijective on objects]] and [[locally fully faithful 2-functor|locally fully faithful]]. If $f\colon A\to B$ is a 1-morphism in $K$, we write its image in $M$ as $B(1,f)\colon A\nrightarrow B$.
* For each arrow $f\colon A\to B$ in $K$, the proarrow $B(1,f)\colon A\to B$ has a [[right adjoint]] in $M$, which we write as $B(f,1)$.
As usual on the nLab, here by [[2-category]] we mean a weak 2-category (aka [[bicategory]]) and by [[functor]] we mean a weak 2-functor (aka [[pseudofunctor]]). However, in many or most examples, $K$ is in fact a [[strict 2-category]].
For a proarrow $H\colon B\to D$ and ordinary arrows $f\colon A\to B$ and $g\colon C\to D$, we write $H(g,f)$ for the composite $D(g,1) \circ H \circ B(1,f)$; it is a proarrow from $A$ to $C$. We also write $U_A$, $A(1,1)$, or simply $A$ for the identity proarrow $A\nrightarrow A$.
Note that in the case where proarrows are profunctors, $H(g, f)$ is not the action on a set of [[heteromorphisms]] by pre- and post- composing with morphisms; instead it is the functor $f$, followed by the profunctor $H$, followed by taking the preimage under the functor $g$, resulting in a profunctor from $A$ to $C$.
#### Terminology
We refer to the entire structure defined above as a **2-category equipped with proarrows** or a **2-category proarrow equipment**. Those working in the field often use the term **proarrow equipment** or simply **equipment** for the entire structure. We prefer "2-category equipped with proarrows" (or, equivalently, "pro-morphisms") for the following reasons:
* It conveys that we are talking about extra stuff added to a 2-category. (Actually, it is "eka-stuff" or "2-stuff" in the sense of [stuff, structure, property](http://ncatlab.org/nlab/show/stuff,+structure,+property).)
* It includes the word "proarrow" which tells people what the extra stuff consists of, namely a generalization of [[profunctors]].
* It includes the word "equipment" which signals what is meant to readers who are familiar with that term.
We do sometimes avail ourselves of the shorter terminology "proarrow equipment," however, once we are clear what is under discussion.
#### Examples
For example, if $V$ is a (Benabou) [[cosmos]], the 2-category $K= V Cat$ of $V$-enriched categories is equipped with proarrows by the 2-category $M=V Mod$ of $V$-enriched profunctors, where $B(1,f)$ and $B(f,1)$ are the two ways of regarding a $V$-functor $f:A\to B$ as a profunctor, namely $B(-,f-)$ and $B(f-,-)$ (hence the notation in the general case). Composition of $V$-profunctors is by [[tensor product]], i.e. [[coends]]; note that we require $V$ to be [[cocomplete category|cocomplete]] with colimits preserved by $\otimes$ on both sides in order to form such associative tensor products. In $V Cat$, $H(g,f)$ is the result of precomposing the profunctor $H:D^{op}\otimes B \to V$ with $g^{op}\otimes f$.
The other most common sorts of generalized category, such as [[internal categories]] in some category with pullbacks, and [[fibered categories]] over some base, are also naturally equipped with proarrows. For internal categories in $S$, we require $S$ to have finite limits and coequalizers preserved by pullback in order for the bicategory of internal profunctors to have associative compositions. (See "virtual equipments," below, for a context which avoids some of these restrictions on $V$ and $S$.)
### Definition as a double category {#DefinitionAsDoubleCategory}
We discuss now how a 2-category with proarrow equipment is equivalently a [[double category]] with extra structure.
Given a 2-category $K$ equipped with proarrows, we can construct a [[double category]] having the same objects as $K$, whose vertical 1-cells are the arrows, whose horizontal 1-cells are the proarrows, and whose squares
$$\array{A & \overset{H}{\to} & C \\
^f\downarrow & \Downarrow & \downarrow^g\\
B& \underset{J}{\to} & D}
$$
are the 2-cells $H \to J(g,f)$. Note that if $K$ is a strict 2-category, then this is a [[pseudo double category]] (vertically strict and horizontally weak) while if $K$ and $M$ are both weak 2-categories, this double category is weak in both directions (like a [[double bicategory]]).
In $V Cat$, for example, the squares of the above form are transformations $H(b,a) \to J(g b,f a)$ natural in $a$ and $b$. Arguably, this double category is a more fundamental and natural object than the 2-functor $V Cat \to V Prof$. More generally, if $K$ is any 2-category equipped with proarrows, we can reconstruct $K$ and its proarrow equipment from the double category defined above, and we can characterize the double categories that arise in this way.
One way to state this characterization is that they are those in which every vertical 1-cell $f\colon A\to B$ has both a [[companion]] (namely $B(1,f)$) and a [[conjoint]] (namely $B(f,1)$). One can then recover $K$ and $M$ as the
[[vertical 2-category|vertical and horizontal 2-categories]], respectively, and the 2-functor $K\to M$ as the functor taking every vertical arrow to its companion. Whenever a vertical arrow has both a companion $B(1,f)$ and a conjoint $B(f,1)$, we automatically have an adjunction $B(1,f)\dashv B(f,1)$, so this defines a proarrow equipment in the first sense.
Another, perhaps even more natural, way to characterize these double categories is as those with the following property: given any "niche"
$$\array{A & & B\\
^f\downarrow & & \downarrow^g\\
C& \underset{J}{\nrightarrow} & D}
$$
there exists a "universal" or "cartesian" filler square
$$
\array{A & \overset{J(g,f)}{\to} & B\\
^f\downarrow & \Downarrow & \downarrow^g\\
C& \underset{J}{\nrightarrow} & D}
$$
with the property that any other square
$$
\array{X & \nrightarrow & Y\\
^{f h}\downarrow & \Downarrow & \downarrow^{g k}\\
C& \underset{J}{\nrightarrow} & D}
$$
factors through the universal one uniquely:
$$
\array{X & \nrightarrow & Y\\
^{h}\downarrow & \exists! \Downarrow & \downarrow^{k}\\
A & \overset{J(g,f)}{\to} & B\\
^f\downarrow & \Downarrow & \downarrow^g\\
C& \underset{J}{\nrightarrow} & D}
$$
The profunctor $J(g,f)$ will, of course, turn out to be the same one we gave that name to earlier. The proarrows $B(1,f)=U_B(id_B,f)$ and $B(f,1) = U_B(f,id_B)$ are then a special case of this construction. We show that they are a companion and conjoint of $f$, respectively, as follows.
By factoring the identity square
$$\array{A & \overset{U_A}{\to} & A\\
^f\downarrow & ^{U_f}\Downarrow & \downarrow^f\\
B & \underset{U_B}{\to} & B}$$
through the universal fillers
$$\array{A & \overset{B(1,f)}{\to} & B\\
^f\downarrow &\Downarrow & \downarrow^{id}\\
B & \underset{U_B}{\to} & B} \qquad\text{and}\qquad
\array{B & \overset{B(f,1)}{\to} & A\\
^{id}\downarrow &\Downarrow & \downarrow^{f}\\
B & \underset{U_B}{\to} & B}
$$
that define $B(1,f)$ and $B(f,1)$, we obtain additional squares
$$\array{A & \overset{U_A}{\to} & A\\
^f\downarrow &\Downarrow & \downarrow^{id}\\
B & \underset{B(f,1)}{\to} & A} \qquad\text{and}\qquad
\array{A & \overset{U_A}{\to} & A\\
^{id}\downarrow &\Downarrow & \downarrow^{f}\\
A & \underset{B(1,f)}{\to} & B}
$$
such that the composites
$$\array{A & \overset{U_A}{\to} & A\\
^{f}\downarrow &\Downarrow & \downarrow^{id}\\
B & \overset{B(f,1)}{\to} & A\\
^{id}\downarrow &\Downarrow & \downarrow^{f}\\
B & \underset{U_B}{\to} & B} \qquad\text{and}\quad
\array{A & \overset{U_A}{\to} & A\\
^{id}\downarrow &\Downarrow & \downarrow^{f}\\
A & \underset{B(1,f)}{\to} & B\\
^f\downarrow &\Downarrow & \downarrow^{id}\\
B & \underset{U_B}{\to} & B}
$$
are both equal to $U_f$. And using the uniqueness of factorization through the universal squares, we can conclude moreover that the composites
$$\array{A & \overset{U_A}{\to} & A & \overset{B(1,f)}{\to} & A\\
^{id}\downarrow &\Downarrow & \downarrow^{f} & \Downarrow & \downarrow^{id}\\
A & \underset{B(1,f)}{\to} & B & \underset{U_B}{\to} & B}
\qquad\text{and}\qquad
\array{B & \overset{B(f,1)}{\to} & A & \overset{U_A}{\to} & A\\
^{id}\downarrow &\Downarrow & \downarrow^{f} & \Downarrow & \downarrow^{id}\\
B & \underset{U_B}{\to} & B & \underset{B(f,1)}{\to} & A}
$$
are equal to the identity squares of $B(1,f)$ and $B(f,1)$ respectively. These are the defining equations of a companion and a conjoint.
Finally, we record the following.
+-- {: .num_lemma}
###### Central Lemma
There is a natural bijection between squares of the form
$$\array{A_0 & \overset{H}{\to} & B_0\\
^{f_1}\downarrow && \downarrow^{g_1}\\
A_1 && B_1\\
^{f_2}\downarrow & \Downarrow & \downarrow^{g_2}\\
A_2 && B_2\\
^{f_3}\downarrow && \downarrow^{g_3}\\
A_3 & \underset{J}{\to} & B_3 }
$$
and squares of the form
$$\array{A_1 & \overset{A_1(f_1 ,1)}{\to} & A_0 & \overset{H}{\to} & B_0 & \overset{B_1(1,g_1)}{\to} & B_1\\
^{f_2}\downarrow && &\Downarrow & && \downarrow^{g_2}\\
A_2 & \underset{A_3(1,f_3)}{\to} & A_3 & \underset{J}{\to} & B_3 & \underset{B_3(g_3 ,1)}{\to} & B_2.}
$$
=--
One way to think of this is as saying that vertical arrows can be "slid around corners" to become horizontal arrows, turning them into the "representable proarrows" $B(1,f)$ or $B(f,1)$ (depending on whether you slid them "backwards" or "forwards" to get around the corner). The bijection is just given by composing with the four special squares in the definition of companions and conjoints. In particular, by a Yoneda argument, for any $f\colon A\to C$, $g\colon B\to D$, and $J\colon C\nrightarrow D$ we have
\[J(g,f) \cong C(1,f) \odot J \odot D(g,1) \label{coyon} \]
which was what we took as the the definition of $J(g,f)$ with the original definition of proarrow equipment. Thus, the companions and conjoints determine the rest of the cartesian squares. Note that this is an equipment-version of [Yoneda reduction](http://ncatlab.org/nlab/show/Yoneda+reduction).
In conclusion, we have three definitions of proarrow equipment:
* A 2-functor which is bijective on objects, locally fully faithful, and maps every 1-morphism to one having a right adjoint.
* A double category in which every vertical arrow has a companion and a conjoint.
* A double category in which every niche has a cartesian filler.
While the first definition is perhaps simpler, for some purposes the second and third definitions are preferable. For instance, the definition of the 3-category of "2-categories equipped with proarrows" is much more naturally defined by viewing them as double categories. (See Dominic Verity's thesis and Mike Shulman's paper on framed bicategories.) It also generalizes better to situations in which not all proarrows have composites; see "virtual equipments," below.
### As a category internal to $Cat$
We discuss how a 2-category with proarrow equipment is an [[internal category]] in the [[(2,1)-category]] [[Cat]] in the sense of the theory of _[[internal (β,1)-categories]]_.
(...)
For the moment see at _[Segal space - Examples - In 1Grpd](http://ncatlab.org/nlab/show/Segal%20space#ExamplesInIGrpd)_.
### Related notions
There are a number of variations on the notion.
* [[virtual proarrow equipment]]
* [[1-category proarrow equipment]]
* [[(1,2)-category proarrow equipment]]
Some related concepts include:
* [[cartesian bicategory]]
* [[bicategory of relations]]
* [[allegory]]
See also at [[Segal space]], the section _[Examples -- In 1Grpd](Segal+space#ExamplesInIGrpd)_.
## Category theory in a proarrow equipment
We now give a few examples of how to do category theory internal to a proarrow equipment.
### Homset definition of adjunctions {#HomsetAdjn}
We start with this: two (vertical) arrows $f\colon A\to B$ and $g\colon B\to A$ are adjoint (in $\mathcal{V}(\underline{X})$) if and only if we have an isomorphism $B(f,1)\cong A(1,g)$. Why? Well, an adjunction $f\dashv g$ comes with a unit and a counit, which (expressed in $\underline{X}$) are of the form
$$\array{A & \overset{U_A}{\to} & A\\
^f\downarrow && \downarrow\\
B & \overset{\eta}{\Leftarrow} & \downarrow^{id} \\
^g\downarrow && \downarrow\\
A& \underset{U_A}{\to} & A} \qquad\text{and}\qquad
\array{B & \overset{U_B}{\to} & B\\
\downarrow && \downarrow^g\\
^{id}\downarrow & \overset{\varepsilon}{\Leftarrow} & A \\
\downarrow && \downarrow^f\\
B& \underset{U_B}{\to} & B.}
$$
Applying the bijection of the Central Lemma, we see that $\eta$ corresponds to a map $B(f,1) \to A(1,g)$, and likewise $\varepsilon$ corresponds to a map $A(1,g)\to B(f,1)$. The triangle identities for $\eta$ and $\varepsilon$ are then equivalent to saying that these two maps are inverse isomorphisms. So we've recovered the classical equivalence between the "diagrammatic" and isomorphism-of-hom-sets definitions of an adjunction, in a purely formal way.
### Fully faithful arrows
An arrow $f:A\to B$ in a 2-category equipped with proarrows is said to be **fully faithful** if the canonical morphism $U_A\to B(f,f)$ is an isomorphism of proarrows $A\to A$. In $V Cat$ this recaptures the correct notion of $V$-fully-faithful $V$-functor.
It is not hard to see, using the fact that $K\to M$ is locally fully faithful, that any fully-faithful arrow $f\colon A\to B$ is, in fact, internally fully-faithful in $K$ in the sense that $K(X,A)\to K(X,B)$ is fully faithful for all $X\in K$. However, the converse fails in general. But it is not hard to show that the two are equivalent if $f\colon A\to B$ has a left or right adjoint, using the above characterization of adjunctions.
### Limits
The notion of limit we end up in a 2-category equipped with proarrows with is actually more general than what you're probably used to, but this extra generality turns out to be useful. Let $d\colon D\to C$ be an arrow and let $J\colon D\nrightarrow A$ be a proarrow; we're going to define what it means for an arrow $\ell\colon A\to C$ to be the *$J$-weighted limit* of $d$. In the proarrow-equipped 2-category $V Cat$ of $V$-enriched categories, if $A$ is the [unit](http://ncatlab.org/nlab/show/unit+enriched+category) $V$-category $I$, then this will recover the usual notion of [weighted limit](http://ncatlab.org/nlab/show/weighted+limit). Of course, in a general proarrow equipment we may not have a "unit object," so that extra generality is unavoidable; it's like using generalized elements in ordinary category theory.
To make things easier, let's assume that our proarrow equipment is *closed*, in the sense that composition of proarrows has adjoints in each variable
$$ \mathcal{H}(\underline{X})(H\odot K,L) \cong \mathcal{H}(\underline{X})(H,K\rhd L) \cong \mathcal{H}(\underline{X})(K,L\lhd H).$$
The Central Lemma implies that analogously to \eqref{coyon}, we have
\[K(g,f) \cong D(1,g)\rhd K \lhd C(f,1). \label{yon} \]
In $V\text{-}Prof$, the adjoints are given by the [ends](http://ncatlab.org/nlab/show/end)
$$ (K\rhd L)(b,a) = \int_{c\in C} [K(c,b), L(c,a)] $$
and
$$ (L \lhd H)(c,b) = \int_{a\in A} [H(b,a), L(c,a)]. $$
Therefore, \eqref{yon} is an abstract form of the Yoneda lemma, just as \eqref{coyon} is an abstract form of Yoneda reduction.
Now recall that for $V$-categories $D$ and $C$, an object $\ell\in C$ is a $J$-weighted limit of $d\colon D\to C$ if we have a natural isomorphism
$$
\begin{aligned}
C(c,\ell) &\cong [D,V](J, C(c,d(-)))\\
&= \int_{x\in D} [J(x), C(c,d(x))].
\end{aligned}
$$
Thus it makes sense to define, in a closed proarrow equipment, an arrow $\ell\colon A\to C$ to be the $J$-weighted limit of $d$ if we have an isomorphism
$$ C(1,\ell) \cong C(1,d) \lhd J. $$
(If our proarrow equipment is not closed, we simply assert that $C(1,\ell)$ has the universal property that $C(1,d) \lhd J$ would have if it existed. In other terminology, we assert that $C(1,\ell)$ is a *right lifting* of $C(1,d)$ along $J$ in the 2-category of proarrows.) In particular, when $A$ is the unit $V$-category, this recovers the classical notion. But even in $V Cat$ there are interesting examples for other values of $A$. If we take $J = D(j,1)$ for some functor $j\colon A\to D$, then \eqref{coyon} and \eqref{yon} imply that
$$
\begin{aligned}
C(1,d) \lhd J &= C(1,d) \lhd D(j,1)\\
& \cong j^* C(1,d)\\
& \cong D(1,j) \odot C(1,d)\\
& \cong C(1,d j)
\end{aligned}
$$
so that $\ell = d j$ is the $J$-weighted limit of $d$. That is, $D(j,1)$-weighted limits are just given by composition with $j$.
More interestingly, one can show that if $J = D(1,k)$ for some $k\colon D\to A$, then $J$-weighted limits specialize to *pointwise* right Kan extensions along $k$. That is, the extra data of a proarrow equipment lets us define the good notion of Kan extension (even in the enriched case) as a special case of a general notion of limit. Thus, in a general 2-category equipped with proarrows, we *define* a "pointwise right Kan extension" along an arrow $k\colon D\to A$ to be a $D(1,k)$-weighted limit. It is easy to show that any pointwise Kan extension is, in particular, an internal Kan extension in $K$, but as we have seen the converse fails in $V Cat$.
### Right adjoints preserve limits
Suppose that $\ell\colon A\to C$ is a $J$-weighted limit of $d\colon D\to C$ in the above sense, and let $g\colon C\to B$ be an arrow with a left adjoint $f\colon B\to C$. We want to show that $g\ell$ is a $J$-weighted limit of $g d$. But we have
$$
\begin{aligned}
B(1,g\ell) &\cong C(1,\ell) \odot B(1,g)\\
&\cong \big(C(1,d) \lhd J\big) \odot C(f,1)\\
&\cong C(1,f) \rhd \big(C(1,d) \lhd J\big)\\
&\cong \big(C(1,f) \rhd C(1,d)\big) \lhd J\\
&\cong \big(C(1,d) \odot C(f,1)\big) \lhd J\\
&\cong \big(C(1,d) \odot B(1,g)\big) \lhd J\\
&\cong B(1,g d) \lhd J.
\end{aligned}
$$
which is what we want.
## Graphs and cographs
As noted above, in the case of ordinary categories, the profunctors can in fact be recovered from the 2-category $Cat$. Specifically, profunctors $A\to B$ can be identified with two-sided discrete fibrations from $B$ to $A$ (that is, spans $B \leftarrow C \to A$ such that $C \to B$ is a [[Grothendieck fibration|(Grothendieck) fibration]], $C\to A$ is an opfibration, the two structures are compatible, and each fiber of $C\to B\times A$ is discrete). The two-sided fibration corresponding to a profunctor is also called its [[graph of a profunctor|graph]]. The same is true for internal categories, but not for enriched categories.
However the $V$-profunctors $A\to B$ *can* be recovered from the 2-category $V Cat$ in a different, and in fact dual, way: they are the two-sided [[codiscrete cofibrations]] from $B$ to $A$, i.e. two-sided discrete fibrations in $(V Cat)^{op}$. The two-sided cofibration corresponding to a profunctor is, unsurprisingly, its [[cograph of a profunctor|cograph]], and also its [[collage]]. This was first noticed by Street and subsequently expanded on by other authors. In particular, one can write down axioms on a 2-category guaranteeing that its codiscrete cofibrations can be used to equip it with proarrows, and axioms on a proarrow equipment guaranteeing that it arises from codiscrete cofibrations.
## Virtual equipments
One can formulate a generalized notion of equipment which includes $V Cat$ for _any_ monoidal category $V$ (and in fact, any [[multicategory]]), and $Cat(S)$ for any category $S$ with pullbacks. The precise definition is complicated, but the basic idea is not: we start from the double-category definition, and instead of composites of the horizontal (pro)arrows, we allow the squares to have domains that are arbitrary composable strings, like so:
$$\array{ & \to \to \dots \to &\\
\downarrow & \Downarrow & \downarrow\\
& \underset{}{\to}& .}$$
So far this is analogous to the generalization from monoidal categories to multicategories, and gives the notion of [[virtual double category]]. We then impose a condition analogous to the existence of companions and conjoints to obtain the notion of [[virtual equipment]]. Most of category theory can be done in a virtual equipment just as well as in an equipment.
## Categories enriched in an equipment {#CategoriesEnrichedInAnEquipment}
For any equipment $W$ one can define a notion of **$W$-enriched category**. (See also at _[[category enriched in a bicategory]]_.) This consists of
* a collection $ob(C)$ of objects,
* for each object $x$ an *extent* $e(x)$ which is an object of $W$,
* for each pair of objects $x,y$ a proarrow $hom_C(x,y):e(x)\to e(y)$,
* composition and identity-assigning 2-cells obeying associativity and unit axioms.
So far we have not used the ordinary arrows, so many authors have studied only the notion of "category enriched in a bicategory." (Note that any 2-category $M$ can be regarded as the proarrow 2-category of an equipment in which the only ordinary arrows are identites.) However, we need the extra structure when we define a *functor* $f:C\to D$ between $W$-enriched categories, which consists of:
* a function $f:ob(C)\to ob(D)$,
* for each object $x$ of $C$ an arrow $f_x:e(x)\to e(f(x))$ in $W$,
* for each pair of objects $x$ and $y$, a square
$$\array{e(y) & \overset{hom_C(x,y)}{\to} & e(x)\\
^{f_y}\downarrow & \Downarrow & \downarrow^{f_x}\\
e(f(y))& \underset{hom_D(f(x),f(y))}{\to} & e(f(x))}$$
* satisfying functoriality axioms.
Finally, we define a *natural transformation* between such functors $f,g:C\to D$ to consist of
* squares
$$\array{e(x) & \overset{U_{e(x)}}{\to} & e(x)\\
^{g_x}\downarrow & \Downarrow & \downarrow^{f_x}\\
e(g(x))& \underset{hom_D(f(x),f(y))}{\to} & e(f(x))}$$
* satisfying a naturality axiom.
By choosing $W$ appropriately, categories enriched in an equipment include most types of generalized category:
* If $W$ is a monoidal category $V$, regarded as a one-object bicategory and thence as an equipment with only one ordinary arrow (so the objects of $V$ are the proarrows of $W$), then $W$-enriched categories, functors, and natural transformations are the same as $V$-enriched ones.
* If $S$ has pullbacks, there is an equipment $Span(S)$ whose objects and arrows are those of $S$ and whose proarrows are spans. A category enriched in $Span(S)$ which has one object is the same as an internal category in $S$, and likewise for functors and transformations. (Here it is essential to use an equipment, rather than merely a bicategory, to get the right notion of functor.)
* Arbitrary $Span(S)$-enriched categories can be identified with "locally small fibrations" over $S$, aka "locally internal categories" over $S$.
Other choices of $W$ give "categories which are both enriched and internal," for example:
* For a suitably nice category $S$ (such as a [[Grothendieck topos]]) there is an equipment $Ab(S)$ whose objects and arrows are those of $S$, and whose proarrows $A\to B$ are internal abelian group objects in $S/(B\times A)$. An $Ab(S)$-enriched category with one object can be regarded as an "internal [[Ab-enriched category]]" in $S$.
## References
* {#Wood82} [[Richard J. Wood]], *Abstract Proarrows I*, Cahiers de topologie et gΓ©omΓ©trie diffΓ©rentielle **23** 3 (1982) 279-290 [[numdam:CTGDC_1982__23_3_279_0](http://www.numdam.org/item/CTGDC_1982__23_3_279_0)]
* {#Wood85} [[Richard J. Wood]], *Proarrows II*, Cahiers de Topologie et GΓ©omΓ©trie DiffΓ©rentielle CatΓ©goriques **26** 2 (1985) 135-168 [[numdam:CTGDC_1985__26_2_135_0](http://www.numdam.org/item/CTGDC_1985__26_2_135_0)]
and "Proarrows II" (the original definitions), and a number of following papers.
* [[Ross Street]], "Fibrations in bicategories" (Construction of $V Mod$ from $V Cat$.)
* [[Aurelio Carboni]], Scott Johnson, [[Ross Street]], and [[Dominic Verity]], "Modulated bicategories" (Improved construction of $V Mod$ from $V Cat$.)
* [[Dominic Verity]], "Enriched categories, internal categories, and change of base", Ph.D. Thesis. (The connection with double categories.)
* {#Shulman} [[Michael Shulman]], "Framed bicategories and monoidal fibrations". (The equivalence with certain double categories, there called *framed bicategories*, and a general way to construct equipments such as $Ab(S)$.)
* [[Geoff Cruttwell]], [[Michael Shulman]], "A unified framework for generalized multicategories" (currently the only reference for virtual equipments).
* Renato Betti, [[Aurelio Carboni]], [[Ross Street]], and Robert Walters, "Variation through enrichment." (Locally small fibrations as $Span$-enriched categories.)
A blog post surveying ideas of proarrow equipments, much of which has been copied over to this page:
* [[Mike Shulman]], _Equipments_ ([blog](http://golem.ph.utexas.edu/category/2009/11/equipments.html))
On a [[string diagram]]-calculus for ([[virtual double category|virtual]]) [[double categories]] with ([[virtual equipment|virtual]]) [[2-category equipped with proarrows|pro-arrow equipments]]:
* {#Myers16} [[David Jaz Myers]], _String Diagrams For Double Categories and (Virtual) Equipments_ [[arXiv:1612.02762](https://arxiv.org/abs/1612.02762)]
* [[David Jaz Myers]], *String Diagrams for (Virtual) Proarrow Equipments* (2017) [slides: [pdf](http://www.mat.uc.pt/~ct2017/slides/myers_d.pdf), [[Myers-StringDiagrams2017.pdf:file]]]
An [[(β,1)-category theory|(β,1)-category theoretic]] version of proarrow equipments is in
* Jaco Ruit, _Formal category theory in β-equipments I_ ([arXiv:2308.03583](https://arxiv.org/abs/2308.03583))
[[!redirects 2-category equipped with proarrows]]
[[!redirects 2-categories equipped with proarrows]]
[[!redirects 2-category equipped with pro-arrows]]
[[!redirects 2-categories equipped with pro-arrows]]
[[!redirects 2-category with proarrow equipment]]
[[!redirects 2-categories with proarrow equipment]]
[[!redirects 2-category with pro-arrow equipment]]
[[!redirects 2-categories with pro-arrow equipment]]
[[!redirects 2-category equipment]]
[[!redirects 2-category equipments]]
[[!redirects equipment]]
[[!redirects equipments]]
[[!redirects proarrow equipment]]
[[!redirects proarrow equipments]]
[[!redirects pro-arrow equipment]]
[[!redirects pro-arrow equipments]]
[[!redirects proarrow]]
[[!redirects proarrows]]
|
2-category of 2-dimensional cobordisms | https://ncatlab.org/nlab/source/2-category+of+2-dimensional+cobordisms | This is a special case of the $(\infty,n)$-[[(infinity,n)-category of cobordisms|category of cobordisms]].
##References#
A complete [[generators and relations]] presentation of the 2-dimensional extended unoriented oriented bordism [[bicategory]] as a [[symmetric monoidal bicategory]] is given in
* Chris Schommer-Pries, _The classification of two-dimensional extended topological field theories_ PhD thesis, UC Berkeley (2009) ([pdf](http://4167562941749007073-a-1802744773732722657-s-sites.googlegroups.com/site/chrisschommerpriesmath/Home/Schommer-Pries-Thesis.pdf?attredirects=0&auth=ANoY7cq2K1zI1CuKhtNciHgk1yh4N_4acDXRydHRgR4No5aXp6TvsR6mR5IkV0Wsr4HhC4qMlLXrD67XtC26RYTPcCml2RTVYNYzpBnPgmNZECcxZNdBJbdjOE2UfdM_ya5ohr8UvugsHcbFqr3gVJg12WQqizOYcUQMkUNM7kgcL5uBU7GImLsPWJ7XWQq0IIodpv7By5Fh4ZrobR4DNWk74o0fQ9o9eamxKmczHCEDr1U8CH8Sj6E%3D))
<img src="http://math.ucr.edu/home/baez/schommer_pries.jpg" alt=""></img>
The 2-category theory in Schommer-Pries' thesis relies on a thesis by a student of Ross Street:
* Paddy McCrudden, _Balanced coalgebroids_ , Theory and Applications of Categories 7 (2000), 71-147. ([tac](http://www.tac.mta.ca/tac/volumes/7/n6/7-06abs.html)) |
2-category of adjunctions | https://ncatlab.org/nlab/source/2-category+of+adjunctions |
+-- {: .rightHandSide}
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###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}
##Idea
For any [[2-category]] $K$, there are two 2-categories (each with several variants) that could be called "the 2-category of adjunctions in $K$".
* The 2-category which here we call $Adj(K)$ has the same objects as $K$, its morphisms are the [[adjunctions]] in $K$ (pointing in the direction of, say, the [[left adjoint]]), and its [[2-morphisms]] are [[mate]]-pairs of 2-morphisms between adjunctions in $K$.
* The 2-category $[Adj,K]$ is the [[functor 2-category]] from the [[walking adjunction]] $Adj$ to $K$. Thus its *objects* are the adjunctions in $K$ --- or more precisely, triples $(x,y,(f,g,\eta,\epsilon))$ where $x,y$ are objects of $K$ and $(f,g,\eta,\epsilon)$ is an adjunction between $x$ and $y$. Its morphisms are pairs of morphisms $x\to x'$ and $y\to y'$ such that certain squares commute (perhaps up to a transformation or isomorphism), and its 2-cells are similarly composed of cylinders.
Note that the *[[1-morphisms]]* of $Adj(K)$ are the *[[objects]]* of $[Adj,K]$.
## Properties
* The morphisms in $Adj\big(Adj(K)\big)$ are the [[adjoint triples]] in $K$.
* The inclusion of $Mnd$, the [[free monad]], in $Adj$ induces a [[2-functor]] from $[Adj,K]$ to $[Mnd,K]$, the [2-category of monads](monad#the_bicategory_of_monads) in $K$. The adjoints to this 2-functor are the [[Kleisli category|Kleisli]] and [[Eilenberg-Moore category|Eilenberg-Moore]] constructions on monads in $K$.
## Related concepts
* [[CatAdj|$Cat_{Adj}$]]
## References
* [[Stephen Schanuel]] and [[Ross Street]], *The free adjunction*, Cahiers de topologie et gΓ©omΓ©trie diffΓ©rentielle catΓ©goriques, tome 27, no 1 (1986), p. 81-83 |
2-category theory | https://ncatlab.org/nlab/source/2-category+theory |
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###Context###
#### 2-Category theory
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#### Higher category theory
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[[!include higher category theory - contents]]
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#Contents#
* table of contents
{:toc}
## Idea
**2-category theory** is the study of [[2-categories]]. It is the "first new level" in [[higher category theory]].
## Related concepts
[[!include table of category theories]]
## References
See the references at *[[2-category]]*.
[[!redirects (2,1)-category theory]] |
2-category theory - contents | https://ncatlab.org/nlab/source/2-category+theory+-+contents | **[[2-category theory]]**
**Definitions**
* [[2-category]]
* [[strict 2-category]]
* [[bicategory]]
* [[enriched bicategory]]
**Transfors between 2-categories**
* [[2-functor]]
* [[pseudofunctor]]
* [[lax functor]]
* [[equivalence of 2-categories]]
* [[2-natural transformation]]
* [[lax natural transformation]]
* [[icon]]
* [[modification]]
* [[Yoneda lemma for bicategories]]
**Morphisms in 2-categories**
* [[fully faithful morphism]]
* [[faithful morphism]]
* [[conservative morphism]]
* [[pseudomonic morphism]]
* [[discrete morphism]]
* [[eso morphism]]
**Structures in 2-categories**
* [[adjunction]]
* [[mate]]
* [[monad]]
* [[cartesian object]]
* [[fibration in a 2-category]]
* [[codiscrete cofibration]]
**Limits in 2-categories**
* [[2-limit]]
* [[2-pullback]]
* [[comma object]]
* [[inserter]]
* [[inverter]]
* [[equifier]]
**Structures on 2-categories**
* [[2-monad]]
* [[lax-idempotent 2-monad]]
* [[pseudomonad]]
* [[pseudoalgebra for a 2-monad]]
* [[monoidal 2-category]]
* [[cartesian bicategory]]
* [[Gray tensor product]]
* [[proarrow equipment]]
|
2-category with contravariance | https://ncatlab.org/nlab/source/2-category+with+contravariance | # 2-categories with contravariance
* table of contents
{:toc}
## Idea
A *2-category with contravariance* is a [[2-category]]-like structure that contains a basic notion of [[contravariant functor|contravariant morphism]]. There are multiple ways to make this precise, depending on what sort of natural transformations we include.
## No mixed-variance transformations
The simplest case is when we include only natural transformations between functors of the same variance. In this case, for any two objects $x$ and $y$ we will have two disjoint hom-categories $hom^+(x,y)$ and $hom^-(x,y)$ whose objects are called "covariant" and "contravariant" respectively, and composition functors
$$ \begin{aligned}
hom^+(y,z) \times hom^+(x,y) &\to hom^+(x,z)\\
hom^+(y,z) \times hom^-(x,y) &\to hom^-(x,z)\\
hom^-(y,z) \times hom^+(x,y)^{op} &\to hom^-(x,z)\\
hom^-(y,z) \times hom^-(x,y)^{op} &\to hom^+(x,z)
\end{aligned} $$
Note that the variances behave like a $\mathbb{Z}/2$-[[graded category|grading]], with covariant functors considered "even" and contravariant functors "odd". Moreover, postcomposing with a contravariant functor is a contravariant operation.
This sort of 2-category with contravariance can be described more abstractly in two ways:
* It is a [[generalized multicategory]] relative to the [[2-monad]] on [[2-Cat]] defined by $ob(T A) = ob(A)$ and $T A(x,y) = A(x,y) + A(x,y)^{op}$.
* It is an [[enriched category]] over the category $V = Cat \times Cat$ with the non-standard (and non-[[symmetric monoidal category|symmetric]]) monoidal structure
$$(A^+,A^-)\otimes (B^+,B^-) = (A^+\times B^+ + A^-\times (B^-)^{op}, A^+\times B^- + A^- \times (B^+)^{op}). $$
The operation "$op$" can then be characterized by a [[universal property]], using the [[representable multicategory|representability]] property of a generalized multicategory or the notion of [[copowers]] in an enriched category. See [Shulman 2016](#Shulman2016) for more details.
### Examples
* Of course, [[Cat]] with its usual notion of [[contravariant functor]].
* If $V$ is any [[symmetric monoidal category]], then $V$-categories have opposites and so we can define contravariant $V$-functors in the usual way.
* 2-categories of [[indexed categories]] and [[fibrations]] have "fiberwise" opposites and hence fiberwise contravariant morphisms.
* More generally, any 2-category with a [[duality involution]] has an underlying 2-category with contravariance, and conversely a 2-category with contravariance that has all "opposites" (characterized by a universal property as above) has a duality involution.
* For instance, if $B$ is a [[compact closed bicategory]], then its sub-bicategory of "maps" ([[left adjoint]] morphisms) has a duality involution given by the duals in $B$, and hence is a "bicategory with contravariance".
## With mixed-variance transformations
It is also possible to define natural transformations from a covariant functor to a contravariant functor or vice versa, as a special case of [[dinatural transformations]]; thus we might ask for a sort of 2-category with contravariance that includes these as well. (This will include many of the examples above, but not $V$-enriched categories unless $V$ is cartesian monoidal, and thus also not the bicategory of maps in $B$ unless $B$ is not just compact closed but a [[cartesian bicategory]].)
Now our hom-objects will not decompose into two disjoint categories $hom^+$ and $hom^-$. Instead we have only one hom-category, but with gradings of "even" (covariant) or "odd" (contravariant) assigned to its objects (the morphisms of our 2-category-like structure). In the example of [[Cat]], the objects of these hom-categories will be all functors, covariant and contravariant alike, with covariant ones considered even and contravariant ones odd.
The $2$-morphisms $F\stackrel{\alpha}{\Rightarrow}G\colon C\to D$ in $Cat$ are as usual $\ob(C)$-indexed families of morphisms $FX\stackrel{\alpha_X}{\to} GX$, but the commutative diagram they will have to satisfy for each $X\stackrel{f}{\to} Y$ in $C$ depends on the combination of variances of $F$ and $G$:
$$
\array{
FX&\stackrel{Ff}{\rightarrow}&FY&&FX&\stackrel{Ff}{\rightarrow}&FY&&FX&\stackrel{Ff}{\leftarrow}&FY&&FX&\stackrel{Ff}{\leftarrow}&FY
\\\alpha_X\downarrow&&\alpha_Y\downarrow&&
\alpha_X\downarrow&&\alpha_Y\downarrow&&
\alpha_X\downarrow&&\alpha_Y\downarrow&&
\alpha_X\downarrow&&\alpha_Y\downarrow
\\GX&\stackrel{Gf}{\rightarrow}&GY&&GX&\stackrel{Gf}{\leftarrow}&GY&&GX&\stackrel{Gf}{\rightarrow}&GY&&GX&\stackrel{Gf}{\leftarrow}&GY
}$$
Note that these are the special cases of [[dinatural transformations]] between functors $C^{op}\times C\to D$ when both functors depend only on one of $C$ or $C^{op}$. It is clear that "vertical" composition of $2$-morphisms makes sense, so we do have hom-categories whose objects are functors of both variances. Note that if we consider transformations between functors of different variance to be graded odd and natural transformation between functors of the same variance to be graded even, then vertical composition is appropriately additive on the grading (the composition of two even morphisms, or two odd ones, is even; while the composition of an even and an odd morphism is odd).
Horizontal composition is trickier to describe. Of course, we can compose functors of both sorts, and this is also additive on the grading. Moreover, we can [[whisker]] a transformation of any sort by a functor of either sort on either side, and postwhiskering by an odd functor reverses the direction of a transformation. There is an appropriate [[interchange law]] (it is just the "naturality square" for the second transformation involved), so we have something that looks sort of like a 2-category.
+--{:.standout}
**Conjecture:** Let $W$ be the category whose objects are categories with $\mathbb{Z}/2$-gradings assigned to their objects, and whose morphisms are functors preserving grading. Then there is a [[closed monoidal category|biclosed]] (non-[[symmetric monoidal category|symmetric]]) [[monoidal category|monoidal structure]] on $W$ (or at least a [[closed category]] or [[multicategory]] structure) such that the above structure on categories, functors of both variances, and transformations assembles into a $W$-enriched category.
=--
## References
* [[Mike Shulman]], *Contravariance through enrichment*, [arXiv](https://arxiv.org/abs/1606.05058), 2016
{#Shulman2016}
[[!redirects 2-categories with contravariance]]
|
2-congruence | https://ncatlab.org/nlab/source/2-congruence |
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###Context###
#### Relations
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[[!include relations - contents]]
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#### 2-Category theory
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[[!include 2-category theory - contents]]
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#### Internal $(\infty,1)$-Categories
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[[!include internal infinity-categories contents]]
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#Contents#
* table of contents
{:toc}
## Idea
The notion of _[[2-congruence]]_ is the generalization of the notion of _[[congruence]]_ from [[category theory]] to [[2-category theory]].
The correct notions of _[[regular 2-category|regularity]] and [[exact 2-category|exactness]]_ for [[2-categories]] is one of the subtler parts of the theory of first-order structure. In particular, we need a suitable replacement for the 1-categorical notion of _[[equivalence relation]]_. The (almost) correct definition was probably first written down in [[StreetCBS]].
One way to express the idea is that in an [[n-category]], every [[object]] is [[internal logic|internally]] a $(n-1)$-category; exactness says that conversely every "internal $(n-1)$-category" is represented by an object. When $n=1$, an "internal 0-category" means an internal _equivalence relation_; thus exactness for 1-categories says that every equivalence relation is a kernel (i.e. is represented by some object). Thus, we need to find a good notion of "[[internal category|internal 1-category]]" in a 2-category.
Of course, there is an obvious notion of an internal category in a 2-category, as a straightforward generalization of [[internal category|internal categories]] in a 1-category. But internal categories in [[Cat]] are [[double categories]], so we need to somehow cut down the [[double categories]] to those that really represent honest [[1-categories]]. These are the _2-congruences_.
## Definition
Before we define 2-congruences below in def. \ref{2Congruence}, we need some preliminaries.
### 2-Congruences
+--{: .num_defn #HomWiseDiscreteInternalCategory}
###### Definition
If $K$ is a finitely complete 2-category, a **homwise-discrete category** in $K$ consists of
* a [[discrete morphism]] $D_1\to D_0\times D_0$, together
* with composition and identity maps $D_0\to D_1$ and $D_1\times_{D_0} D_1\to D_1$ in $K/(D_0\times D_0)$,
which satisfy the usual axioms of an [[internal category]] up to [[isomorphism]].
Together with the evident notions of **[[internal functor]]** and **internal [[natural transformation]]** there is a [[2-category]] $HDC(K)$ of hom-wise discrete 2-categories in $K$.
=--
+--{: .num_remark}
###### Remark
Since $D_1\to D_0\times D_0$ is discrete, the structural isomorphisms will automatically satisfy any [[coherence]] axioms one might care to impose.
=--
+--{: .num_remark}
###### Remark
The transformations between functors $D\to E$ are a version of the notion for internal categories, thus given by a morphism $D_0\to E_1$ in $K$. The 2-cells in $K(D_0,E_0)$ play no explicit role, but we will recapture them below.
=--
+--{: .num_remark}
###### Remark
By homwise-discreteness, any "[[modification]]" between transformations is necessarily a unique isomorphism, so (after performing some quotienting, if we want to be pedantic) we really have a 2-category $HDC(K)$ rather than a [[3-category]].
=--
+--{: .num_defn #Kernel}
###### Definition
If $f:A\to B$ is any morphism in $K$, there is a canonical homwise-discrete category $(f/f) \to A\times A$, where $(f/f)$ is the [[comma object]] of $f$ with itself. We call this the **kernel** $ker(f)$ of $f$ (the "comma [[kernel pair]]" or "comma [[Cech nerve]]" of $f$).
In particular, if $f=1_A$ then $(1_A/1_A) = A^{\mathbf{2}}$, so we have a canonical homwise-discrete category $A^{\mathbf{2}} \to A\times A$ called the **kernel** $ker(A)$ of $A$.
=--
+--{: .num_remark}
###### Remark
It is easy to check that taking kernels of objects defines a functor $\Phi:K \to HDC(K)$; this might first have been noticed by [Street](http://www.springerlink.com/content/407n62422140864p/). See prop. \ref{InclusionOfKernelsInCongruences} below.
=--
+--{: .num_theorem #Folding}
###### Theorem
If $D_1\,\rightrightarrows\, D_0$ is a homwise-discrete category in $K$, the following are equivalent.
1. $D_0 \leftarrow D_1 \to D_0$ is a [[two-sided fibration]] in $K$.
1. There is a functor $\ker(D_0)\to D$ whose object-map $D_0\to D_0$ is the identity.
=--
Actually, homwise-discreteness is not necessary for this result, but we include it to avoid worrying about [[coherence]] isomorphisms, and since that is the case we are most interested in here.
+--{: .proof}
###### Proof
We consider the case $K=Cat$; the general case follows because all the notions are defined representably. A homwise-discrete category in $Cat$ is, essentially, a [[double category]] whose horizontal 2-category is homwise-discrete (hence equivalent to a [[1-category]]). We say "essentially" because the pullbacks and diagrams only commute up to isomorphism, but up to equivalence we may replace $D_1\to D_0\times D_0$ by an [[isofibration]], obtaining a (pseudo) double category in the usual sense. Now the key is to compare both properties to a third: the existence of a [[companion pair|companion]] for any vertical arrow.
Suppose first that $D_0 \leftarrow D_1 \to D_0$ is a two-sided fibration. Then for any (vertical) arrow $f:x\to y$ in $D_0$ we have [[cartesian morphism|cartesian]] and [[opcartesian morphisms]] (squares) in $D_1$:
\[
\array{
x & \overset{id}{\to} & x
& \qquad &
x & \overset{f_1}{\to} & y'
\\
{}^{\mathllap{\cong}}\downarrow & opcart & \downarrow^{\mathrlap{f}}
& \qquad &
{}^{\mathllap{f}}\downarrow & cart & \downarrow^{\mathrlap{\cong}}
\\
x' & \overset{f_2}{\to} & y
& \qquad &
y & \overset{id}{\to} & y
}
\]
The vertical arrows marked as isomorphisms are so by one of the axioms for a two-sided fibration. Moreover, the final compatibility axiom for a 2-sided fibration says that the square
\[
\array{ x & \overset{f_1}{\to} & y'\\
\cong \downarrow & & \downarrow\cong \\
x' & \overset{f_2}{\to} & y,}
\]
induced by factoring the horizontal identity square of $f$ through these cartesian and opcartesian squares, must be an isomorphism. We can then show that $f_1$ (or equivalently $f_2$) is a [[companion pair|companion]] for $f$ just as in ([Shulman 07, theorem 4.1](#Shulman07)). Conversely, from a [[companion pair]] we can show that $D_0 \leftarrow D_1 \to D_0$ is a two-sided fibration just as as in [loc cit](#Shulman07).
The equivalence between the existence of companions and the existence of a functor from the kernel of $D_0$ is essentially found in ([Fiore 06](#Fiore06)), although stated only for the "edge-symmetric" case. In their language, a kernel $ker(A)$ is the double category $\Box A$ of commutative squares in $A$, and a functor $ker(D_0)\to D$ which is the identity on $D_0$ is a _thin structure_ on $D$. In one direction, clearly $ker(D_0)$ has companions, and this property is preserved by any functor $ker(D_0)\to D$. In the other direction, sending any vertical arrow to its horizontal companion is easily checked to define a functor $ker(D_0)\to D$.
=--
In particular, we conclude that up to isomorphism, there can be at most one functor $ker(D_0)\to D$ which is the identity on objects.
+--{: .num_defn #2Congruence}
###### Definition
A **2-congruence** in a finitely complete [[2-category]] $K$ is a homwise-discrete category, def. \ref{HomWiseDiscreteInternalCategory} in $K$ satisfying the equivalent conditions of Theorem \ref{Folding}.
=--
+--{: .num_example}
###### Example
The kernel $ker(A)$, def. \ref{Kernel} of any object is a 2-congruence.
More generally, the kernel $ker(f)$ of any morphism is also a 2-congruence.
=--
### 2-Forks and Quotients
The idea of a 2-fork is to characterize the structure that relates a morphism $f$ to its kernel $ker(f)$. The kernel then becomes the universal 2-fork on $f$, while the _quotient_ of a 2-congruence is the couniversal 2-fork constructed from it.
+--{: .num_defn}
###### Definition
A **2-fork** in a 2-category consists of a 2-congruence $s,t:D_1\;\rightrightarrows\; D_0$, $i:D_0\to D_1$, $c:D_1\times_{D_0} D_1\to D_1$, and a morphism $f:D_0\to X$, together with a 2-cell $\phi:f s \to f t$ such that $\phi i = f$ and such that
$$\array{
D_1\times_{D_0} D_1 & \to & D_1 & = & D_1\\
\downarrow && \downarrow & \Downarrow_\phi & \downarrow\\
D_1 & \to & D_0 && D_0\\
|| &\Downarrow_\phi && \searrow^f & \downarrow f\\
D_1 & \to & D_0 & \overset{f}{\to} & X
} \qquad = \qquad
\array{ D_1\times_{D_0} D_1 \\
& \searrow^c\\
&& D_1 & \to & D_0\\
&& \downarrow & \Downarrow_\phi & \downarrow f\\
&& D_1 & \overset{f}{\to} & X.
}$$
=--
The comma square in the definition of the kernel of a morphism $f:A\to B$ gives a canonical 2-fork
$$(f/f) \;\rightrightarrows\; A \overset{f}{\to} B.$$
It is easy to see that any other 2-fork
$$D_1 \;\rightrightarrows\; D_0 = A \overset{f}{\to} B$$
factors through the kernel by an essentially unique functor $D \to ker(f)$ that is the identity on $A$.
If $D_1 \;\rightrightarrows\; D_0 \overset{f}{\to} X$ is a 2-fork, we say that it equips $f$ with an **action** by the 2-congruence $D$. If $g:D_0\to X$ also has an action by $D$, say $\psi:g s \to g t$, a 2-cell $\alpha:f\to g$ is called an **action 2-cell** if $(\alpha t).\phi= \psi . (\alpha s)$. There is an evident category $Act(D,X)$ of morphisms $D_0\to X$ equipped with actions.
+--{: .num_defn}
###### Definition
A **quotient** for a 2-congruence $D_1\;\rightrightarrows\; D_0$ in a 2-category $K$ is a 2-fork $D_1 \;\rightrightarrows\; D_0 \overset{q}{\to} Q$ such that for any object $X$, composition with $q$ defines an equivalence of categories
$$K(Q,X) \simeq Act(D,X).$$
=--
A quotient can also, of course, be defined as a suitable [[nLab:2-categorical limit|2-categorical limit]].
+-- {: .num_lem}
###### Lemma
The quotient $q$ in any 2-congruence is [[nlab:eso morphism|eso]].
=--
+-- {: .proof}
###### Proof
If $m\colon A\to B$ is ff, then the square we must show to be a pullback is
$$\array{Act(D,A) & \overset{}{\to} & Act(D,B)\\
\downarrow && \downarrow\\
K(D_0,A)& \underset{}{\to} & K(D_0,B)}$$
But this just says that an action of $D$ on $A$ is the same as an action of $D$ on $B$ which happens to factor through $m$, and this follows directly from the assumption that $m$ is ff.
=--
+--{: .num_defn}
###### Definition
A 2-fork $D_1 \;\rightrightarrows\; D_0 \overset{f}{\to} X$ is called **exact** if $f$ is a quotient of $D$ _and_ $D$ is a kernel of $f$.
=--
This is the 2-categorical analogue of the notion of exact fork in a 1-category, and plays an analogous role in the definition of a [[regular 2-category]] and an [[exact 2-category]].
### The 2-category of 2-conguences
{#2CategoryOf2Congruences}
There is an evident but naive [[2-category]] of 2-congruences in
any 2-category. And there is a refined version where internal [[functors]]
are replaced by internal [[anafunctors]].
+--{: .num_defn #2CongWithOrdinaryFunctors}
###### Definition
For $K$ a [[2-category]], write $2Cong_s(K)$ for the [[full sub-2-category]]
of that of hom-wise discrete internal categories, def. \ref{HomWiseDiscreteInternalCategory} on the 2-congruences, def. \ref{2Congruence}
$$
2 Cong_s(K) \hookrightarrow HDC(K)
\,.
$$
=--
+--{: .num_prop #InclusionOfKernelsInCongruences}
###### Proposition
There is a [[2-functor]]
$$
\Phi : K\to 2Cong_s(K)
$$
sending each object to its kernel, def. \ref{Kernel}.
=--
+--{: .num_defn #2CongWithAnafunctors}
###### Definition
Let the 2-category $K$ be equipped with the structure of a [[2-site]].
With this understood, write
$$
2 Cong(K)
$$
for the [[2-category]] of 2-congruences with morphisms the [[anafunctors]] between them.
=--
+--{: .num_remark }
###### Remark
The evident inclusion
$$
2 Cong_s(K) \hookrightarrow 2 Cong(K)
$$
is a homwise-full sub-2-category closed under finite limits.
=--
## Properties
### Opposites
The opposite of a homwise-discrete category is again a homwise-discrete category. However, the opposite of a 2-congruence in $K$ is a 2-congruence in $K^{co}$, since [[nLab:opposite 2-category|2-cell duals]] interchange fibrations and opfibrations. Likewise, passage to opposites takes 2-forks in $K$ to 2-forks in $K^{co}$, and preserves and reflects kernels, quotients, and exactness.
### Regularity
{#Regularity}
We discuss that when the ambient 2-category $K$ has finite [[2-limits]],
then its 2-category $2 Cong_s(K)$ of 2-congruences, def. \ref{2CongWithOrdinaryFunctors} is a _[[regular 2-category]]_. This is theorem \ref{regex} below. A sub-2-category of $Cong_s(K)$ is the _regular completion_ of $K$.
In the following and throughout, "$n$" denotes either of (see [[(n,r)-category]])
$$
n = (0,1), (1,1), (2,1), (2,2)
\,.
$$
+--{: .num_lemma}
###### Lemma
Suppose that $K$ has finite [[2-limits]]. Then:
1. $HDC(K)$ (def. \ref{HomWiseDiscreteInternalCategory}) has finite limits.
1. $n Cong_s(K)$ is closed under finite limits in $HDC(K)$.
1. The 2-functor $\Phi : K \to 2 Cong_s(K)$, prop. \ref{InclusionOfKernelsInCongruences}, is [[full and faithful 2-functor|2-fully-faithful]] (that is, an equivalence on hom-categories) and preserves finite limits.
=--
+--{: .proof}
###### Proof
It suffices to deal with finite products, inserters, and equifiers. Evidently $\Phi(1)$ is a terminal object. If $D$ and $E$ are homwise-discrete categories, define $P_0 = D_0\times E_0$ and $P_1 = D_1\times E_1$; it is easy to check that then $P_1 \;\rightrightarrows\; P_0$ is a homwise-discrete category that is the product $D\times E$ in $HDC(K)$. Since $(D_0\times E_0) ^{\mathbf{2}} \simeq (D_0) ^{\mathbf{2}} \times (E_0) ^{\mathbf{2}}$, and products preserve ffs, we see that $P$ is an $n$-congruence if $D$ and $E$ are and that $\Phi$ preserves products.
For inserters, let $f,g:C \;\rightrightarrows\; D$ be functors in $HDC(K)$, define $i_0:I_0\to C_0$ by the pullback
$$\array{I_0 & \to & D_1\\ i_0 \downarrow && \downarrow \\
C_0 & \overset{(f_0,g_0)}{\to} & D_0\times D_0,}$$
and define $i_1:I_1 \to C_1$ by the pullback
$$\array{I_1 & \to & X\\
i_1\downarrow && \downarrow\\
C_1 & \overset{(f_1,g_1)}{\to} & D_1\times D_1}$$
where $X$ is the "object of commutative squares in $D$." Then $I_1 \;\rightrightarrows\; I_0$ is a homwise-discrete category and $i:I\to C$ is an inserter of $f,g$. Also, $I$ is an $n$-congruence if $C$ is, and $\Phi$ preserves inserters.
Finally, for equifiers, suppose we have functors $f,g:C \;\rightrightarrows\; D$ and 2-cells $\alpha,\beta:f \;\rightrightarrows\; g$ in $HDC(K)$, represented by morphisms $a,b:C_0 \;\rightrightarrows\; D_1$ such that $(s,t) a \cong (f_0,g_0)\cong (s,t) b$. Let $e_0:E_0\to C_0$ be the universal morphism equipped with an isomorphism $\phi:a e_0 \cong b e_0$ such that $(s,t)\phi$ is the given isomorphism $(s,t) a\cong (s,t) b$ (this is a finite limit in $K$.) Note that since $(s,t):D_1\to D_0\times D_0$ is discrete, $e_0$ is ff. Now let $E_1 = (e_0\times e_0)^*C_1$; then $E_1 \;\rightrightarrows\; E_0$ is a homwise-discrete category and $e:E\to C$ is an equifier of $\alpha$ and $\beta$ in $HDC(K)$. Also $E$ is an $n$-congruence if $C$ is, and $\Phi$ preserves equifiers.
For any morphism $f:A\to B$ in $K$, $\Phi(f)$ is the functor $ker(A)\to ker(B)$ that consists of $f:A\to B$ and $f^{\mathbf{2}}: A^{\mathbf{2}} \to B^{\mathbf{2}}$. A transformation between $\Phi(f)$ and $\Phi(g)$ is a morphism $A\to B ^{\mathbf{2}}$ whose composites $A\to B ^{\mathbf{2}} \;\rightrightarrows\; B$ are $f$ and $g$; but this is just a transformation $f\to g$ in $K$. Thus, $\Phi$ is homwise fully faithful. And homwise essential-surjectivity follows from the essential uniqueness of thin structures, or equivalently a version of Prop 6.4 in [FBMF][].
=--
Moreover, we have:
+--{: .num_theorem #regex}
###### Theorem
If $K$ is an $n$-category with finite limits, then $n Cong_s(K)$ is [[regular 2-category|regular]].
=--
+--{: .proof}
###### Proof
It is easy to see that a functor $f:C\to D$ between $n$-congruences is ff in $n Cong_s(K)$ iff the square
$$\array{C_1 & \to & D_1\\ \downarrow && \downarrow\\
C_0\times C_0 & \to & D_0\times D_0}$$
is a pullback in $K$.
We claim that if $e:E\to D$ is a functor such that $e_0:E_0\to D_0$ is split (that is, $e_0 s\cong 1_{D_0}$ for some $s:D_0\to E_0$), then $e$ is eso in $n Cong_s(K)$. For if $e\cong f g$ for some ff $f:C\to D$ as above, then we have $g_0 s:D_0 \to C_0$ with $f_0 g_0 s \cong e_0 s \cong 1_{D_0}$, and so the fact that $C_1$ is a pullback induces a functor $h:D\to C$ with $h_0=g_0 s$ and $f h\cong 1_D$. But this implies $f$ is an equivalence; thus $e$ is eso.
Moreover, if $e_0:E_0\to D_0$ is split, then the same is true for any pullback of $e$. For the pullback of $e:E\to D$ along some $k:C\to D$ is given by a $P$ where $P_0 = E_0 \times_{D_0} D_{iso} \times_{D_0} C_0$; here $D_{iso}\hookrightarrow D_1$ is the "object of isomorphisms" in $D$. What matters is that the projection $P_0\to C_0$ has a splitting given by combining the splitting of $e_0$ with the "identities" morphism $D_0\to D_{iso}$.
Now suppose that $f:D\to E$ is any functor in $n Cong_s(K)$. It is easy to see that if we define $Q_0=D_0$ and let $Q_1$ be the pullback
$$\array{ Q_1 & \to & E_1 \\ \downarrow && \downarrow\\
Q_0 \times Q_0 & \overset{f_0\times f_0}{\to} & E_0\times E_0}$$
then $f \cong m e$ where $e:D \to Q$ and $m:Q\to E$ are the obvious functors. Moreover, clearly $m$ is ff, and $e$ satisfies the condition above, so any pullback of it is eso. It follows that if $f$ itself were eso, then it would be equivalent to $e$, and thus any pullback of it would also be eso; hence esos are stable under pullback.
Since $m$ is ff, the kernel of $f$ is the same as the kernel of $e$, so to prove $K$ regular it remains only to show that $e$ is a quotient of that kernel. If $C \;\rightrightarrows\; D$ denotes $ker(f)$, then $C$ is the comma object $(f/f)$ and thus we can calculate
$$C_0 = D_0\times_{E_0} E_1 \times_{E_0} D_0 \cong Q_1.$$
Therefore, if $g:D\to X$ is equipped with an action by $ker(f)$, then the action 2-cell is given by a morphism $Q_1=C_0\to X_1$, and the action axioms evidently make this into a functor $Q\to X$. Thus, $Q$ is a quotient of $ker(f)$, as desired.
=--
+--{: .num_remark }
###### Remark
There are three "problems" with the 2-category $n Cong_s(K)$.
1. It is too big. It is not necessary to include every $n$-congruence in order to get a regular category containing $K$, only those that occur as kernels of morphisms in $K$.
1. It is too small. While it is regular, it is not [[exact 2-category|exact]].
1. It doesn't remember information about $K$. If $K$ is already regular, then passing to $n Cong_s(K)$ destroys most of the esos and quotients already present in $K$.
=--
The solution to the first problem is straightforward.
+--{: .num_defn }
###### Definition
If $K$ is a 2-category with finite limits, define
$$
K_{reg/lex} \hookrightarrow 2 Cong_s(K)
$$
to be the sub-2-category of $2 Cong_s(K)$ spanned by the 2-congruences which occur as kernels of morphisms in $K$.
=--
+--{: .num_remark }
###### Remark
If $K$ is an $n$-category then any such kernel is an $n$-congruence, so in this case $K_{reg/lex}$ is contained in $n Cong_s(K)$ and is an $n$-category. Also, clearly $\Phi$ factors through $K_{reg/lex}$.
=--
+--{: .num_theorem}
###### Theorem
For any finitely complete 2-category $K$, the 2-category $K_{reg/lex}$ is [[refular 2-category|regular]], and the functor $\Phi:K\to K_{reg/lex}$ induces an equivalence
$$Reg(K_{reg/lex},L) \simeq Lex(K,L)$$
for any regular 2-category $K$.
=--
Here $Reg(-,-)$ denotes the 2-category of regular functors, transformations, and modifications between two regular 2-categories, and likewise $Lex(-,-)$ denotes the 2-category of finite-limit-preserving functors, transformations, and modifications between two finitely complete 2-categories.
+--{: .proof}
###### Proof
It is easy to verify that $K_{reg/lex}$ is closed under finite limits in $2 Cong_s(K)$, and also under the eso-ff factorization constructed in Theorem \ref{regex}; thus it is regular. If $F:K\to L$ is a lex functor where $L$ is regular, we extend it to $K_{reg/lex}$ by sending $ker(f)$ to the quotient in $L$ of $ker(F f)$, which exists since $L$ is regular. It is easy to verify that this is regular and is the unique regular extension of $F$.
=--
In particular, if $K$ is a [[nlab:regular category|regular 1-category]], $K_{reg/lex}$ is the ordinary regular completion of $K$. In this case our construction reduces to one of the usual constructions (see, for example, the [[nlab:Elephant|Elephant]]).
To solve the second and third problems with $n Cong_s(K)$, we need to modify its morphisms.
### Exactness
{#Exactness}
Let now the ambient [[2-category]] $K$ be equipped with the structure of a [[2-site]]. Recall from def. \ref{2CongWithAnafunctors} the 2-category $2Cong(K)$ whose objects are 2-congruences in $K$, and whose morpisms are internal [[anafunctors]] between these, with respect to the given 2-site structure.
Notice that when $K$ is a [[regular 2-category]] it comes with a canonical structure of a 2-site: its [[regular coverage]].
+--{: .num_theorem}
###### Theorem
For any [[subcanonical topology|subcanonical]] and finitely complete [[2-site]] $K$ (such as a [[regular coverage]]), the 2-category $2Cong(K)$ from
def. \ref{2CongWithAnafunctors}
* is finitely complete;
* contains $2Cong_s(K)$, def. \ref{2CongWithOrdinaryFunctors} as a homwise-full sub-2-category (that is, $2Cong_s(K)(D,E)\hookrightarrow 2Cong(K)(D,E)$ is ff) closed under finite limits.
=--
+--{: .proof}
###### Proof
It is easy to see that products in $2 Cong_S(K)$ remain [[products]] in $n Cong(K)$. Before dealing with [[inserters]] and [[equifiers]], we observe that if $A\leftarrow F \to B$ is an anafunctor in $2 Cong(K)$ and $e:X_0\to F_0$ is any eso, then pulling back $F_1$ to $X_0\times X_0$ defines a new congruence $X$ and an anafunctor $A \leftarrow X \to B$ which is isomorphic to the original in $2 Cong(K)(A,B)$. Thus, if $A\leftarrow F\to B$ and $A\leftarrow G\to B$ are parallel anafunctors in $2 Cong(K)$, by pulling them both back to $F\times_A G$ we may assume that they are defined by spans with the same first leg, i.e. we have $A\leftarrow X \;\rightrightarrows\; B$.
Now, for the inserter of $F$ and $G$ as above, let $E\to X$ be the inserter of $X \;\rightrightarrows\; B$ in $2 Cong_s(K)$. It is easy to check that the composite $E\to X \to A$ is an inserter of $F,G$ in $2 Cong(K)$. Likewise, given $\alpha,\beta: F \;\rightrightarrows\; G$ with $F$ and $G$ as above, we have transformations between the two functors $X \;\rightrightarrows\; B$ in $2 Cong_s(K)$, and it is again easy to check that their equifier in $2 Cong_s(K)$ is again the equifier in $2 Cong(K)$ of the original 2-cells $\alpha,\beta$. Thus, $2 Cong(K)$ has finite limits. Finally, by construction clearly the inclusion of $2 Cong_s(K)$ preserves finite limits.
=--
+--{: .num_theorem #nCongExidempotent}
###### Theorem
If $K$ is a [[subcanonical coverage|subcanonical]] finitely complete $n$-site, then the functor $\Phi:K\to n Cong(K)$, prop. \ref{InclusionOfKernelsInCongruences},
is [[full and faithful 2-functor|2-fully-faithful]].
If $K$ is an $n$-[[exact 2-category|exact]] $n$-category equipped with its [[regular coverage]], then
$$
\Phi : K \to n Cong(K)
$$
is an [[equivalence of 2-categories]].
=--
+--{: .proof}
###### Proof
Since $\Phi:K \to n Cong_s(K)$ is 2-fully-faithful and $n Cong_s(K)\to n Cong(K)$ is homwise fully faithful, $\Phi:K \to n Cong(K)$ is homwise fully faithful. For homwise essential-surjectivity, suppose that $ker(A) \leftarrow F \to ker(B)$ is an anafunctor. Then $h:F_0 \to A$ is a cover and $F_1$ is the pullback of $A ^{\mathbf{2}}$ along it; but this just says that $F_1 = (h/h)$. The functor $F\to B$ consists of morphisms $g:F_0\to B$ and $F_1 = (h/h) \to B ^{\mathbf{2}}$, and functoriality says precisely that the resulting 2-cell equips $g$ with an action by the congruence $F$. But since $F$ is precisely the kernel of $h:F_0\to A$, which is a cover in a subcanonical 2-site and hence the quotient of this kernel, we have an induced morphism $f:A\to B$ in $K$. It is then easy to check that $f$ is isomorphic, as an anafunctor, to $F$. Thus, $\Phi$ is homwise an equivalence.
Now suppose that $K$ is an $n$-exact $n$-category and that $D$ is an $n$-congruence. Since $K$ is $n$-exact, $D$ has a quotient $q:D_0\to Q$, and since $D$ is the kernel of $q$, we have a functor $D \to ker(Q)$ which is a weak equivalence. Thus, we can regard it either as an anafunctor $D\to ker(Q)$ or $ker(Q)\to D$, and it is easy to see that these are inverse equivalences in $n Cong(K)$. Thus, $\Phi$ is essentially surjective, and hence an equivalence.
=--
Note that by working in the generality of 2-sites, this construction includes the previous one.
+--{: .num_remark}
###### Remark
If $K$ is a finitely complete 2-category equipped with its minimal coverage, in which the covering families are those that contain a [[split epimorphism]], then
$$
n Cong(K) \simeq n Cong_s(K)
\,.
$$
=--
+--{: .proof}
###### Proof
This is immediate from the proof of Theorem \ref{regex}, which implies that the first leg of any anafunctor relative to this coverage is both eso and ff in $n Cong_s(K)$, and hence an equivalence.
=--
+--{: .num_theorem #nCongOnGroupoidsAndDiscretes}
###### Theorem
If $K$ is a 2-exact 2-category with [[core in a 2-category|enough groupoids]], then
$$
K\simeq 2 Cong(gpd(K))
\,.
$$
Likewise, if $K$ is 2-exact and has enough discretes, then
$$
K\simeq 2 Cong(disc(K))
\,.
$$
=--
+--{: .proof}
###### Proof
Define a functor $K\to 2Cong(gpd(K))$ by taking each object $A$ to the kernel of $j:J\to A$ where $j$ is eso and $J$ is groupoidal (for example, it might be the [[core]] of $A$). Note that this kernel lives in $2Cong(gpd(K))$ since $(j/j)\to J\times J$ is discrete, hence $(j/j)$ is also groupoidal. The same argument as in Theorem \ref{nCongExidempotent} shows that this functor is 2-fully-faithful for any regular 2-category $K$ with enough groupoids, and essentially-surjective when $K$ is 2-exact; thus it is an equivalence. The same argument works for discrete objects.
=--
In particular, the 2-exact 2-categories having enough discretes are precisely the 2-categories of internal categories and anafunctors in 1-exact 1-categories.
Our final goal is to construct the $n$-exact completion of a regular $n$-category, and a first step towards that is the following.
+--{: .num_theorem #almostExreg}
###### Theorem
If $K$ is a regular $n$-category, so is $n Cong(K)$. The functor $\Phi:K\to n Cong(K)$ is regular, and moreover for any $n$-exact 2-category $L$ it induces an equivalence
$$Reg(n Cong(K), L) \to Reg(K,L).$$
=--
+--{: .proof}
###### Proof
We already know that $n Cong(K)$ has finite limits and $\Phi$ preserves finite limits. The rest is very similar to Theorem \ref{regex}. We first observe that an anafunctor $A \leftarrow F \to B$ is an equivalence as soon as $F\to B$ is also a weak equivalence (its reverse span $B\leftarrow F \to A$ then provides an inverse.) Also, $A \leftarrow F \to B$ is ff if and only if
$$\array{F_1 & \to & B_1\\ \downarrow && \downarrow \\
F_0\times F_0 & \to & B_0\times B_0}$$
is a pullback.
Now we claim that if $A\leftarrow F \to B$ is an anafunctor such that $F_0\to B_0$ is eso, then $F$ is eso. For if we have a composition
$$\array{ &&&& F \\
&&& \swarrow && \searrow\\
&& G &&&& M\\
& \swarrow && \searrow && \swarrow && \searrow\\
A &&&& C &&&& B}$$
such that $M$ is ff, then $F_0\to B_0$ being eso implies that $M_0\to B_0$ is also eso; thus $M\to B$ is a weak equivalence and so $M$ is an equivalence. Moreover, by the construction of pullbacks in $n Cong(K)$, anafunctors with this property are stable under pullback.
Now suppose that $A \leftarrow F \to B$ is any anafunctor, and define $C_0=F_0$ and let $C_1$ be the pullback of $B_1$ to $C_0\times C_0$ along $C_0 = F_0 to B_0$. Then $C$ is an $n$-congruence, $C\to B$ is ff in $n Cong_s(K)$ and thus also in $n Cong(K)$, and $A \leftarrow F \to B$ factors through $C$. (In fact, $C$ is the image of $F\to B$ in $n Cong_s(K)$.) The kernel of $A\leftarrow F\to B$ can equally well be calculated as the kernel of $F\to B$, which is the same as the kernel of $F\to C$.
Finally, given any $A\leftarrow G \to D$ with an action by this kernel, we may as well assume (by pullbacks) that $F=G$ (which leaves $C$ unchanged up to equivalence). Then since the kernel acting is the same as the kernel of $F\to C$, regularity of $n Cong_s(K)$ gives a descended functor $C\to D$. Thus, $A\leftarrow F \to C$ is the quotient of its kernel; so $n Cong(K)$ is regular.
Finally, if $L$ is $n$-exact, then any functor $K\to L$ induces one $n Cong(K) \to n Cong(L)$, but $n Cong(L)\simeq L$, so we have our extension, which it can be shown is unique up to equivalence.
=--
When $K$ is a regular 1-category, it is well-known that $1 Cong(K)$ (which, in that case, is the category of internal equivalence relations and functional relations) is the 1-exact completion of $K$ (the reflection of $K$ from regular 1-categories into 1-exact 1-categories). Theorem \ref{almostExreg} shows that in general, $n Cong(K)$ will be the $n$-exact completion of $K$ whenver it is $n$-exact. However, in general for $n\gt 1$ we need to "build up exactness" in stages by iterating this construction.
It is possible that the iteration will converge at some finite stage, but for now, define $n Cong^r(K) = n Cong(n Cong^{r-1}(K))$ and let $K_{n ex/reg} = colim_r n Cong^r(K)$.
+--{: .num_theorem}
###### Theorem
For any regular $n$-category $K$, $K_{n ex/reg}$ is an $n$-exact $n$-category and there is a 2-fully-faithful regular functor $\Phi:K\to K_{n ex/reg}$ that induces an equivalence
$$Reg(K_{n ex/reg},L) \simeq Reg(K,L)$$
for any $n$-exact 2-category $L$.
=--
+--{: .proof}
###### Proof
Sequential colimits preserve 2-fully-faithful functors as well as functors that preserve finite limits and quotients, and the final statement follows easily from Theorem \ref{almostExreg}. Thus it remains only to show that $K_{n ex/reg}$ is $n$-exact. But for any $n$-congruence $D_1 \;\rightrightarrows\; D_0$ in $K_{n ex/reg}$, there is some $r$ such that $D_0$ and $D_1$ both live in $n Cong^r(K)$, and thus so does the congruence since $n Cong^r(K)$ sits 2-fully-faithfully in $K_{n ex/reg}$ preserving finite limits. This congruence in $n Cong^r(K)$ is then an object of $n Cong^{r+1}(K)$ which supplies a quotient there, and thus also in $K_{n ex/reg}$.
=--
## Examples
{#Examples}
### In $Grpd$
{#ExamplesInGrpd}
> Under construction.
Let $K :=$ [[Grpd]] be the 2-category of [[groupoids]].
We would like to see that the following statement is true:
The [[2-category]] of 2-congruences in $Grpd$ is [[equivalence of 2-categories|equivalent]] to the 2-category [[Cat]] of [[small categories]].
$$
2Cong(Grpd)
\simeq
Cat
\,.
$$
Let's check:
For $C$ a [[small category]], construct a 2-congruence $\mathbb{C}$ in $Grpd$ as follows.
* let $\mathbb{C}_0 := Core(C) \in Grpd$ be the [[core]] of $C$;
* let $\mathbb{C}_1 := Core(C^{\Delta[1]}) \in Grpd$ be the core of the [[arrow category]] of $C$;
* let $(s,t) : \mathbb{C}_1 \to \mathbb{C}_0$ be image under $Core : Cat \to Grpd$ of the endpoint evaluation [[functor]]
$$
C^{\Delta[0] \coprod \Delta[0] \to \Delta[1]}
:
C^{\Delta[1]} \to C^{\Delta[0] \coprod \Delta[0]}
=
C \times C
\,.
$$
(Here we are using the canonical embedding $\Delta \hookrightarrow Cat$ of the [[simplex category]].)
This is clearly a [[faithful functor]]. Moreover, every morphism in [[Grpd]] is trivially a [[conservative morphism]]. So $\mathbb{C}_1 \to \mathbb{C}_0 \times \mathbb{C}_0$ is a [[discrete morphism]] in [[Grpd]].
Since [[Grpd]] is a [[(2,1)-category]], the [[2-pullbacks]] in [[Grpd]] are [[homotopy pullbacks]]. Using that $(s,t)$ is (under the [[right adjoint]] [[nerve]] embedding $N : Grpd \hookrightarrow sSet$) a [[Kan fibration]] (by direct inspection, but also as a special case of standard facts about the [[model structure on simplicial sets]]), the object of composable morphisms is found to be
$$
\mathbb{C}_1 \times_{\mathbb{C}_0} \mathbb{C}_1
\simeq
Core(C^{\Delta[2]})
\,.
$$
Accordingly, let the internal composition in $\mathbb{C}$ be induced by the given composition in $C$:
$$
\mathbb{C}_1 \times_{\mathbb{C}_0} \mathbb{C}_1
\simeq
Core(C^{\Delta[2]})
\stackrel{}{\to}
Core(C^{\Delta[1]})
\simeq
\mathbb{C}_1
\,.
$$
This is clearly [[associativity|associative]] and [[unitality|unital]] and hence makes $\mathbb{C}$ a hom-wise discrete category, def. \ref{HomWiseDiscreteInternalCategory}, internal to $Grpd$.
Observe next (for instance using the discussion and examples at [[homotopy pullback]], see also _[[path object]]_) that
$$
ker(\mathbb{C}_0) =
( \mathbb{C}_0^{\Delta[1]} \stackrel{\to}{\to} \mathbb{C}_0)
\,.
$$
Notice that up to [[equivalence of categories|equivalence of groupoids]], this is just the [[diagonal]] $\Delta : \mathbb{C}_0 \to \mathbb{C}_0 \times \mathbb{C}_0$.
Therefore there is an evident [[internal functor]] $ker(\mathbb{C}_0) \to \mathbb{C}$, which on the first equivalent incarnation of $ker(\mathbb{C}_0)$ given by the inclusion
$$
ker(\mathbb{C}_0)
\simeq
\mathbb{C}_0^{\Delta[1]}
\simeq
Core(C)^{\Delta[1]}
\hookrightarrow
Core(C^{\Delta[1]})
\,,
$$
but which in the second version above simply reproduces the [[identity-assigning morphism]] of the internal category $\mathbb{C}$.
It follows that $\mathbb{C}$ is indeed a 2-congruence, def. \ref{2Congruence}.
Conversely, given a 2-congruence $\mathbb{C}$ in $Grpd$, define a category $C$ as follows:
(...)
+--{: .num_remark}
###### Remark
In the notation of the above proof, we can also form _internally_ the [[core]] of $\mathbb{C}$. This is evidently the internally discrete category $\mathbb{C}_0 \stackrel{\overset{id}{\to}}{\underset{id}{\to}} \mathbb{C}_0$.
This means that the 2-congruences $\mathbb{C}$ in the above proof are [[complete Segal spaces]]
$$
\mathbb{C} : [n] \mapsto Core(C^{\Delta[n]})
\,,
$$
hence are [[internal categories in an (β,1)-category]] in the [[(2,1)-category]] [[Grpd]].
=--
### In a general $(2,1)$-category
{#ExamplesInA2Comma1Category}
(...)
## Related concepts
* [[congruence]]
* [[n-congruence]]
## References
The above material is taken from
* [[Mike Shulman]], _[[michaelshulman:2-congruence]]_
and
* [[Mike Shulman]], _[[michaelshulman:exact completion of a 2-category]]_
Some lemmas are taken from
* [[Mike Shulman]], _Framed bicategories and monoidal fibrations_ ([arXiv:0706.1286](http://arxiv.org/abs/0706.1286))
{#Shulman07}
and
* [[Thomas Fiore]], _Pseudo Algebras and Pseudo Double Categories_ ([arXiv:0608760](http://arxiv.org/abs/math/0608760))
{#Fiore06}
[[!redirects 2-congruences]]
[[!redirects 2-fork]]
[[!redirects 2-forks]]
|
2-crossed complex | https://ncatlab.org/nlab/source/2-crossed+complex |
# Contents
* automatic table of contents goes here
{:toc}
##Idea#
[[crossed complex|Crossed complexes]] are a useful extension of [[crossed module]]s allowing not only the encoding of an algebraic model for the [[homotopy 2-type]], but also information on the '[[complex of chains on the universal cover]]'. The category of crossed complexes is a [[monoidal closed category]] equivalent to various types of [[strict infinity-groupoid]].
To model the [[homotopy 3-type]] of a space, we can use either a [[2-crossed module]] or a [[crossed square]] (or various other algebraic models to be added some time in the future). A [[crossed complex]] is a 'hybrid', part [[crossed module]] but with a 'tail' which is a [[chain complex]]. What would be the 'hybrid' between a 2-crossed module and a chain complex? Are there examples that are easily constructed? What sort of information do they encode? Are they easy to analyse, understand, ... and useful?
##Definition##
A __2-crossed complex__ is a [[normal complex of groups]]
$$\ldots \to C_n \stackrel{\partial_n}{\longrightarrow} C_{n-1} \longrightarrow \ldots \longrightarrow C_0,$$
together with a 2-crossed module structure given on $C_2\to C_1\to C_0$ by a Peiffer lifting function $\{ -,-\} : C_1\times C_1 \to C_2$, such that, on writing $\pi = Coker(C_1\to C_0)$,
1. each $C_n$, $n\geq 3$ and $Ker \,\partial_2$ are $\pi$-modules and the $\partial_n$ for $n\geq 4$, together with the codomain restriction of $\partial_3$, are $\pi$-module homomorphisms;
1. the $ \pi $-module structure on $Ker \partial_2$ is the action induced from the $C_0$-action on $C_2$ for which the action of $\partial_1 C_1$ is trivial.
A 2-crossed complex morphism is defined in the obvious way, being compatible with all the actions, the pairings and Peiffer liftings.
We will denote by $2 Crs$, the corresponding category.
##Examples:#
* Any [[2-crossed module]] clearly gives a 2-crossed complex (with trivial 'tail').
* _From simplicial groups to 2-crossed complexes_. If $G$ is a simplicial group, then
$$ \ldots \to C(G)_3 \to \frac{\mathcal{N}G_2}{d_0(\mathcal{N}G_3\cap D_3)} \to \mathcal{N}G_1\to \mathcal{N}G_0,$$
has the structure of a 2-crossed complex, where $\mathcal{N}G$ is the [[Moore complex]] of $G$, $D_n$ is the subgroup of $G_n$ generated by the degenerate elements, and, for $n\gt2$,
$${C}(G)_{n} = \frac{\mathcal{N}G_n}{(\mathcal{N}G_n\cap D_n)d_0(\mathcal{N}G_{n+1}\cap D_{n+1})},$$
is the $n$-dimensional term of the crossed complex, $C(G)$, associated to the simplicial group $G$ as in the entry [[crossed complex]] (in the section **From simplicial group(oid)s to crossed complexes**.)
(There is an obvious extension of the group based definition above to a groupoid based one, and of this construction to one which takes as input a simplicially enriched groupoid.)
The Moore complex of a simplicial group $G$ has the structure of a 2-crossed complex if and only if for each $n\gt 2$, $\mathcal{N}G_n\cap D_n$ is trivial. This means that the axioms of a [[group T-complex]] are almost satisfied, but not necessarily in dimension 2.
* A quadratic chain complex as defined by H.J. Baues is a special case of a 2-crossed complex, satisfying additional (pre-crossed module) nilpotency condition at the level of the underlying pre-crossed module. (In fact the category of quadratic chain complexes is a reflective subcategory of the category of 2-crossed complexes.) In Baues' book referenced below, there is the construction of the fundamental quadratic chain complex of a pointed CW-complex. The reflection (or cotruncation) of this to the category of quadratic modules (i.e. 3-truncated quadratic chain complexes) faithfully represents the homotopy 3-type of a CW-space (at the level of spaces and maps between them).
* Graham Ellis defined the fundamental squared complex of a CW-complex from triad homotopy groups and generalised Whitehead products, and showed how Baues fundamental quadratic chain complex of a CW-complex can be obtained from it. A homotopy 2-crossed complex of a CW-complex can also be defined is the same way, see the work of João Faria Martins below.
##Crossed complexes and 2-crossed complexes.
Any [[crossed complex]] can be given the structure of a 2-crossed complex simply by defining a trivial Peiffer lifting, $\{-,-\}$. As the Peiffer lifting covers the Peiffer commutators in $C_1$, and these are trivial (since the bottom of the crossed complex is a crossed module), this trivial Peiffer lifting works and gives a 2-crossed complex structure. This defines a functor from the category of crossed complexes to that of 2-crossed complexes.
Any 2-crossed complex which has a Peiffer lifting that is trivial $\{x,y\} = 1$, for all $x,y \in C_1$) is isomorphic to a crossed complex in this sense.
This functor, from $Crs$ to $2-Crs$, has a left adjoint which is the identity on the subcategory of $2-Crs$ with trivial Peiffer liftings, so $Crs$ is equivalent to a reflective subcategory of $2-Crs$
##References##
* See the [[Crossed Menagerie]], chapter 5.
* [[H.-J. Baues]], Combinatorial Homotopy and 4-Dimensional Complexes , de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991).
* [[Graham Ellis]], Crossed squares and combinatorial homotopy. Mathematische Zeitschrift
Volume 214, Number 1, 93-110, DOI: 10.1007/BF02572393
* [[Joao Faria Martins]], _The fundamental 2-crossed complex of a reduced CW-complex_, Homology Homotopy Appl. 13(2): 129-157 (2011) (web [pdf] (https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-13/issue-2/The-fundamental-2-crossed-complex-of-a-reduced-CW-complex/hha/1335806746.full).)
[[!redirects 2-crossed complexes]] |
2-crossed module | https://ncatlab.org/nlab/source/2-crossed+module |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
##Idea#
A _2-crossed module_ encodes a semistrict 3-group -- a [[Gray-group]] -- in generalization of how a [[crossed module]] encodes a [[strict 2-group]].
A [[simplicial group]] whose [[Moore complex]] has length $1$ (that is, at most stuff in dimensions $0$ and $1$) will be the internal [[nerve]] of a strict $2$-[[2-group|group]] and the Moore complex will be the corresponding [[crossed module]]. What if we have a simplicial group whose Moore complex has at most stuff in dimensions $0$, $1$, and $2$; can we describe its structure in some similar way? Yes, and Conduché provided a neat description of the structure involved. From the structure one can rebuild a simplicial group, a type of internal $2$-[[nerve]] construction.
In other words, a $2$-crossed module _is_ the Moore complex of a $2$-[[truncated]] simplicial group.
## Definition
A __$2$-crossed module__ is a [[normal complex of groups]]
$$L\stackrel{\partial_2}{\to} M \stackrel{\partial_1}{\to}N,$$
together with an [[action]] of $N$ on all three groups and a mapping
$$\{ - ,- \} : M\times M \to L$$
such that
1. the action of $N$ on itself is by conjugation, and $\partial_2$ and $\partial_1$ are $N$-equivariant;
1. for all $m_0,m_1 \in M$,
$$\partial_2\{m_0,m_1\} = \,^{\partial_1 m_0}m_1 . m_0m_1^{-1}m_0^{-1};$$
1. if $\ell_0,\ell_0 \in L$, then
$$\{\partial_2\ell_0,\partial_2\ell\} = [\ell_1,\ell_0];$$
1. if $\ell \in L$ and $m\in M$, then
$$\{m,\partial \ell\}\{\partial \ell,m\} = \,^{\partial m}\ell.\ell^{-1};$$
1. for all $m_0,m_1,m_2 \in M$,
* $\{m_0,m_1m_2\} = \{m_0,m_1\}\{ \partial \{m_0,m_2\},(m_0m_1m_0^{-1})\}\{m_0,m_2\}$;
* $\{m_0m_1,m_2\} = \,^{\partial m_0}\{m_1,m_2\}\{m_0,m_1m_2m_1^{-1}\}$;
1. if $n\in N$ and $m_0,m_1 \in M$, then
$$ \,^{n} \{m_0,m_1\} = \{ \,^{n}m_0, \,^{n}m_1\}.$$
The pairing $\{ - ,- \} : M\times M \to L$ is often called the **Peiffer lifting** of the $2$-crossed module.
## Remarks
* In a $2$-crossed module as above the structure $\partial_2: L \to M$ is a [[crossed module]], but $\partial_1: M\to N$ may not be one, as the Peiffer identity need not hold. The _[[Peiffer commutator]]_, which measures the failure of that identity, may not be trivial, but it will be a boundary element and the Peiffer lifting gives a structured way of getting an element in $L$ that maps down to it.
* It is sometimes useful to consider a [[crossed module]] as being a [[crossed complex]] of length 1 (i.e. on possibly non-trivial morphism only). Likewise one can consider a 2-crossed module as a special case of a [[2-crossed complex]]. Such a gadget is intuitively a 2-crossed module with a 'tail', which is a chain complex of modules over the $\pi_0$ of the base 2-crossed module, much as a crossed complex is a crossed module together with a 'tail'.
* A quadratic module, as developed by [[H.-J. Baues]], is a special case of a 2-crossed module, satisfying nilpotency conditions at the level of the underlying pre-crossed module (which is a close to being a crossed module as possible.) The fundamental quadratic module of a CW-complex yields an equivalence of categories between the category of pointed 3-types and the category of quadratic modules.
* A functorial fundamental 2-crossed module of a CW-complex can also be defined, by using Graham Ellis fundamental crossed square of a CW-complex; this is explained in the article of João Faria Martins, below. We can also define this fundamental 2-crossed module of a CW-complex, by using Kan's fundamental simplicial group of a CW-complex, and by applying the usual reflection from simplicial groups to simplicial groups of Moore complex of lenght two, known to be equivalent to 2-crossed modules.
* The homotopy theory of 2-crossed modules can be addressed by noting that 2-crossed modules, inducing a reflective subcategory of the category of simplicial groups, inherit a natural Quillen model structure, as explored in the article of Cabello and Garzon below. A version very close to the usual homotopy theory of crossed complexes was developed in the article of Joao Faria Martins below in a parallel way to the homotopy theory of [[quadratic module]]s and [[quadratic complex]]es as introduced by [[H. J. Baues]].
## Examples
Any [[crossed module]],
$
G_2 \stackrel{\delta }{\to}{G_1}
$ gives a 2-crossed module, $L\stackrel{\partial_2}{\to} M \stackrel{\partial_1}{\to}N,$ by setting $L = 1$, the trivial group, and, of course, $M = G_2$, $N = G_1$. Conversely any 2-crossed module having trivial top dimensional group ($L=1$) 'is' a crossed module. This gives an inclusion of the category of crossed modules into that of 2-crossed modules, as a [[reflective subcategory]].
The reflection is given by noting that, if
$$L\stackrel{\partial_2}{\longrightarrow} M \stackrel{\partial_1}{\longrightarrow}N$$
is a 2-crossed module, then $Im\, \partial_2$ is a normal subgroup of $M$, and then there is an obvious induced crossed module structure on
$$\partial_1 : \frac{M}{Im\, \partial_2} \to N.$$
But we can do better than this. More generally, let
$$\ldots \to 1 \to 1 \to C_3\stackrel{\partial_3}{\longrightarrow} C_2 \stackrel{\partial_2}{\longrightarrow}C_1,$$
be a truncated [[crossed complex]] (of groups) in which all higher dimensional terms are trivial, then taking $L = C_3$, $M = C_2$ and $N = C_1$, with trivial Peiffer lifting, gives one a 2-crossed complex. Conversely suppose we have a 2-crossed module with trivial Peiffer lifting: $\{m_1,m_2\} = 1$ for all $m_1$, $m_2 \in M$, axiom 3 then shows that $L$ is an Abelian group, and similarly the other axioms can be analysed to show that the result is a truncated crossed complex.
This gives:
+-- {: .un_prop}
###### Proposition
The category $Crs_{2]}$ of crossed complexes of length 2 is equivalent to the full subcategory of $2-CMod$ given by those 2-crossed modules with trivial Peiffer lifting.
=--
Of course, the resulting 'inclusion' has a left adjoint, which is quite fun to check out! (You kill off the subgroup of $L$ generated by the Peiffer lifting, .... is that all?)
### From simplicial groups to 2-crossed modules
If $G$ is a simplicial group then
$$\frac{\mathcal{N}G_2}{d_0(\mathcal{N}G_3)} \to \mathcal{N}G_1\to \mathcal{N}G_0,$$
is a 2-crossed module. (You are invited to find the Peiffer lifting!)
### From crossed squares to 2-crossed modules
Both [[crossed square|crossed squares]] and 2-crossed modules model all connected homotopy 3-types so one naturally asks how to pass from one description to the other. Going from crossed squares to 2-crossed modules is easy, so will be given here (going back is harder).
Let
$$\array{& L & {\to}^\lambda & M & \\
\lambda^\prime & \downarrow &&\downarrow & \mu\\
&N & {\to}_{\nu}& P & \\
}$$
be a crossed square then $N$ acts on $M$ via $P$, so ${}^n m := {}^{\nu(n)}m$, and so we can form $M\rtimes N$ and the sequence
$$L\stackrel{((\lambda')^{-1},\lambda)}{\longrightarrow}M\rtimes N\stackrel{\mu\nu}{\longrightarrow}P$$
is then a 2-crossed complex.
(And, yes, these are actually group homomorphisms: $(\mu,\nu)(m,n) = \mu(m)\nu(n)$, the product of the two elements! Try it!)
The full result and an explanation of what is going on here is given in
* D. Conduché, _Simplicial Crossed Modules and Mapping Cones_, Georgian Math. J., 10, (2003),
623--636
## Related concepts
* [[group]]
* [[2-group]], [[crossed module]], [[differential crossed module]]
* [[3-group]], **2-crossed module** / [[crossed square]], [[differential 2-crossed module]]
* [[n-group]]
* [[β-group]], [[simplicial group]], [[crossed complex]], [[hypercrossed complex]]
[[!redirects 2-crossed modules]]
## References:
* [[H. J. Baues]]: _Combinatorial homotopy and $4$-dimensional complexes._ With a preface by Ronald Brown. de Gruyter Expositions in Mathematics, 2. Walter de Gruyter \& Co., Berlin, 1991.
* Julia G. Cabello, [[Antonio R. GarzΓ³n]]: _Quillen's theory for algebraic models of $n$-types_. Extracta Math. 9 (1994), no. 1, 42--47. ([EuDML](https://eudml.org/doc/38395))
* [[P. Carrasco]] and [[T. Porter]], _Coproduct of 2-crossed modules. Applications to a definition of a tensor product for 2-crossed complexes_, Collectanea Mathematica, DOI:10.1007/s13348-015-0156-9.
* [[Daniel ConduchΓ©]], _Modules croisés généralisés de longueur $2$_, in:
Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983). J. Pure Appl. Algebra 34 (1984), no. 2-3, 155--178.
* [[Joao Faria Martins]], _The fundamental 2-crossed complex of a reduced CW-complex_, Homology Homotopy Appl. 13(2): 129-157 (2011) (web [pdf](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-13/issue-2/The-fundamental-2-crossed-complex-of-a-reduced-CW-complex/hha/1335806746.full).)
* [[Graham Ellis]], _Crossed squares and combinatorial homotopy_, Math. Z. 214 (1993), no. 1, 93--110.
|
2-dinatural transformation | https://ncatlab.org/nlab/source/2-dinatural+transformation | # 2-dinatural transformations
* table of contents
{: toc}
## Idea
A *2-dinatural transformation* is a [[categorification]] of a [[dinatural transformation]]. It is more complicated since there are three kinds of [[opposite 2-category]], and also because it could be [[lax natural transformation|lax]], colax, or pseudo as well as strict.
## Definition
Let $C$ and $D$ be [[2-categories]] and let
$$F,G: C^{coop}\times C^{op}\times C^{co} \times C\to D$$
be [[2-functors]]. A **lax 2-dinatural transformation** $\alpha:F\to G$ consists of
1. For each $x\in C$, a 1-morphism component
$$ \alpha_x : F(x,x,x,x) \to G(x,x,x,x) $$
2. For each 1-morphism $f:x\to y$ in $C$, a 2-morphism component from the composite
$$ F(y,y,x,x) \xrightarrow{F(f,f,1,1)} F(x,x,x,x) \xrightarrow{\alpha_x} G(x,x,x,x) \xrightarrow{G(1,1,f,f)} G(x,x,y,y) $$
to the composite
$$ F(y,y,x,x) \xrightarrow{F(1,1,f,f)} F(y,y,y,y) \xrightarrow{\alpha_y} G(y,y,y,y) \xrightarrow{G(f,f,1,1)} G(x,x,y,y).$$
(This is a generalization of the hexagon identity for an ordinary [[dinatural transformation]].)
3. For any 2-morphism $\mu:f\to g:x\to y$ in $C$, the two composites
$$ G(1,1,g,f)\circ \alpha_x \circ F(g,f,1,1)
\xrightarrow{G(1,1,\mu,1) \ast 1 \ast F(\mu,1,1,1)}
G(1,1,f,f) \circ \alpha_x \circ F(f,f,1,1)
\xrightarrow{\alpha_f}
G(f,f,1,1) \circ \alpha_y \circ F(1,1,f,f)
\xrightarrow{G(1,\mu,1,1) \ast 1 \ast F(1,1,1,\mu)}
G(f,g,1,1) \circ \alpha_y \circ F(1,1,f,g)
$$
and
$$ G(1,1,g,f)\circ \alpha_x \circ F(g,f,1,1)
\xrightarrow{G(1,1,1,\mu) \ast 1 \ast F(1,\mu,1,1)}
G(1,1,g,g) \circ \alpha_x \circ F(g,g,1,1)
\xrightarrow{\alpha_g}
G(g,g,1,1) \circ \alpha_y \circ F(1,1,g,g)
\xrightarrow{G(\mu,1,1,1) \ast 1 \ast F(1,1,\mu,1)}
G(f,g,1,1) \circ \alpha_y \circ F(1,1,f,g)
$$
are equal.
4. Each 2-morphism component $\alpha_{1_x}$ is the identity $1_{\alpha_x}$.
5. For any composable 1-morphisms $x\xrightarrow{f} y \xrightarrow{g} z$, the composite of $\alpha_f$ and $\alpha_g$ (together with two functoriality commutative squares for $F$ and $G$) is equal to $\alpha_{g f}$.
## Examples
If one of $F$ and $G$ is constant and the other depends only on $C^{op}\times C$, we obtain a notion of **2-extranatural transformation**.
If $F$ and $G$ each depend on only one factor in $C^{coop}\times C^{op}\times C^{co} \times C$, we obtain a notion of strict 2-natural transformation from a 2-functor of any arbitrary variance to a 2-functor of any other arbitrary variance. For example, if $F:C^{co}\to D$ and $G:C^{op}\to D$, then a lax 2-dinatural transformation $\alpha:F\to G$ consists of an $\ob(\mathbf{C})$-indexed family of $1$-morphisms $FX\stackrel{\alpha_X}{\to} GX$ in $\mathbf{D}$, and for each two objects $X,Y$ of $\mathbf{C}$, an $\ob[X,Y]$-indexed family of $2$-morphisms $\alpha_f$, so that for every $2$-morphism $f\stackrel{\gamma}{\Rightarrow} g$, we have the commutative diagram of $2$-morphisms in $\mathbf{D}$:
$$
\array{
&&FX&\stackrel{Ff}{\rightarrow}&FY\\
&&\alpha_X\downarrow&\stackrel{\alpha_f}{\Rightarrow}&\downarrow\alpha_Y&&FX\\
&&GX&\stackrel{Gf}{\leftarrow}&GY&&\downarrow Ff\\
FX&\stackrel{id}{\neArrow}&&&& \stackrel{G\gamma.(\alpha_Y\circ Ff)}{\seArrow}&FY\\
\alpha_X\downarrow&&&&&&\downarrow\alpha_Y\\
FY&\stackrel{id}{\seArrow}&&&&\stackrel{(Gg\circ\alpha_Y).F\gamma}{\neArrow}&GY\\
&&FX&\stackrel{Fg}{\rightarrow}&FY&&\downarrow Gg\\
&&\alpha_X\downarrow&\stackrel{\alpha_g}{\Rightarrow}&\downarrow\alpha_Y&&GX\\
&&GX&\stackrel{Gg}{\leftarrow}&GY
}$$
where . is whiskering/horizontal composition. Furthermore, given composable $1$-morphisms $X\stackrel{f}{\rightarrow}Y\stackrel{h}{\rightarrow} Z$, the $2$-morphisms $\alpha_X\stackrel{\alpha_f}{\Rightarrow}Gf\circ\alpha_Y\circ Ff$ and $\alpha_Y\stackrel{\alpha_h}{\Rightarrow}Gh\circ\alpha_Z\circ Fh$ are related via the formula $\alpha_{h\circ f}=(Gf.\alpha_h.Ff)\circ\alpha_f$, which says that the [[pasting diagram]] of $2$-morphisms:
$$
\array{
FX&\stackrel{Ff}{\rightarrow}&FY&\stackrel{Fh}{\rightarrow}&FZ\\
\alpha_X\downarrow&\stackrel{\alpha_f}{\Rightarrow}&\downarrow\alpha_Y&\stackrel{\alpha_h}{\Rightarrow}&\downarrow\alpha_Z\\
GX&\stackrel{Gf}{\leftarrow}&GY&\stackrel{Gh}{\leftarrow}&GZ
}$$
reduces to
$$
\array{
FX&\stackrel{Fh\circ Ff}{\rightarrow}&FZ\\
\alpha_X\downarrow&\stackrel{\alpha_{h\circ f}}{\Rightarrow}&\downarrow\alpha_Z\\
GX&\stackrel{Gh\circ Gf}{\leftarrow}&GZ
}$$
This sort of transformation appears in the [[category of V-enriched categories]], which is a $2$-category which comes with a [[unit enriched category]] $\mathcal{I}$ and either a lax natural transformation $[\mathcal{I},-]^{op}\Rightarrow[[\mathcal{I},-],V_0]$ (in the case of $\mathcal{V}$ a [[monoidal category|monoidal structure]] on $V$), or a lax natural transformation $[\mathcal{I},-]^{op}\Rightarrow[-,V^e]$ (in the case of $\mathcal{V}$ a [[closed category|closed structure]] on $[\mathcal{I},V^e]\cong V_0$).
[[!redirects 2-dinatural transformations]]
[[!redirects strict 2-dinatural transformation]]
[[!redirects strict 2-dinatural transformations]]
[[!redirects lax 2-dinatural transformation]]
[[!redirects lax 2-dinatural transformations]]
[[!redirects pseudo 2-dinatural transformation]]
[[!redirects pseudo 2-dinatural transformations]]
[[!redirects colax 2-dinatural transformation]]
[[!redirects colax 2-dinatural transformations]]
|
2-functor | https://ncatlab.org/nlab/source/2-functor | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A __$2$-functor__ is the [[categorification]] of the notion of a [[functor]] to the setting of [[2-category|2-categories]]. At the 2-categorical level there are several possible versions of this notion one might want depending on the given setting, some of which collapse to the standard definition of a functor between categories when considered on $2$-categories with discrete hom-categories (viewed as $1$-categories). The least restrictive of these is a [[lax functor]], and the strictest is (appropriately) called a [[strict 2-functor]].
For the various separate definitions that do collapse to standard functors, see:
* [[strict 2-functor]]
* [[pseudo functor]]
There is also a notion of '[[lax functor]]', however this notion does not necessarily yield a standard functor when considered on discrete hom-categories.
For the generalisation of this to higher categories, see [[semistrict higher category]].
## Definition
Here we present explicitly the definition for the middling notion of a pseudofunctor, and comment on alterations that yield the stronger and weaker notions.
#### Pseudofunctor between strict $2$-categories
Let $\mathfrak{C}$ and $\mathfrak{D}$ be strict [[2-categories]]. A pseudofunctor $P:\mathfrak{C}\to\mathfrak{D}$ consists of
* A function $P:Ob_\mathfrak{C}\to Ob_\mathfrak{D}$.
* For each pair of objects $A,B\in Ob_\mathfrak{C}$ a functor
$$
P_{A,B}:\mathfrak{C}(A,B)\to\mathfrak{D}(P(A),P(B)).
$$
We will generally write the function and functors as $P$.
* For each triplet of objects $A,B,C\in Ob_\mathfrak{C}$, a natural isomorphism
\begin{centre}
\begin{xymatrix@C20mm}
\mathfrak{C}(B,C)\times\mathfrak{C}(A,B) \rtwocell^{P\circ\Gamma}_{\Gamma\circ(P\times P)}{\gamma} & \mathfrak{D}(P(A),P(C))
\end{xymatrix}
\end{centre}
whose components are $2$-cell isomorphisms $\gamma_{f,g}:P(f\circ g) \Rightarrow P(f)\circ P(g)$ as below
\begin{centre}
\begin{xymatrix@C20mm}
P(A) \rtwocell_{P(f)\circ P(g)}^{P(f\circ g)}{\;\;\;\;\gamma_{f,g}} & P(C)
\end{xymatrix}
\end{centre}
* For each object object $A\in Ob_\mathfrak{C}$, a natural isomorphism
\begin{centre}
\begin{xymatrix@C20mm}
1 \rtwocell^{P\circ id_A\;\;\;\;\;\;}_{id_{P(A)}\;\;\;\;\;\;}{\iota\;\;\;\;\;\;\;\;\;\;} & \mathfrak{D}(P(A),P(A))
\end{xymatrix}
\end{centre}
where $1$ denotes the terminal category and $id_A$ is the identity-selecting functor at $A$. Its component is a $2$-cell isomorphism $\iota_{_*}:P(1_A)\Rightarrow 1_{P(A)}$ as below
\begin{centre}
\begin{xymatrix@C20mm}
P(A) \rtwocell_{1_{P(A)}}^{P(1_A)}{\;\;\;\;\iota_*} & P(A)
\end{xymatrix}
\end{centre}
These are subject to the following axioms:
1. For any composable triplet of $1$-cells $(f,g,h)\in Ob_{\mathfrak{C}(C,D)}\times Ob_{\mathfrak{C}(B,C)}\times Ob_{\mathfrak{C}(A,B)}$ we have that
$$
(\gamma_{f,g}\star 1_{P(h)})\circ\gamma_{f\circ g,h}=(1_{P(f)}\star\gamma_{g,h})\circ\gamma_{f,g\circ h},
$$
where $\circ$ denotes vertical composition and $\star$ denotes horizontal composition, as illustrated by the following commutative $2$-cell diagram in $\mathfrak{D}(P(A),P(D))$:
\begin{centre}
\begin{xymatrix@R20mm@C20mm}
P(f)\circ P(g)\circ P(h) & \ar@2{->}[l]_{1_{P(f)}\star\gamma_{g,h}} P(f)\circ P(g\circ h) \\ P(f\circ g)\circ P(h) \ar@2{->}[u]^{\gamma_{f,g}\star 1_{P(h)}} & \ar@2{->}[l]^{\gamma_{f\circ g,h}} \ar@2{->}[u]_{\gamma_{f,g\circ h}} P(f\circ g\circ h)
\end{xymatrix}
\end{centre}
2. For any composable $1$-cells $(f,g)\in Ob_{\mathfrak{C}(B,C)}\times Ob_{\mathfrak{C}(A,B)}$ we have that
$$
\iota_B\star 1_{P(g)}=\gamma_{1_B,g}^{-1},
$$
$$1_{P(f)}\star\iota_B=\gamma_{f,1_B}^{-1},
$$
as illustrated by the commutative $2$-cell diagrams below
\begin{centre}
\begin{xymatrix@R30mm@C35mm}
P(A)\rtwocell^{P(g)}_{P(g)}{\;\;\;\;\;\;1_{P(g)}} \drtwocell<5.5>_{P(g)\;\;\;\;\;}^{\;\;\;\;\;\;\;\;\;\;\;\;\;\; P(1_B)\circ P(g)}{\;\;\;\;\;\;\;\gamma_{1_B,g}^{-1}} & P(B) \dtwocell^{\;\;\;\;\;\;P(1_B)}_{1_{P(B)\;\;\;\;\;\;}}{\iota_B} \\ & P(B)
\end{xymatrix} \begin{xymatrix@R30mm@C35mm}
P(B) \rtwocell^{P(1_B)}_{1_{P(B)}}{\;\;\;\;\iota_B} \drtwocell<5.5>_{P(f)\;\;\;\;\;}^{\;\;\;\;\;\;\;\;\;\;\;\;\;\; P(f)\circ P(1_B)}{\;\;\;\;\;\;\;\gamma_{f,1_B}^{-1}} & P(B) \dtwocell<4>^{\;\;\;\;P(f)}_{P(f)\;\;\;\;}{1_{P(f)}} \\ & P(C)
\end{xymatrix}
\end{centre}
#### Lax 2-Functor
To obtain the notion of a __lax functor__ we only require that the coherence morphisms $\gamma_{f,g}$ and $\iota_A$ be $2$-cells, not necessarily $2$-cell isomorphisms. This prevents us from going back and forth between preimages and images of identity $1$-cells and horizontally composed $1$-cells/$2$-cells. Similarly, to obtain an __oplax functor__ we reverse the direction of these 2-cells.
#### Strict 2-Functor
To obtain the notion of a strict $2$-functor we require that $\gamma_{f,g}$ and $\iota_A$ are identity arrows, so horizontal composition and $1$-cell identities literally factor through each functor in the same way vertical composition and $2$-cell identities do.
\begin{remark}
There is a notion of a 'weak 2-category', however it usually doesn\'t make sense to speak of strict $2$-functors between weak $2$-categories[^1], but it does make sense to speak of lax (or 'weak') $2$-functors between strict $2$-categories. Indeed, the weak $3$-[[3-category|category]] [[Bicat]] of bicategories, pseudofunctors, [[pseudonatural transformations]], and [[modifications]] is [[equivalence of categories|equivalent]] to its full sub-3-category spanned by the strict 2-categories. However, it is not equivalent to the $3$-category [[Str2Cat]] of strict $2$-categories, *strict* $2$-functors, transformations, and modifications. (For discussion of the terminological choice "$2$-functor" and $n$-functor in general, see [[higher functor]].)
\end{remark}
## Properties
\begin{prop}\label{CharacterizationOfEquivalencesOf2Categories}
**(recognition of [[equivalences of 2-categories]] assuming the [[axiom of choice]])**
\linebreak
Assuming the [[axiom of choice]], a 2-functor $F \,\colon\, \mathcal{C} \xrightarrow{\;} \mathcal{D}$ is an [[equivalence of 2-categories]] precisely if it is
1. **essentially surjective**:
[[surjection|surjective]] on [[equivalence in a 2-category|equivalence]] [[equivalence classes|classes]] of [[objects]]: $\pi_0(F) \;\colon\; \pi_0(\mathcal{C}) \twoheadrightarrow \pi_0(\mathcal{D})\;$,
1. **fully faithful** (e.g. [Gabber & Ramero 2004, Def. 2.4.9 (ii)](#GabberRamero04)):
for each [[pair]] of [[objects]] $X,\, Y \in \mathcal{C}$ the component functor is an [[equivalence of categories|equivalence of]] [[hom-categories]] $F_{X,Y} \,\colon\, \mathcal{C}(X,Y) \xrightarrow{\simeq} \mathcal{D}\big(F(X), F(Y)\big)$,
which by the analogous theorem for 1-functors ([this Prop.](equivalence+of+categories#ViaEssentiallySurjectiveAndFullyFaithful)) means equivalently that $F$ is (e.g. [Johnson & Yau 2020, Def. 7.0.1](#JohnsonYau20))
1. **essentially full on 1-cells**:
namely that each component functor $F_{X,Y}$ is an [[essentially surjective functor]];
1. **fully faithful on 2-cells**:
namely that each component functor $F_{X,Y}$ is a [[fully faithful functor]].
\end{prop}
This is classical [[folklore]]. It is made explicit in, e.g. [Gabber & Ramero 2004, Cor. 2.4.30](#GabberRamero04); [Johnson & Yau 2020, Thm. 7.4.1](#JohnsonYau20).
## Related concepts
* [[function]]
* [[functor]]
* **2-functor** / [[pseudofunctor]] / [[(2,1)-functor]]
* [[n-functor]]
* [[(β,1)-functor]]
* [[(β,n)-functor]]
[[!include properties of functors -- contents]]
## References
Textbook accounts:
* {#GabberRamero04} [[Ofer Gabber]], [[Lorenzo Ramero]], Def. 2.1.14 in: *Foundations for almost ring theory* ([arXiv:math/0409584](https://arxiv.org/abs/math/0409584))
* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], _2-Dimensional Categories_, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))
[^1]: Although there are certain contexts in which it does. For instance, there is a [[model structure]] on the category of [[bicategories]] and strict 2-functors between them, which models the homotopy theory of bicategories and weak 2-functors.
[[!redirects 2-functors]]
[[!redirects strict 2-functor]]
[[!redirects strict 2-functors]] |
2-gerbe | https://ncatlab.org/nlab/source/2-gerbe |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
### Nonabelian 2-gerbes
For $\mathcal{X}$ an [[(β,1)-topos]], a **2-gerbe** $P$ in $\mathcal{X}$ is an object which is
1. [[1-connective]];
1. [[2-truncated]].
The first condition says that it is an [[(β,1)-sheaf]] with values in [[2-groupoid]]s. The second says that $P \to *$ is an [[effective epimorphism]] and that the [[categorical homotopy groups in an (β,1)-topos|0-th homotopy sheaf]] is the terminal sheaf. In the literature this is often stated as saying that $P$ is a) _locally connected_ and b) _locally non-empty_ .
### Abelian 2-gerbes
For $\mathcal{X}$ an [[(β,1)-topos]], an **abelian 2-gerbe** $P$ in $\mathcal{X}$ is an object which is
1. [[2-truncated]];
1. [[2-connected]].
## Related concepts
* [[principal bundle]] / [[torsor]] / [[associated bundle]]
* [[principal 2-bundle]] / [[gerbe]] / [[bundle gerbe]]
* [[principal 3-bundle]] / **2-gerbe** / [[bundle 2-gerbe]]
* [[principal β-bundle]] / [[associated β-bundle]] / [[β-gerbe]]
## References
A comprehensive discussion of nonabelian 2-gerbes is in
* [[Lawrence Breen]], _On the classification of 2-gerbes and 2-stacks_ , Astérisque 225 (1994) ([numdam:AST_1994__225__1_0](http://www.numdam.org/item/?id=AST_1994__225__1_0))
A more expository discussion is in
* [[Lawrence Breen]], _Notes on 1- and 2-gerbes_ in [[John Baez]], [[Peter May]] (eds.) _[[Towards Higher Categories]]_ ([arXiv:math/0611317](http://arxiv.org/abs/math/0611317)).
{#Breen}
Abelian 2-gerbes are a special case (see [[β-gerbe]]) of the discussion in section 7.2.2 of
* [[Jacob Lurie]], _[[Higher Topos Theory]]_
See also
* [[Ettore Aldrovandi]], _2-Gerbes bound by complexes of gr-stacks, and cohomology_ Journal of Pure and Applied Algebra 212 (2008), 994–103 ([pdf](http://www.math.fsu.edu/~aluffi/archive/paper271.pdf))
[[!redirects 2-gerbes]] |
2-Giraud theorem | https://ncatlab.org/nlab/source/2-Giraud+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _2-Giraud theorem_ is the generalization of [[Giraud's theorem]] from [[topos theory]] to [[2-topos theory]].
## Statement
The following theorem, which generalizes the classical [[Giraud theorem]], is due to [[StreetCBS]].
+-- {: .num_theorem #Giraud}
###### Theorem
For a [[2-category]] $K$, the following are equivalent.
* $K$ is equivalent to the 2-category of [[2-sheaves]] on a small [[2-site]].
* $K$ is an infinitary [[2-pretopos]] with a small [[eso-generator]].
* $K$ is a [[reflective sub-2-category]] of a category $[C^{op},Cat]$ of [[2-presheaves]] with left-exact reflector.
=--
In fact, it is not hard to prove the same theorem for [[n-categories]], for any $1\le n\le 2$.
+-- {: .num_theorem #Giraud}
###### Theorem
For an [[n-category]] $K$, the following are equivalent.
* $K$ is equivalent to the $n$-category of [[n-sheaves]] on a small [[n-site]].
* $K$ is an infinitary [[n-pretopos]] with a small [[eso-generator]].
* $K$ is a reflective sub-$n$-category of a category $[C^{op},n Cat]$ of $n$-presheaves with left-exact reflector.
=--
For $n=2$ this is Street's theorem; for $n=1$ it is the classical theorem. The other values included are of course $n=(1,2)$ and $n=(2,1)$.
## References
* [[Mike Shulman]], _[[michaelshulman:2-Giraud theorem]]_
|
2-Grothendieck construction | https://ncatlab.org/nlab/source/2-Grothendieck+construction |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Generalization of the [[Grothendieck construction]] from [[categories]] to [[2-categories]] and [[bicategories]].
## Related concepts
* [[(infinity,1)-Grothendieck construction|$(\infty,1)$-Grothendieck construction]]
## References
Definition of a bicategory of elements construction for pseudo-distributors/pseudo-profunctors:
* [[Ross Street]], Β§1.10 of: _Fibrations in bicategories_, [[Cahiers de Topologie et GΓ©omΓ©trie DiffΓ©rentielle CatΓ©goriques]], **21** 2 (1980) 111-160 [[numdam](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1980__21_2_111_0)]
More exhaustive references:
* [[Igor BakoviΔ]], *Grothendieck construction for bicategories* [[pdf](https://www2.irb.hr/korisnici/ibakovic/sgc.pdf)]
* [[Igor BakoviΔ]], *Fibrations of bicategories* [[pdf](https://www2.irb.hr/korisnici/ibakovic/groth2fib.pdf)]
* [[Mitchell Buckley]]. _Fibred 2-categories and bicategories_, Journal of pure and applied algebra **218** 6 (2014) 1034-1074 [[doi](https://doi.org/10.1016/j.jpaa.2013.11.002)]
See also:
* [[Luca Mesiti]], *The 2-Set-enriched Grothendieck construction and the lax normal conical 2-limits* [[arXiv:2302.04566](https://arxiv.org/abs/2302.04566)]
[[!redirects 2-Grothendieck constructions]]
[[!redirects Grothendieck construction for 2-categories]]
[[!redirects Grothendieck constructions for 2-categories]]
[[!redirects Grothendieck construction for bicategories]]
[[!redirects Grothendieck constructions for bicategories]]
|
2-group | https://ncatlab.org/nlab/source/2-group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of _2-group_ is a [[vertical categorification]] of the notion of _[[group]]_.
It is the special case of an [[n-group]] for $n=2$, equivalently an [[β-group]] which is [[n-truncated|1-truncated]]. Under the [[looping and delooping]]-equivalence, 2-groups are equivalent to [[pointed object|pointed]] [[n-connected|connected]] [[homotopy n-type|homotopy 2-types]].
Somewhat more precisely, a _$2$-group_ is a [[group object]] in the [[(2,1)-category]] of [[groupoids]]. Equivalently, it is a [[monoidal groupoid]] in which the [[tensor product]] with any [[object]] has an [[inverse]] up to [[isomorphism]]. Also equivalently, by the [[looping and delooping]]-equivalence, it is a [[pointed object|pointed]] [[2-groupoid]] with a single [[equivalence class]] of objects.
Like other notions of [[higher category theory]], $2$-groups come in weak and strict forms, depending on how you interpret the above.
### Strict $2$-groups
The earliest version studied is that of [[strict 2-group]]s.
A __strict $2$-group__ consists of:
* a collection of [[group]] homomorphisms of the form
$$
C_1 \stackrel{s,t}{\to} C_0 \stackrel{i}{\to} C_1
$$
such that the composites $s\cdot i$ and $t\cdot i$ are the identity morphisms on $C_0$, and
such that, writing $C_1 \times_{t,s} C_1$ for the pullback,
$$
\array{
C_1 \times_{t,s} C_1 &\to& C_1
\\
\downarrow && \downarrow^{t}
\\
C_1 &\stackrel{s}{\to}& C_0
}
$$
there is, in addition, a homomorphism
$$
C_1 \times_{t,s} C_1 \stackrel{comp}{\to} C_1
$$
"respecting $s$ and $t$";
* such that the _composition_ $comp$ is associative
and unital with respect to $i$ "in the obvious way".
See [[strict 2-group]] for further discussion and examples.
### Weak $2$-groups
A __weak $2$-group__, or simply __$2$-group__, is a (weak) [[monoidal category]] where every morphism is invertible and *such that*:
* given any object $x$, there exists an object $x^{-1}$ such that the monoidal products $x \otimes x^{-1}$ and $x^{-1} \otimes x$ are each [[isomorphism|isomorphic]] to the monoidal unit $1$.
A __coherent $2$-group__ is a monoidal category where every morphism is invertible and *equipped with*:
* for each object $x$ a specific object $x^{-1}$ and specific [[isomorphism]]s from $x \otimes x^{-1}$ and $x^{-1} \otimes x$ to $1$ which form an [[adjoint equivalence]].
A theorem in HDA V (see references) shows that every weak $2$-group may be made coherent. For purposes of [[internalization]], one probably wants to use the coherent version.
## Definition
+-- {: .num_defn}
###### Definition
The [[(2,1)-category]] $2Grp$ of **2-groups** is equivalently
* the [[full sub-(β,1)-category|full sub-(2,1)-category]] of that of [[monoidal categories]] and [[strong monoidal functors]] on those that are [[groupoids]] and whose [[tensor product]] has weak [[inverses]] for each object;
* the [[full sub-(β,1)-category]] of that of [[β-groups]] on the [[n-truncated|1-truncated]] objects;
* the [[full sub-(β,1)-category]] of that of [[group object in an (β,1)-category|group objects]] in [[βGrpd]] on the [[n-truncated|1-truncated]] objects;
* the [[full sub-(β,1)-category]]
$$
\infty Grpd_{1 \leq \bullet \leq 2}^{*/}
\hookrightarrow
\infty Grpd^{*/}
$$
of [[βGrpd]]$^{*/}$ on those objects which are both [[n-connected|connected]] as well as [[n-truncated|2-truncated]].
=--
+-- {: .num_remark}
###### Remark
The last equivalent characterization is related to the previous ones by the [[looping and delooping]]-equivalence
$$
\array{
Grp(\infty Grpd)
&\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}&
\infty Grpd^{*/}_{\geq 1}
\\
\uparrow^{\mathrlap{full\;inc.}}
&&
\uparrow^{\mathrlap{full\;inc.}}
\\
2 Grp
&\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}&
2Grpd_{\geq 1}^{*/}
}
\,.
$$
Here $(-)^{*/}$ denotes taking [[pointed objects]], hence the [[over-(β,1)-category|slice under]] the point, and $(-)_{\geq}$ denotes the full [[sub-(β,1)-category|full inclusion]] on [[n-connected|connected]] objects.
=--
By replacing in the last of these equivalent characterizations the ambient [[(β,1)-topos]] [[βGrpd]] with any other one, to be denoted $\mathbf{H}$, obtains notions of 2-groups with extra structure. For instance for $\mathbf{H} = $ [[SmoothβGrpd]] the $(\infty,1)$-topos of [[smooth β-groupoids]] one obtains:
+-- {: .num_defn}
###### Definition
The [[(2,1)-category]] $Smooth2Grp$ of **smooth 2-groups** is
$$
\array{
Grp(Smooth \infty Grpd)
&\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}&
Smooth\infty Grpd^{*/}_{\geq 1}
\\
\uparrow^{\mathrlap{full\;inc.}}
&&
\uparrow^{\mathrlap{full\;inc.}}
\\
Smooth 2 Grp
&\stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}&
Smooth2Grpd_{\geq 1}^{*/}
}
\,.
$$
=--
Below in [presentation by crossed modules](#PresentationByCrossedModules) are discussed more explict presentations of $2Grp$ and $Smooth2Grpd$ etc. by explicit algebraic data.
## Properties
### Presentation by crossed modules
{#PresentationByCrossedModules}
By the discussion there, every _[[β-group]]_ has a _presentation_ by a [[simplicial group]]. More precisely, the [[(β,1)-category]], $\infty Grp$, is [[presentable (β,1)-category|presented]] by the [[model structure on simplicial groups]] (for instance under [[simplicial localization]])
$$
\infty Grpd \simeq L_W Grp^{\Delta^{op}}
\,.
$$
Moreover, if $G \in Grp^{\Delta^{op}}$ is an [[n-group]], then it is equivalent to a [[coskeleton|n-coskeletal]] simplicial group. For $n = 2$ one finds that these are naturally identified with _[[crossed modules]]_ of groups (see there for more details).
In conclusion, this means that
+-- {: .num_prop #PresentationOf2GrpByCrossedModules}
###### Proposition
The [[(2,1)-category]] $2Grp$ of 2-groups is [[equivalence of (infinity,1)-categories|equivalent]] to the [[simplicial localization]] of the [[category with weak equivalences]] whose
* objects are [[crossed modules]]
* morphisms are [[homomorphisms]] of crossed modules;
* weak equivalences are those morphisms of crossed modules which correspond to [[weak homotopy equivalences]] of the corresponding simplicial groups.
=--
A straightforward analysis shows that
+-- {: .num_prop #HomotopyGroupsOfCrossedModule}
###### Proposition
For $(G_1 \stackrel{\delta}{\to} G_0, G_0 \stackrel{\alpha}{\to} Aut(G_1))$ a [[crossed module]], the [[homotopy groups]] of the corresponding [[2-group]]/[[simplicial group]] are
* $\pi_0 = G_0 / im(\delta)$ (the [[quotient]] of $G_0$ by the [[image]] of $\delta$, which is necessarily a [[normal subgroup]] of $G_0$);
* $\pi_1 = ker(\delta)$ (the [[kernel]] of $\delta$).
Accordingly, a weak equivalence of crossed modules $f : G \to H$ is a morphism of crossed modules which induces an [[isomorphism]] of kernel and cokernel of $\delta_G$ with that of $\delta_H$.
=--
Similar statements hold for 2-groups with extra structure. For instance the $(2,1)$-category $Smooth2Grp$ of smooth 2-groups is equivalent to the [[simplicial localization]] of the category whose
* objects are [[sheaves]] of [[crossed modules]] on [[CartSp]]${}_{smooth}$;
* weak equivalences are those morphisms of sheaves of crossed modules which on every [[stalk]] induce weak equivalences of crossed modules as above.
(See the discussion at [[SmoothβGrpd]] for more on this.)
## Examples
### Specific examples
#### Picard 2-group
* [[Picard 2-group]]
#### Automorphism 2-groups
For $C$ any [[2-category]] and $c \in C$ any object of it, the category $Aut_C(c) \subset Hom_C(c,c)$ of auto-equivalences of $c$ and invertible 2-morphisms between these is naturally a 2-group, whose group product comes from the horizontal composition in $C$.
If $C$ is a [[strict 2-category]] there is the notion of strict [[automorphism 2-group]]. See there for more details on that case.
For instance if $C = Grp_2 \subset Grpd$ is the 2-category of [[group]] obtained by regarding groups as one-object [[groupoid]]s, then for $H \in Grp$ a group, its automorphism 2-group obtained this way is the strict 2-group
$$
AUT(H) := Aut_{Grp_2}(H)
$$
corresponding to the [[crossed module]] $(H \stackrel{Ad}{\to} Aut(H))$, where $Aut(H)$ is the ordinary [[automorphism group]] of $H$.
#### Inner automorphism 2-groups
See [[inner automorphism 2-group]].
#### String 2-group
See [[string 2-group]].
#### Platonic 2-group
See _[[Platonic 2-group]]_
### Equivalences of 2-groups
{#ExamplesForEquivalencesOf2Groups}
We discuss some weak equivalences in the [[category with weak equivalences]] of [[crossed modules]] and crossed module homomorphisms, which presents $2Grp$ by the discussion [above](#PresentationByCrossedModules).
#### From inclusions of normal subgroups
{#FromInclusionsOfNormalSubgroups}
Let $G$ be a [[group]] and $N \hookrightarrow G$ the inclusion of a [[normal subgroup]]. Equipped with the canonical [[action]] of $G$ on $N$ by [[conjugation]], this inclusion constitutes a [[crossed module]]. There is a canonical morphism of crossed modules from $(N \hookrightarrow G)$ to $(1 \to G/N)$, hence to the ordinary [[quotient]] group, regarded as a crossed module.
+-- {: .num_prop}
###### Observation
The morphism $(N \hookrightarrow G) \to G/N$ is a weak equivalence of crossed modules, prop. \ref{PresentationOf2GrpByCrossedModules}. Accordingly, it presents an [[equivalence in an (infinity,1)-category|equivalence]] of 2-groups.
=--
+-- {: .proof}
###### Proof
The canonical morphism in question is given by the commuting diagram of groups
$$
\array{
N &\stackrel{f_1}{\to}& 1
\\
\downarrow && \downarrow
\\
G &\stackrel{f_0}{\to}& G/N
}
\,.
$$
By prop. \ref{HomotopyGroupsOfCrossedModule} we need to check that this induces an isomorphism on the [[kernel]] and [[cokernel]] of the vertical morphisms.
The kernel of the left vertical morphism is the trivial group, because $N \hookrightarrow G$ is an inclusion, by definition. Clearly also the kernel of the right vertical morphisms is the trivial group. Hence $f_1$ restricted to the kernels is the unique morphism from the trivial group to itself, hence is an isomrphism.
Moreover, the cokernel of the left vertical morphism is of course the quotient $G/N$ and $f_0$, being the quotient map, is manifestly an isomorphism on cokernels.
=--
This class of weak equivalence plays an important role as constituting [[β-anafunctor|2-anafunctors]] that exhibit long [[fiber sequence]] extensions of [[short exact sequences]] of [[central extensions]] of groups.
+-- {: .num_prop}
###### Observation
Let $A \to \hat G$ be the inclusion of a [[center|central]] subgroup, exhibiting a [[central extension]] $A \to \hat G \to G$ with $G := \hat G/A$. Then this [[short exact sequence]] of groups extends to a long [[fiber sequence]] of 2-groups
$$
A \to \hat G \to G \to \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \to \mathbf{B}^2 A
\,,
$$
where $\mathbf{B}A$ denotes the 2-group given by the [[crossed module]] $(A \to 1)$, and similarly for the other cases.
Here the [[connecting homomorphism]] $G \to \mathbf{B}A$ is presented in the category of crossed modules by a zig-zag / [[anafunctor]] whose left leg is the above weak equivalence:
$$
(1 \to G) \stackrel{\simeq}{\leftarrow}
(A \to \hat G)
\to
(A \to 1)
\,.
$$
=--
+-- {: .num_example}
###### Example
For smooth 2-groups, useful examples of the above are smooth refinements of various [[universal characteristic classes]]:
* the second [[Stiefel-Whitney class]]
$$
w_2 : \mathbf{B}SO \to \mathbf{B}^2\mathbb{Z}_2
$$
is induced this way from the central extension $\mathbb{Z}_2 \to Spin \to SO$ of the [[special orthogonal group]] by the [[spin group]];
* the first [[Chern class]]
$$
c_1 : \mathbf{B}U(1) \to \mathbf{B}^2 \mathbb{Z}
$$
induced from the central extension $\mathbb{Z} \to \mathbb{R} \to U(1)$.
=--
## Related concepts
* [[group]]
* [[groupoid]], [[monoidal groupoid]]
* **2-group**, [[crossed module]], [[differential crossed module]]
* [[braided 2-group]], [[symmetric 2-group]]
* [[quantum 2-group]]
* [[3-group]], [[2-crossed module]] / [[crossed square]], [[differential 2-crossed module]]
* [[n-group]]
* [[β-group]], [[simplicial group]], [[crossed complex]], [[hypercrossed complex]]
* [[group stack]]
[[smooth 2-group]]
* [[higher gauge symmetry]]
* [[generalized global symmetry]]
[[!include homotopy n-types - table]]
## References
{#References}
### Original articles
{#OriginalArticles}
The notion of 2-groups first appears in
* {#Solian72} [[Alexandru Solian]], *Groupe dans une catΓ©gorie*, C. R. Acad. Sc. Paris **275** (1972) [[ark:/12148/bpt6k57310477/f7](https://gallica.bnf.fr/ark:/12148/bpt6k57310477/f7.item)]
* {#Solian80} [[Alexandru Solian]], *Coherence in categorical groups*, Communications in Algebra **9** 10 (1980) 1039-1057 [[doi:10.1080/00927878108822631](https://doi.org/10.1080/00927878108822631)]
and in
* {#Sinh73} [[HoΓ ng XuΓ’n SΓnh]], _Gr_-catΓ©gories, PhD thesis, Hanoi (1973, 1975) [[web](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh.html), [pdf](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh_I.pdf), [[Sinh_Gr-Categories.pdf:file]]]
[Sinh (1973)](#Sinh73) was supervised by [[Alexander Grothendieck]] and showed that 2-groups are classified, up to equivalence, by quadruples consisting of:
* a group $G$
* an abelian group $A$
* an action of $G$ as automorphisms of $A$
* an element of $H^3(G,A)$, often called the *Sinh invariant*.
She later published two papers on the subject:
* {#Sinh78} [[HoΓ ng XuΓ’n SΓnh]], *Gr-catΓ©gories strictes*, Acta Mathematica Vietnamica **3** 2 (1978) 47-59 [[web](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh.html), [pdf](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/Hoang_Xuan_Sinh_Gr_categories_strictes.pdf)]
* {#Sinh83} [[HoΓ ng XuΓ’n SΓnh]], *CatΓ©gories de Picard restreintes*, Acta Mathematica Vietnamica **7** 1 (1983) 117-122 [[pdf](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/Hoang_Xuan_Sinh_Categories_de_Picard_restreintes.pdf), [web](https://pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh.html)]
[Sinh 1978](Sinh78) appears to prove that every 2-group is equivalent to a [[strict 2-group]] arising from a [[crossed module]]. The second calls a [[symmetric 2-group]] (i.e. a [[symmetric monoidal category]] with all objects and morphisms invertible) a *[[Picard category]]*, and calls a Picard category *restrained* if the [[braiding]] $B_{x,x} \colon x \otimes x \to x \otimes x$ is the [[identity morphism|identity]] for all objects $x$. The article then proves that every Picard category is equivalent to one arising from a 2-term [[chain complex]] of abelian groups.
Computational enumeration of geometrically [[discrete group|discrete]] 2-groups using the computer program [XMod](http://pages.bangor.ac.uk/~mas023/chda/xmod/xmod244.html):
* Murat Alp, Christopher Wensley, _Enumeration of $Cat^1$-groups of low order_, Int. J. Algebra Comput. 10, 407 (2000) [[doi:10.1142/S0218196700000170](http://www.worldscientific.com/doi/abs/10.1142/S0218196700000170)]
* [[Graham Ellis]], Luyen Van Le, *Homotopy 2-types of Low order*, Experimental Mathematics **23** 4 (2014) [[doi:10.1080/10586458.2014.912059](https://doi.org/10.1080/10586458.2014.912059), [pdf](http://hamilton.nuigalway.ie/preprints/2t.pdf)]
### Review
An early textbook account on [[strict 2-groups]] and explaining the relation to [[crossed modules]]:
* [[Saunders MacLane]], Β§XII.8 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]
Exposition of general 2-groups as [[monoidal categories]] with all objects and morphisms invertible (sometimes called [[Picard 2-groups]]):
* {#BaezLauda03} [[John Baez]], [[Aaron Lauda]], _HDA V: 2-Groups_, Theory and Applications of Categories **12** (2004) 423-491 [[arXiv:math.QA/0307200](http://arxiv.org/abs/math.QA/0307200), [tac:12-14](http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html)]
### Geometric 2-groups
Beware that most of the above discussion is about [[discrete infinity-group|geometrically discrete]] 2-groups.
Discussion of geometrically structured 2-groups (notably [[smooth 2-groups]], hence "Lie 2-groups"):
* {#Schommer-Pries11} [[Chris Schommer-Pries]], *Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group* Geometry & Topology **15** (2011) 609-676 [[arXiv:0911.2483](http://arxiv.org/abs/0911.2483), [doi:10.2140/gt.2011.15.609](https://doi.org/10.2140/gt.2011.15.609)]
and via [[group stacks]]:
* [[Urs Schreiber]], Β§2.6.5 & Β§3.4.2 of: _[[schreiber:differential cohomology in a cohesive topos]]_ [[arXiv:1310.7930](https://arxiv.org/abs/1310.7930)]
For more on this see the references at *[[string 2-group]]*.
### Examples
A key example due to its universality for [[higher central extensions]] of [[compact Lie groups]] by the [[circle 2-group]] to [[smooth 2-groups]] is
* the *[[string 2-group]]* -- see [there](string+2-group#References) for references.
Further on [[2-group]]-[[higher central extensions|extensions]] by the [[circle 2-group]]:
of [[tori]] (see also at *[[T-duality 2-group]]*):
* [[Nora Ganter]], _Categorical Tori_, SIGMA 14 (2018), 014, 18 ([arXiv:1406.7046](https://arxiv.org/abs/1406.7046))
of [[finite subgroups of SU(2)]] (to [[Platonic 2-groups]]):
* {#EpaGanter16} [[Narthana Epa]], [[Nora Ganter]], _Platonic and alternating 2-groups_, Higher Structures 1(1):122-146, 2017 ([arXiv:1605.09192](http://arxiv.org/abs/1605.09192), [hs:30](https://journals.mq.edu.au/index.php/higher_structures/article/view/30))
{#ReferencesInCondensedMatterPhysics} Arguments that 2-groups play a role in [[symmetry protected topological phases of matter]]:
* [[Anton Kapustin]], [[Ryan Thorngren]], *Higher symmetry and gapped phases of gauge theories*, in *Algebra, Geometry, and Physics in the 21st Century*, Progress in Mathematics, **324** (2017) 177-202 [[arXiv:1309.4721](https://arxiv.org/abs/1309.4721), [doi:10.1007/978-3-319-59939-7_5](https://doi.org/10.1007/978-3-319-59939-7_5)]
* {#BBCW19} [[Maissam Barkeshli]], [[Parsa Bonderson]], [[Meng Cheng]], [[Zhenghan Wang]], *Symmetry Fractionalization, Defects, and Gauging of Topological Phases*, Phys. Rev. B **100** 115147 (2019) [[arXiv:1410.4540](https://arxiv.org/abs/1410.4540), [doi:10.1103/PhysRevB.100.115147](https://doi.org/10.1103/PhysRevB.100.115147), [talk pdf](http://helper.ipam.ucla.edu/publications/stq2015/stq2015_12401.pdf)]
In this vein, 2-groups are now discussed by a vocal group of physicists under the term "[[generalized global symmetry|generalized symmetries]]" or similar.
[[!redirects 2-groups]]
[[!redirects weak 2-group]]
[[!redirects weak 2-groups]] |
2-groupoid | https://ncatlab.org/nlab/source/2-groupoid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Idea
A **$2$-groupoid** is
* a [[(n,r)-category|(2,0)-category]]
* a [[2-category]] in which all morphisms are [[equivalence]]s
* an [[n-groupoid]] for $n = 2$
* a $2$-[[truncated]] [[β-groupoid]].
## Definition
Fix a meaning/model of [[β-groupoid]], however weak or strict you wish. Then a __$2$-groupoid__ is an $\infty$-groupoid such that all [[parallel pair]]s of $j$-morphisms are [[equivalence|equivalent]] for $j \geq 3$. Thus, up to [[equivalence of categories|equivalence]], there is no point in mentioning anything beyond $2$-morphisms, except whether two given parallel $2$-morphisms are equivalent. This definition may give a concept more general than your preferred definition of $2$-groupoid, but it will be equivalent; basically, you may have to rephrase equivalence of $2$-morphisms as [[equality]].
## Specific models
There are various objects that model the abstract notion of $2$-groupoid.
### Bigroupoids
A **[[bigroupoid]]** is a [[bicategory]] in which all morphisms are [[equivalences]].
Bigroupoids may equivalently be thought of in terms of their [[Duskin nerve]]s. These are precisely the [[Kan complex]]es that are 2-[[hypergroupoid]]s.
### 2-Hypergroupoids
A $2$-[[hypergroupoid]] is a model for a 2-groupoid. This is a [[simplicial set]], whose vertices, edges, and 2-[[simplices]] we identify with the [[object]]s, [[morphism]]s and [[2-morphism]]s of the form
$$
\array{
&& y
\\
& \nearrow &\Downarrow& \searrow
\\
x &&\stackrel{}{\to}&& z
}
$$
in the 2-groupoid, respectively.
Moreover, the 3-[[simplices]] in the simplicial set encode the composition operation: given three composable 2-simplex faces of a tetrahedron (a 3-[[horn]])
$$
\array{
y &\to& &\to& z
\\
\uparrow &\seArrow& &\nearrow& \downarrow
\\
\uparrow &\nearrow& &\Downarrow& \downarrow
\\
x &\to&&\to& w
}
\;\;\;
\;\;\;
\array{
y &\to& &\to& z
\\
&\searrow& &\swArrow& \downarrow
\\
&& &\searrow& \downarrow
\\
&&&& w
}
$$
the unique composite of them is is a fourth face $\kappa$ and a 3-cell $comp$ filling the resulting hollow tetrahedron:
$$
\array{
y &\to& &\to& z
\\
\uparrow &\seArrow& &\nearrow& \downarrow
\\
\uparrow &\nearrow& &\Downarrow& \downarrow
\\
x &\to&&\to& w
}
\;\;\;
\stackrel{comp}{\to}
\;\;\;
\array{
y &\to& &\to& z
\\
\uparrow &\searrow& &\swArrow& \downarrow
\\
\uparrow &{}_\kappa\Downarrow& &\searrow& \downarrow
\\
x &\to&&\to& w
}
\,.
$$
The 3-[[coskeletal]]-condition says that every boundary of a 4-[[simplex]] made up of five such tetrahedra has a unqiue filler. This is the [[associativity]] [[coherence law]] on the comoposition operation:
$$
\array{\arrayopts{\rowalign{center}}
\array{\begin{svg}
[[!include monoidal category > pentagonator]]
\end{svg}}
}
$$
This says that any of the possible ways to use several of the 3-simpleces to compose a bunch of compsable 2-morphisms are actually equal.
### Homotopy 2-types
More generally one may consider a [[Kan complex]] that are just [[homotopy equivalent]] to a $3$-[[coskeletal]] one as a $2$-groupoid -- precisely: as representing the same [[homotopy type]], namely a [[homotopy 2-type]].
## Strict $2$-groupoids
The general notion of $2$-groupoid above is also called __weak $2$-groupoid__ to distinguish from the special case of **[[strict 2-groupoids]]**. A strict $2$-groupoid is a [[strict 2-category]] in which all morphisms are strictly invertible. This is equivalently a certain type of [[Grpd]]-[[enriched category]].
## Examples
* For $A$ an [[abelian group]], there is its double [[delooping]] 2-groupoid $\mathbf{B}^2 A$. This is given by the [[strict]] 2-groupoid that comes from the [[crossed complex]] $A \to 0 \stackrel{\to}{\to} 0$. As a [[Kan complex]] this is the image under the [[Dold-Kan correspondence]] of the [[chain complex]] $[A \to 0 \to 0]$.
* The [[fundamental 2-groupoid]] of a [[topological space]].
* The [[fundamental infinity-groupoid|fundamental $\infty$-groupoid]] of a [[topological space]] that is itself a [[homotopy 2-type]].
* The [[path 2-groupoid]] of a [[smooth manifold]].
## Related concepts
* [[Grpd-enriched category]]
[[!include homotopy n-types - table]]
[[!redirects 2-groupoid]]
[[!redirects 2-groupoids]]
[[!redirects weak 2-groupoid]]
[[!redirects weak 2-groupoids]]
|
2-groupoid of Lie 2-algebra valued forms | https://ncatlab.org/nlab/source/2-groupoid+of+Lie+2-algebra+valued+forms |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}
## Idea
For $\mathfrak{g}$ a [[Lie 2-algebra]] the **2-groupoid of $\mathfrak{g}$-valued forms** is the [[2-groupoid]] whose objects are [[differential form]]s with values in $\mathfrak{g}$, whose morphisms are [[gauge transformation]]s between these, and whose 2-morphisms are _higher order gauge transformations_ of those.
This naturally refines to a non-[[concrete sheaf|concrete]] [[Lie 2-groupoid]] is the 2-[[truncated]] [[β-Lie groupoid]] whose $U$-parameterized smooth families of objects are smooth [[differential form]]s with values in a [[Lie 2-algebra]], and whose morphisms are gauge transformations of these.
This is the [[higher category theory|higher category]] generalization of the [[groupoid of Lie-algebra valued forms]].
A [[cocycle]] with coefficients in this 2-groupoid is a [[connection on a 2-bundle]].
## Definition
### For strict Lie 2-algebras
Consider a Lie [[strict 2-group]] $G$ corresponding to a Lie [[crossed module]] $(G_2 \stackrel{\delta}{\to} G_1)$ with action $\alpha : G_1 \to Aut(G_2)$. Write $\mathbf{B}G$ for the corresponding [[delooping]] 2-groupoid, the one coming from the [[crossed complex]]
$$
[\mathbf{B}G]
=
(G_2 \stackrel{\delta}{\to} G_1 \stackrel{\to}{\to} *)
\,.
$$
Write $[\mathfrak{g}_2 \stackrel{\delta_*}{\to} \mathfrak{g}_1]$ for the corresponding [[differential crossed module]] with action $\alpha_* : \mathfrak{g}_1 \to der(\mathfrak{g}_2)$
+-- {: .un_def }
###### Definition
The 2-groupoid of Lie 2-algebra valued forms is defined to be the 2-stack
$$
\bar \mathbf{B}G : CartSp{}^{op} \to 2Grpd
$$
which assigns to $U \in CartSp$ the following 2-groupoid:
* An [[object]] is a pair
$$
A \in \Omega^1(U,\mathfrak{g}_1)\,, \;\;\;
B \in \Omega^2(U,\mathfrak{g}_2)
\,.
$$
* A 1-[[morphism]] $(g,a) : (A,B) \to (A',B')$ is a pair
$$
g \in C^\infty(U,G_1)\,,\;\;\; a \in \Omega^1(U,\mathfrak{g}_2)
$$
such that
$$
A' = Ad_{g^{-1}}\left( A + \delta_* a \right) + g^{-1} d g
$$
and
$$
B' = \alpha_{g^{-1}}(
B + d a + [a \wedge a] + \alpha_*(A \wedge a)
)
\,.
$$
The composite of two 1-morphisms
$$
(A,B) \stackrel{(g_1,a_1)}{\to} (A',B') \stackrel{(g_2,a_2)}{\to}
(A'', B'')
$$
is given by the pair
$$
(g_1 g_2, a_1 + (\alpha_{g_2})_* a_2)
\,.
$$
* a [[2-morphism]] $f : (g,a) \Rightarrow (g', a'):(A,B)\to (A',B')$ is a function
$$
f \in C^\infty(U,G_2)
$$
such that
$$
g' = \delta(f)^{-1} \cdot g
$$
and
$$
a' = Ad_{f^{-1}} \left(a + (r_f^{-1} \circ \alpha_f)_*(A)\right) + f^{-1} d f
$$
and composition is defined as follows: vertical composition is given by pointwise multiplication ([[David Roberts|DR]]: the order still needs sorting out!) and horizontal composition is given as horizontal composition in the one-object 2-groupoid $\mathbf{B}G)$.
=--
### For general Lie 2-algebras
{#ForGeneralLie2Algebras}
We consider now $\mathfrak{g}$ a general [[Lie 2-algebra]].
Let $\mathfrak{g}_0$ and $\mathfrak{g}_1$ be the two vector spaces involved and let
$$
\{t^a\} \,, \;\;\; \{b^i\}
$$
be a dual basis, respectively. The structure of a Lie 2-algebra is conveniently determined by writing out the most general [[Chevalley-Eilenberg algebra]]
$$
CE(\mathfrak{g}) \in cdgAlg_\mathbb{R}
$$
with these generators.
We thus have
$$
d_{CE(\mathfrak{g})} t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c - r^a{}_i b^i
$$
$$
d_{CE(\mathfrak{g})} b^i = -\alpha^i_{a j} t^a \wedge b^j
-
r_{a b c} t^a \wedge t^b \wedge t^c
\,,
$$
for collections of structure constants $\{C^a{}_{b c}\}$ (the bracket on $\mathfrak{g}_0$) and $\{r^i_a\}$ (the differential $\mathfrak{g}_1 \to \mathfrak{g}_0$) and $\{\alpha^i{}_{a j}\}$ (the [[action]] of $\mathfrak{g}_0$ on $\mathfrak{g}_1$) and $\{r_{a b c}\}$ (the "Jacobiator" for the bracket on $\mathfrak{g}_0$).
These constants are subject to constraints (the weak [[Jacobi identity]] and its higher [[coherence law]]s) which are precisely equivalent to the condition
$$
(d_{CE(\mathfrak{g})})^2 = 0
\,.
$$
Over a test space $U$ a $\mathfrak{g}$-valued form datum is a morphism
$$
\Omega^\bullet(U) \leftarrow W(\mathfrak{g})
:
(A,B)
$$
from the [[Weil algebra]] $W(\mathfrak{g})$.
This is given by a 1-form
$$
A \in \Omega^1(U, \mathfrak{g}_0)
$$
and a 2-form
$$
B \in \Omega^2(U, \mathfrak{g}_1)
\,.
$$
The [[curvature]] of this is $(\beta, H)$, where the 2-form component ("fake curvature") is
$$
\beta^a = d_{dR} A^a + \frac{1}{2}C^a{}_{b c} A^b \wedge A^c
+ r^a{}_i B^i
$$
and whose 3-form component is
$$
H^i
=
d_{dR} B^i
+
\alpha^i{}_{a j} A^a \wedge B^j
+
t_{a b c} A^a \wedge A^b \wedge A^c
\,.
$$
## Properties
+-- {: .un_def }
###### Proposition
**(flat Lie 2-algebra valued forms)**
The full sub-2-groupoid on _flat_ Lie 2-algebra valued forms, i.e. those pairs $(A,B)$ for which the 2-form [[curvature]]
$$
\delta_* B - d A + [A \wedge A] = 0
$$
and the 3-form [[curvature]]
$$
d B + [A \wedge B] = 0
$$
vanishes is a resolution of the underlying discrete Lie 2-groupoid $\mathbf{\flat} \mathbf{B}G$ of the Lie 2-groupoid $\mathbf{B}G$.
=--
This is discussed at [[β-Lie groupoid]] in the section <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#DiffCoeffsForLie2Group">strict Lie 2-groups -- differential coefficients</a>.
+-- {: .un_def }
###### Proposition
Let $\mathbf{\Pi}_2 : CartSp \to 2LieGrpd$ be the smooth 2-[[fundamental groupoid]] functor and let $P_2 : CartSp \to 2LieGrpd$ be the [[path n-groupoid|path 2-groupoid]] functor, taking values in the 2-catgeory $2Grpd(Difeol)$ of 2-groupoids [[internal category|internalization]] to [[diffeological space]]s. Then
* the 2-groupoid of Lie 2-algebra valued forms for which both 2- and 3-form curvature vanish is canonically equivalent to
$$
Hom_{2Grpd(Diffeol)}(\Pi_2(-), \mathbf{B}G) : CartSp^{op} \to 2Grpd
\,;
$$
* the 2-groupoid of Lie 2-algebra valued forms for which the 2-form curvature vanishes is canonically equivalent to
$$
Hom_{2Grpd(Diffeol)}(P_2(-), \mathbf{B}G) : CartSp^{op} \to 2Grpd
\,;
$$
=--
The equivalence is given by 2-dimensional [[parallel transport]].
A proof is in [SchrWalII](http://arxiv.org/abs/0802.0663).
The following proposition asserts that the Lie 2-groupoid of Lie 2-algebra valued forms is the coefficient object for for _differential nonabelian cohomology_ in degree 2, namely for _connections_ on [[principal 2-bundle]]s and in particular on [[gerbe]]s.
+-- {: .un_def }
###### Proposition
**(2-bundles with connection)**
For $X$ a [[paracompact space|paracompact]] [[smooth manifold]] and $\{U_i \to X\}$ a [[good open cover]] the 2-groupoid, let $X \stackrel{\simeq}{\leftarrow} C(\{U_i\})$ be the corresponding [[Cech nerve]] smooth 2-groupoid. Then
$$
Hom_{2Grpd(Diffeol)}( C(\{U_i\}), \bar \mathbf{B}G)
$$
is equivalent to the [[2-groupoid]] of $G$-[[principal 2-bundle]]s with [[connection on a 2-bundle|2-connection]].
=--
This is discussed and proven in
[SchrWalII](http://arxiv.org/abs/0802.0663) for the case where the 2-form curvature is restricted to vanish. In this case the above can be written as
$$
Hom( C(\{U_i\}), Hom(P_2(-), \mathbf{B}G))
\simeq
Hom(P_2(C(\{\U_i\})), \mathbf{B}G)
\,,
$$
where $P_2(C(\{\U_i\}) \in 2LieGrpd$ is a resolution of the [[path 2-groupoid]] of $X$.
## Related concepts
* [[groupoid of Lie-algebra valued forms]]
* **2-groupoid of Lie 2-algebra valued forms**
* [[nonabelian Stokes theorem]]
* [[3-groupoid of Lie 3-algebra valued forms]]
* [[β-groupoid of β-Lie-algebra valued forms]]
## References
The 2-groupoid of Lie 2-algebra valued forms described in [definition 2.11](http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.0663v3.pdf#page=27) of
* Schreiber, Waldorf, _Smooth functors versus differential forms_ (<a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+references#SWII">web</a>). {#SchWalII}
There are many possible conventions. The one reproduced above is supposed to describe the _bidual_ [[opposite 2-category]] of the 2-groupoid as defined in that article, with the direction of 1- and 2-morphisms reversed.
See also
<a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+survey#ConnectionOn2Bundle">differential cohomology in an (β,1)-topos -- survey - connections on 2-bundles</a>.
[[!redirects Lie 2-algebra valued 2-form]]
[[!redirects Lie 2-algebra valued forms]]
[[!redirects Lie 2-algebra valued differential forms]]
[[!redirects Lie 2-algebra valued form]] |
2-Hilbert space | https://ncatlab.org/nlab/source/2-Hilbert+space |
#Contents#
* table of contents
{:toc}
## Idea
The concept of $2$-Hilbert space is supposed to be a [[categorification]] of that of [[Hilbert space]], or at least of [[finite-dimensional vector space|finite dimensional]] such: [[inner product spaces]].
One way to define this is as a [[Kapranov-Voevodsky 2-vector space]] where the [[hom-functor]] plays the role of the [[categorification|categorified]] [[inner product]] ([Baez 96](#Baez96)).
More generally this works for [[semisimple categories]] (...)
## Related concepts
* [[Schur's lemma]]
* [[2-vector space]]
## References
See
* {#Baez96} [[John Baez]]; _Higher-Dimensional Algebra II: 2-Hilbert Spaces_ ([arXiv](http://arxiv.org/abs/q-alg/9609018))
for one approach.
[[!redirects 2-Hilbert spaces]] |
2-isomorphism | https://ncatlab.org/nlab/source/2-isomorphism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
\tableofcontents
\section{Idea}
In a [[higher category theory|higher category]], [[inverse|invertibility]] of [[n-morphisms]] in its highest dimension is always considered strictly. Thus in an ordinary [[category]], we have [[1-morphisms]] which can be [[isomorphism|isomorphisms]]. In a [[2-category]], it is the [[2-morphisms]] for which, when it comes to invertibility, we always ask for a strict inverse, whereas for [[1-morphisms]] we typically ask only for an [[equivalence]]. These 2-morphisms which admit an inverse are known as _2-isomorphisms_.
\section{Definition}
\begin{defn} Let $\mathcal{A}$ be a [[2-category]]. A _2-isomorphism_ in $\mathcal{A}$ is a 2-arrow $\phi : f \rightarrow f'$ of $\mathcal{A}$ which admits a (strict) inverse, that is to say, there is a 2-arrow $\phi^{-1}: f' \rightarrow f$ of $\mathcal{A}$ such that $\phi^{-1} \circ \phi = id(f)$ and $\phi \circ \phi^{-1} = id\left(f' \right)$. \end{defn}
\section{Related concepts}
A 2-category in which every 2-arrow is a 2-isomorphism is known as a [[(2,1)-category]].
[[!redirects 2-isomorphisms]]
|
2-Lawvere theory | https://ncatlab.org/nlab/source/2-Lawvere+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of _2-Lawvere theory_ or _Lawvere 2-theory_ is the [[categorification]] of [[Lawvere theory]].
See also at _[[doctrine]]_.
## Related concepts
* [[algebraic theory]] / [[Lawvere theory]] / **2-Lawvere theory** [[(β,1)-algebraic theory]]
*[[enriched Lawvere theory]]
* [[monad]] / [[2-monad]] / [[(β,1)-monad]]
* [[operad]] / [[(β,1)-operad]]
## References
* [[Noson Yanofsky]], _The syntax of coherence_, Cahiers de Topologie et GΓ©omΓ©trie DiffΓ©rentielle CatΓ©goriques, 2000. [Cahiers pdf](http://www.numdam.org/article/CTGDC_2000__41_4_255_0.pdf), [arXiv](https://arxiv.org/abs/math/9910006)
* [[John Gray]], _2-algebraic theories and triples_, in Colloques sur lβalgebre des categories. Amiens 1973. Resumes des conferences vol. 14, pp. 178β180.
* [[John Bourke]], [[Nick Gurski]], _The Gray tensor product via factorisation_, Applied Categorical Structures, Volume 25, Issue 4, pp 603-624, (2017). [ACS](https://link.springer.com/article/10.1007/s10485-016-9467-6), [arXiv](https://arxiv.org/abs/1508.07789)
* [[Jonathan Cohen]], _Coherence for rewriting 2-theories_, PhD thesis, Australian National University, (2009). [arXiv](https://arxiv.org/abs/0904.0125)
* [[John Power]], [[Stephen Lack]], _Lawvere 2-theories_, 2007. ([slides](http://www.mat.uc.pt/~categ/ct2007/slides/lack.pdf))
* [[Noson Yanofsky]], _Coherence, Homotopy and 2-Theories_, K-Theory 23: Pgs 203 - 235. (2001) ([arXiv:math.CT/0007033](https://arxiv.org/abs/math/0007033))
[[!redirects 2-Lawvere theories]]
[[!redirects Lawvere 2-theory]]
[[!redirects Lawvere 2-theories]]
[[!redirects 2-algebraic theory]]
[[!redirects 2-algebraic theories]] |
2-limit | https://ncatlab.org/nlab/source/2-limit |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $\infty$-Limits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
----
#### 2-Category theory
+-- {: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
A **2-limit** is the type of [[limit]] that is appropriate in a (weak) [[2-category]]. (Since general 2-categories are often called _[[bicategories]]_, 2-limits are often called _[[bilimits]]_.)
There are three notable changes when passing from ordinary 1-limits to 2-limits:
1. In order to satisfy the [[principle of equivalence]], the "cones" in a 2-limit are required to commute only up to [[2-morphism|2-isomorphism]].
2. The [[universal property]] of the limit is expressed by an [[equivalence of categories]] rather than a [[bijection]] of [[sets]]. This means that
1. every other cone over the diagram that commutes up to isomorphism factors through the limit, up to isomorphism, and
2. every transformation _between_ cones also factors through a 2-cell in the limit. We will give some examples below.
3. Since 2-categories are [[enriched category|enriched]] over [[Cat]] (this is precise in the [[strict 2-category|strict]] case, and [[bicategory|weakly]] true otherwise), $Cat$-[[weighted limit]]s become important. This means that both the diagrams we take limits of and the shape of "cones" that limits represent can involve $2$-cells as well as $1$-cells.
## Definition
Let $K$ and $D$ be [[2-categories]], and $J\colon D\to Cat$ and $F\colon D\to K$ be [[2-functors]]. A **$J$-weighted (2-)limit of $F$** is an object $L\in K$ equipped with a [[pseudonatural equivalence]]
$$ K(X,L) \simeq [D,Cat](J,K(X,F-)). $$
where $[D,Cat]$ denotes the 2-category of [[2-functors]] $D\to Cat$, [[pseudonatural transformations]] between them, and [[modifications]] between those.
A 2-limit in the [[opposite 2-category]] $K^{op}$ is called a **2-colimit** in $K$. Everything below applies dually to 2-colimits, the higher analogues of [[colimits]]. (But somebody might want to make a separate page that gives appropriate examples of these.)
### Strictness and terminology
{#Terminology}
If $K$ and $D$ are [[strict 2-categories]], $J$ and $F$ are [[strict 2-functors]], and if we replace this pseudonatural equivalence by a (strictly 2-natural) isomorphism *and* the 2-category $[D,Cat]$ by the 2-category $[D,Cat]_{strict}$ of strict 2-functors and strict 2-natural transformations, then we obtain the definition of a **[[strict 2-limit]]**. This is precisely a Cat-weighted limit in the sense of ordinary [[enriched category]] theory. See [[strict 2-limit]] for details.
On the other hand, if $K$, $D$, $J$, and $F$ are strict as above, and we replace the equivalence by an isomorphism but keep the weak meaning of $[D,Cat]$, then we obtain the notion of a **strict pseudolimit**. Strict pseudolimits are, in particular, 2-limits, whereas strict 2-limits are not always (although some, such as [[PIE-limits]] and [[flexible limits]], are). In a strict 2-category, these types of strict limits are often technically useful in constructing the "up-to-isomorphism" 2-limits we consider here.
When we know we are working in a (weak) 2-category, the only type of limit that makes sense is a (non-strict) 2-limit. Therefore, we usually call these simply "limits." To emphasize the distinction with the strict 2-limits in a strict 2-category, the "up-to-isomorphism" 2-limits were historically often called _bilimits_ (by analogy with [[bicategory]] for "weak 2-category"). However, this terminology is somewhat unfortunate, not only because it doesn't generalize well to $n$, but because it leads to words like "biproduct," which also has the [[biproduct|completely unrelated meaning]] of an object that is both a product and a coproduct (which is common in [[additive category|additive categories]]).
Unfortunately, we probably shouldn't use "weak limit" to emphasize the "up-to-isomorphism" nature of these limits, because that also has the [[weak limit|completely unrelated meaning]] of an object in a 1-category satisfying the existence, but not the uniqueness property of an ordinary limit.
## Examples
### 2-limits over diagrams of special shape
Any ordinary type of limit can be "2-ified" by boosting its ordinary universal property up to a 2-categorical one. In the following examples we work in a 2-category $K$.
* A **[[terminal object]]** in $K$ is an object 1 such that $K(X,1)$ is equivalent to the [[terminal category]] for any object $X$. This means that for any $X$ there is a morphism $X\to 1$ and for any two morphisms $f,g:X\to 1$ there is a unique morphism $f\to g$, and this morphism is an isomorphism.
* A **[[product]]** of two objects $A,B$ in $K$ is an object $A\times B$ together with a natural equivalence of categories $K(X,A\times B) \simeq K(X,A)\times K(X,B)$. This means that we have projections $p:A\times B\to A$ and $q:A\times B\to B$ such that (1) for any $f:X\to A$ and $g:X\to B$, there exists an $h:X\to A\times B$ and isomorphisms $p h\cong f$ and $q h\cong g$, and (2) for any $h,k:X\to A\times B$ and 2-cells $\alpha:p h \to p k$ and $\beta: q h \to q k$, there exists a unique $\gamma:h \to k$ such that $p \gamma = \alpha$ and $q \gamma = \beta$.
* A **[[pullback]]** of a [[co-span]] $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ consists of an object $A\times_C B$ and projections $p:A\times_C B\to A$ and $q:A\times_C B\to B$ together with an isomorphism $\phi:f p \cong g q$, such that (1) for any $m:X\to A$ and $n:X\to B$ with an isomorphism $\psi:f m \cong g n$, there exists an $h:X\to A\times_C B$ and isomorphisms $\alpha:p h \cong m$ and $\beta:q h \cong n$ such that $g\beta . \phi h . f \alpha^{-1} = \psi$, and (2) given any two morphisms $h,k:X\to A\times_C B$ and 2-cells $\mu:p h \to p k$ and $\nu:q h \to q k$ such that $f \mu = g \nu$ (modulo the given isomorphism $f p \cong g q$), i.e., $\phi k . f\mu = g\nu . \phi h$, there exists a unique 2-cell $\gamma:h\to k$ such that $p \gamma = \mu$ and $q \gamma = \nu$. This is sometimes called the _pseudo-pullback_ but that term more properly refers to a particular [[strict 2-limit]].
* An **[[equalizer]]** of $f,g:A\to B$ consists of an object $E$ and a morphism $e:E\to A$ together with an isomorphism $f e \cong g e$, which is universal in a sense the reader should now be able to write down. This is sometimes called the _pseudo-equalizer_ but that term more properly refers to a particular [[strict 2-limit]]. Note that frequently, such as in the construction of all limits from basic ones, equalizers need to be replaced by [[descent object]]s.
There are also various important types of 2-limits that involve 2-cells in the diagram shape or in the weight, and hence are not just "boosted-up" versions of 1-limits.
* The **[[comma object]]** of a cospan $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ is a universal object $(f/g)$ and projections $p:(f/g)\to A$ and $q:(f/g)\to B$ together with a transformation (not an isomorphism) $f p \to g q$. In [[Cat]], [[comma objects]] are [[comma category|comma categories]]. Comma objects are sometimes called _lax pullbacks_ but that term more properly refers to the lax version of a pullback; see "lax limits" below.
* The **[[inserter]]** of a pair of parallel arrows $f,g:A \;\rightrightarrows\; B$ is a universal object $I$ equipped with a map $i:I\to A$ and a 2-cell $f i \to g i$.
* The **[[equifier]]** of a pair of parallel 2-cells $\alpha,\beta: f\to g: A\to B$ is a universal object $E$ equipped with a map $e:E\to A$ such that $\alpha e = \beta e$.
* The **[[inverter]]** of a 2-cell $\alpha:f\to g:A\to B$ is a universal object $V$ with a map $v:V\to A$ such that $\alpha v$ is invertible.
* The **[[power]]** of an object $A$ by a category $C$ is a universal object $A^C$ equipped with a functor $C\to K(A^C,A)$. Of particular importance is the case when $C$ is the [[walking arrow]] $\mathbf{2}$.
### Finite 2-Limits
A 2-limit is called **finite** if its diagram shape and its weight are both "finitely presentable" in a suitable sense (defined in terms of [[computads]]; see [Street's article](#StreetLimitsIndexed) _Limits indexed by category-valued 2-functors_ ). Pullbacks, comma objects, inserters, equifiers, and so on are all finite limits, as are powers by any finitely presented category. All finite limits can be constructed from pullbacks, a terminal object, and powers with $\mathbf{2}$.
### $(2,1)$-limits
{#(2,1)limit}
If the ambient [[2-category]] is in fact a [[(2,1)-category]] in that all [[2-morphism]]s are invertible then there is a rich set of tools available for handling the 2-limits in this context. We may say **$(2,1)$-limits** and **$(2,1)$-colimits** in this case.
These are then a special case of the more general [[(β,1)-limit]]s and [[(β,1)-colimit]]s in a [[(β,1)-category]]. A [[(2,1)-category]] is a special case of an [[(β,1)-category]].
Traditionally, [[(β,1)-limit]]s are best known in terms of the presentation of $(\infty,1)$-categories by [[categories with weak equivalences]] in general and [[model categories]] in particular. (2,1)-limits can often also be viewed in this way. The corresponding presentation of the $(\infty,1)$-limits / $(2,1)$-limits is called **[[homotopy limit]]s** and **[[homotopy colimit]]s**.
For instance 2-limits in the [[(2,1)-category]] [[Grpd]] of [[groupoid]]s, [[functor]]s and (necessarily) [[natural isomorphism]]s. Are equivalently computed as [[homotopy limit]]s in the [[model structure on simplicial sets]] $sSet_{Quillen}$ of diagrams of [[1-truncated]] [[Kan complex]]es. (The equivalence of homotopy limits with $(\infty,1)$-limits is discussed at [[(β,1)-limit]]).
Or for instance, more generally, the 2-limits in any [[(2,1)-sheaf]](=[[stack]]) [[(2,1)-topos]] may be computed as [[homotopy limit]]s in a [[model structure on simplicial presheaves]] over the given [[(2,1)-site]] of diagrams of [[1-truncated]] [[simplicial presheaves]]. This includes as examples [[big topos|big (2,1)-toposes]] such as those over the large sites [[Top]] or [[SmoothMfd]] where computations with [[topological groupoid]]s/[[topological stack]]s, [[Lie groupoid]]s/[[differentiable stack]]s etc. take place.
### Lax limits
{#lax}
A **lax limit** can be defined like a 2-limit, except that the triangles in the definition of a cone are required only to commute up to a specified _transformation_, not necessarily an isomorphism. In other words, in place of the 2-category $[D,Cat]$ we use the 2-category $[D,Cat]_l$ whose morphisms are [[lax natural transformations]]; thus the lax limit $L$ of a diagram $F$ weighted by $J$ is equipped with a universal lax natural transformation $J\to K(L,F-)$.
This may look like a different concept, but in fact, for any weight $J$ there is another weight $Q_l(J)$ such that lax $J$-weighted limits are the same as $Q_l(J)$-weighted 2-limits. Here $Q_l$ is the [[lax morphism classifier]] for 2-functors. Therefore, lax limits are really a special case of 2-limits. Similarly, **oplax limits**, in which we use oplax natural transformations, are also a special case of 2-limits.
There is a further simplification of lax limits in the case of "conical" lax limits where the weight $J=\Delta 1$ is constant at the [[terminal category]]. In this case, it is easy to check that a lax natural transformation $\Delta 1 \to K(X,F-)$ is the same as a lax natural transformation $\Delta X \to F$. Thus, a conical lax limit of $F$ is a representing object for such lax transformations.
Here are some examples.
* Lax terminal objects and lax products are the same as ordinary ones, since there are no commutativity conditions on the cones.
* The **lax limit of an arrow** $f:A\to B$ is a universal object $L$ equipped with projections $p:L\to A$ and $q:L\to B$ and a 2-cell $f p \to q$. Note that this is equivalent to a comma object $(f/1_B)$.
* The **lax pullback** of a cospan $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ is a universal object $P$ equipped with projections $p:P\to A$, $q:P\to B$, $r:P\to C$, and 2-cells $f p \to r$ and $g q \to r$.
Note that lax pullbacks are _not_ the same as [[comma objects]]. In general comma objects are much more useful, but there are 2-categories that admit all lax limits but do not admit comma objects, so using "lax pullback" to mean "comma object" can be misleading.
A **lax colimit** in $K$ is, of course, a lax limit in $K^{op}$. Thus, it is a representing object for lax natural transformations $J \to K(F-,L)$. There is a subtlety here, however, because in the case $J=\Delta 1$, a lax natural transformation $\Delta 1 \to K(F-,L)$ is the same as an *oplax* natural transformation $F \to \Delta L$. Thus, it is easy to mistakenly say "lax colimit" when one really means "oplax colimit" and vice versa.
+-- {: .un_remark}
###### Remark
With this in mind, one might be tempted to switch the meanings of "lax colimit" and "oplax colimit", but there is a good reason not to. Recall that a lax $J$-weighted limit is the same as an ordinary $Q_l(J)$-weighted limit. Standard terminology in enriched category theory is that a $W$-weighted colimit in an enriched category $K$ is the same as a $W$-weighted limit in $K^{op}$, and indeed in that generality there is no other option. Thus, a lax $J$-weighted colimit in $K$ should be an ordinary $Q_l(J)$-weighted colimit in $K$, hence a $Q_l(J)$-weighted limit in $K^{op}$, and thus a lax $J$-weighted limit in $K^{op}$.
=--
Here are some examples of lax and oplax colimits:
* A [[Kleisli object]] is a lax colimit of a [[monad]], regarded as a diagram in a 2-category.
* The [[collage]] of a [[profunctor]] is its lax colimit, regarded as a diagram in the 2-category [[Prof]].
* When $C$ is a category, the [[Grothendieck construction]] of a functor $C\to Cat$ is the same as its *oplax* colimit; see [here](http://ncatlab.org/nlab/show/Grothendieck+construction#AsALaxColimit).
### 2-Colimits in $Cat$
{#2ColimitsInCat}
As shown [here](http://ncatlab.org/nlab/show/Grothendieck+construction#AsALaxColimit), if $C$ is an ordinary category and $F \colon C \to Cat$ is a [[pseudofunctor]], then the [[oplax colimit]] of $F$ is given by the [[Grothendieck construction]] $\int F$ --- and its [[pseudo-colimit]] is given by [[localization|formally inverting]] the [[cartesian morphism|opcartesian]] morphisms in $\int F$. This yields a construction of certain pseudo 2-colimits in $Cat$.
Moreover, a similar result holds more generally when $C$ is a [[bicategory]]. In this case, $\int F$ is also a bicategory: a 2-cell from $(m \colon c \to d, f \colon m_*x \to y)$ to $(n \colon c \to d, g \colon n_*x \to y)$ is given by a 2-cell $\mu \colon m \Rightarrow n$ in $C$ such that $\mu_* x$ is a morphism $f \to g$ over $y$.
Let $\pi_*$ denote the functor that sends a bicategory $K$ to the category whose objects are those of $K$ and whose hom-sets are the [[connected category|connected components]] of the hom-categories of $K$; let also $d_*$ denote the functor that sends a category $X$ to the corresponding locally discrete bicategory. Then there is an equivalence of categories
$$ [K, d_* X] \simeq [\pi_* K, X] $$
It is straightforward to check that the first of the above facts extends to the bicategorical case:
$$ Lax(F, \Delta X) \simeq [{\textstyle \int} F, d_* X] $$
as does the fact that a lax transformation on the left is pseudo if and only if the corresponding functor on the right inverts the opcartesian morphisms in $\int F$. It is almost trivial that the adjunction $\pi_* \dashv d_*$ holds when restricted to the functor $[-, -]_{S^{-1}}$ that takes two categories or bicategories to the full subcategory of functors that invert the class $S$ of morphisms. Taking $S$ to be the opcartesian morphisms in $\int F$, then, we have
$$
Ps(F, \Delta X) \simeq [{\textstyle \int} F, d_* X]_{S^{-1}}
\simeq [\pi_* {\textstyle \int} F, X]_{S^{-1}}
\simeq [(\pi_* {\textstyle \int} F)[S^{-1}], X]
$$
Hence the pseudo colimit of $F$ is got by taking its bicategory of elements, applying the 'local $\pi_0$' functor, and then inverting the (images of the) opcartesian morphisms as usual.
## Related concepts
* [[limit]]
* **2-limit**
* [[strict 2-limit]]
* [[marked 2-limit]]
* [[double limit]]
* [[2-limits and 2-colimits in 2-categories of 2-algebras]]
* [[(β,1)-limit]]
* [[homotopy limit]]
* [[lax (β,1)-colimit]]
* [[quasi-limit]]
* [[coherence for bicategories with finite limits]]
* [[representable 2-category]]
## References
* {#Street74} [[Ross Street]], *Elementary cosmoi I* (Β§6) in *Category Seminar*, Lecture Notes in Mathematics **420**, Springer (1974) [[doi:10.1007/BFb0063103](https://doi.org/10.1007/BFb0063103)]
* {#StreetLimitsIndexed} [[Ross Street]], _Limits indexed by category-valued 2-functors_ Journal of Pure and Applied Algebra **8**, Issue 2 (1976) pp 149-181. doi:[10.1016/0022-4049(76)90013-X](https://doi.org/10.1016/0022-4049(76%2990013-X)
* [[Max Kelly]], _Elementary observations on 2-categorical limits_, Bulletin of the Australian Mathematical Society (1989), 39: 301-317, doi:[10.1017/S0004972700002781](https://doi.org/10.1017/S0004972700002781)
* [[Ross Street]], _Fibrations in Bicategories_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume **21** (1980) no. 2, pp 111-160. [Numdam](http://www.numdam.org/item?id=CTGDC_1980__21_2_111_0 ) and correction, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume **28** (1987) no. 1, pp 53-56 [Numdam](http://www.numdam.org/item?id=CTGDC_1987__28_1_53_0 )
Section 6, page 37 in
* [[Steve Lack]], _A 2-categories companion_. In: Baez J., May J. (eds) Towards Higher Categories. The IMA Volumes in Mathematics and its Applications, vol **152** 2010 Springer, New York, NY. doi:[10.1007/978-1-4419-1524-5_4](https://doi.org/10.1007/978-1-4419-1524-5_4), arXiv:[math.CT/0702535](http://arxiv.org/abs/math.CT/0702535).
* G. J. Bird, [[Max Kelly]], [[John Power]], [[Ross Street]], _Flexible limits for 2-categories_, Journal of Pure and Applied Algebra **61** Issue 1 (1989) pp 1-27. doi:[10.1016/0022-4049(89)90065-0](http://dx.doi.org/10.1016/0022-4049(89%2990065-0)
Chapters 3,4,5 in
* [[Thomas Fiore]], _Pseudo Limits, Biadjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory_, Mem. Amer. Math. Soc. **182** (2006), no. 860 ([arXiv:math/0408298](http://arxiv.org/abs/math/0408298)) ([AMS page](http://bookstore.ams.org/memo-182-860), [Google Books](https://books.google.com.au/books?id=y_DUCQAAQBAJ))
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[[!redirects (2,1)-colimit]]
[[!redirects (2,1)-colimits]]
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2-limits and 2-colimits in 2-categories of 2-algebras | https://ncatlab.org/nlab/source/2-limits+and+2-colimits+in+2-categories+of+2-algebras | +-- {: .rightHandSide}
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###Context###
#### Limits and colimits
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#### Higher algebra
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#Contents#
* tic
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## Introduction
In moving from one dimension to two dimensions, there are a proliferation of concepts, with a choice of weakness for each one. For one, there is the notion of (strict) [[2-category]] and [[bicategory]], (strict) [[2-monad]] and [[pseudomonad]], and (strict) 2-algebras and pseudoalgebras. Therefore, while it is clear that "2-algebras for a 2-monad on a 2-category inherit 2-limits (and certain 2-colimits) from the base", it can be tricky to recall which level of strictness is appropriate in each case. On this page, we list the various limit and colimit creation properties for two-dimensional algebras.
## Limits
We shall use the conventional terminology of "2-" for strict concepts and "pseudo" for weak concepts to make it easier to compare with the references.
* The 2-category of 2-algebras and strict morphisms for a 2-monad on a 2-category inherits all 2-limits (this follows from $Cat$-enriched category theory).
* The 2-category of 2-algebras and pseudo morphisms for a 2-monad on a 2-category inherits all [[PIE 2-limits]] (Β§3 of [BKP89](#BKP89).
* The 2-category of 2-algebras and lax morphisms for a 2-monad on a 2-category inherits all [[oplax limits]], limits of strict morphisms, [[equifiers]] and [[inserters]] where one morphism is strict, [[EilenbergβMoore objects]] for [[comonads]], [[products]] and [[powers]] (see [Lack05](#Lack05) and [LS11](#LS11)). The page [[rigged limit]] contains more details.
* The 2-category of 2-algebras and pseudo morphisms for a [[flexible 2-monad]] on a 2-category inherits all [[flexible limits]] (see Remark 7.2 of [BKPS89](#BKPS89)).
* The 2-category of pseudoalgebras and pseudo morphisms for a [[pseudomonad]] on a 2-category inherits all [[bilimits]] (see Theorem 6.3.1.6 of [Osmond21](#Osmond21)).
## Related pages
* [[2-monad]]
* [[colimits in categories of algebras]]
## References
* {#BKP89} R. Blackwell, G. M. Kelly, and A. J. Power, _Two-dimensional monad theory_, Jour. Pure Appl. Algebra 59 (1989), 1--41
* {#BKPS89} G. J. Bird, [[Max Kelly]], [[John Power]], [[Ross Street]], _Flexible limits for 2-categories_, Journal of Pure and Applied Algebra **61** Issue 1 (1989) pp 1-27. doi:[10.1016/0022-4049(89)90065-0](http://dx.doi.org/10.1016/0022-4049(89%2990065-0)
* {#Lack05} [[Stephen Lack]], _Limits for lax morphisms_, *Appl. Categ. Structures*,
13(3):189--203, 2005
* {#LS11} [[Stephen Lack]] and [[Mike Shulman]], _Enhanced 2-categories and limits for lax morphisms_, [arXiv](http://arxiv.org/abs/1104.2111).
* {#Osmond21} Axel Osmond, _A categorical study of spectral dualities_, PhD thesis, UniversitΓ© Paris CitΓ©, 2021. |
2-line | https://ncatlab.org/nlab/source/2-line |
#Contents#
* table of contents
{:toc}
## Idea
Given a [[2-ring]] $R$, a _2-line_ over $R$ is an $R$-[[2-module]] which is [[equivalence|equivalent]] to $R$ as a 2-module.
## Related concepts
* [[line]]
* [[2-line bundle]]
[[!redirects 2-lines]] |
2-logic | https://ncatlab.org/nlab/source/2-logic |
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#### Type theory
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=--
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#Contents#
* table of contents
{:toc}
## Idea
As a [[category]] has an [[internal logic]] so a [[2-category]] has internal _2-logic_
## Related concepts
* [[type theory]], [[logic]]
* [[2-type theory]], **2-logic**
* [[(β,1)-type theory]], [[(β,1)-logic]]
## References
* [[Mike Shulman]], _[[michaelshulman:2-categorical logic]]_
* [[michaelshulman:internal logic of a 2-category]]
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2-monad | https://ncatlab.org/nlab/source/2-monad |
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### Context
#### Categorical algebra
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#### Higher algebra
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=--
=--
#Contents#
* table of contents
{:toc}
## Definition
### 2-Monad
A **2-monad** is a [[monad]] on a [[2-category]], or more generally a monad _in_ a [[3-category]]. This concept manifests at varying levels of strictness:
* For a _strict 2-monad_ (which classically is called a simply a "2-monad"), the 2-category $K$ is a [[strict 2-category]], the functor $T:K\to K$ is a [[strict 2-functor]], and the transformations $\mu$ and $\eta$ are [[strict 2-natural transformation]]s and satisfy their laws strictly. This is the same as a $Cat$-[[enriched category theory|enriched]] monad. Strict 2-monads live naturally in [[strict 3-category|strict 3-categories]].
* For a fully _weak 2-monad_, $K$ is a weak 2-category (such as a [[bicategory]]), $T$ is a weak (aka pseudo) 2-functor, and $\mu$ and $\eta$ are pseudo natural transformations that satisfy their laws up to specified isomorphisms satisfying coherence conditions. Weak 2-monads live naturally in fully weak 3-categories (or [[tricategory|tricategories]])
* In between we have various notions that are sometimes called _pseudomonads_. For instance, we could require $K$ to be a strict 2-category and $T$ a strict 2-functor, but $\mu$ and $\eta$ only pseudo natural. This sort of pseudomonad lives naturally in a [[Gray-category]].
### Algebras and pseudoalgebras over a 2-monad
One can consider various 2-categories of algebras/modules for a 2-monad, depending on whether the algebras satisfy their laws strictly or weakly, and whether the morphisms commute with the algebra structure strictly or weakly.
At the strictest level, for a strict 2-monad $T$ we can consider the $Cat$-enriched [[Eilenberg-Moore category]], which consists of strict algebras (see [[algebra over a monad]]), strict morphisms, and strict transformations between these. In 2-categorical literature, it is usually denoted $T Alg_s$. Many common types of structure on categories are specified by strict algebras for a strict 2-monad, but usually the strict morphisms are too strict.
There are three types of weak morphism: _pseudo_ (which preserve the structure up to a specified coherent isomorphism), _lax_ (which preserve it up to a noninvertible transformation) and _colax_ or _oplax_ (for which the transformation goes the other direction). See [[lax morphism]] for further discussion. With strict algebras and these various types of morphism, we obtain 2-categories $T Alg$ (the pseudo case), $T Alg_l$, and $T Alg_c$ for lax and colax respectively. One can also define an [[F-category]] of strict and lax morphisms together (or strict and pseudo, or pseudo and lax), and a [[double category]] which includes both lax and colax morphisms.
For example, ordinary (non-strict) [[monoidal category|monoidal categories]] are the _strict_ algebras for a strict 2-monad $T_{MC}$ on $Cat$, but usually we care about pseudo, lax, and oplax [[monoidal functor]]s rather than strict ones. _Strict_ monoidal categories are the strict algebras for a _different_ strict 2-monad $T_{StrMC}$ on $Cat$.
We can, however, also consider *pseudo* algebras for a 2-monad; see [[pseudoalgebra for a 2-monad]]. If the 2-monad is not strict, then this is usually the only sensible course. Pseudoalgebras for a strict 2-monad $T$ usually give an "unbiased" weaker notion of the structure specified by $T$. For example, the pseudoalgebras for $T_{StrMC}$ are, not ordinary monoidal categories, but [[bias|unbiased]] monoidal categories. (It is true, however, that the 2-category $Ps T_{StrMC} Alg$ of unbiased monoidal categories and strong monoidal functors is strictly 2-equivalent, i.e. Cat-enriched equivalent, to the 2-category of ordinary biased monoidal categories and strong monoidal functors.)
There are also 2-monads that specify [[property-like structure]]. For instance, there is a 2-monad whose algebras are categories with finite products. Actually, its algebras are categories equipped with _specified_ finite products, the strict morphisms of these algebras preserve these specified finite products on the nose, and the pseudo morphisms preserve them in the usual sense of "preserving finite products." In this case, _every_ functor between algebras is an oplax morphism, since there is always a canonical comparison map $F(A\times B) \to F(A)\times F(B)$.
For limit and colimit properties of algebras, see [[2-limits and 2-colimits in 2-categories of 2-algebras]].
## Properties
### Relation to doctrines
2-monads (particularly on [[Cat]]) are also sometimes called _[[doctrines]]_, with the intuition in mind that they are an "algebraic theory" of structure on a category just as a monad (on $Set$) is an [[algebraic theory]] of structure on a set. However, this use of terminology is arguably at variance with the original intuitive meaning of "doctrine."
### Relation of strict 2-monads to 1-monads
A strict 2-monad $T$ has an underlying [[monad]] $T_0$, such that strict $T$-algebras and strict $T$-morphisms are the same as $T_0$-algebras and $T_0$-morphisms. (This is a special case of the the general theory of underlying ordinary categories for [[enriched categories]].) Moreover, if a strict 2-category $A$ admits [[powers]] or [[copowers]] with the [[interval category]], then any monad on its underlying ordinary category $A_0$ has at most one enrichment to a strict 2-monad. Thus, in this case "being a 2-monad" is a mere [[property]] of a monad; see the "unicity" paper of John Power below.
## Related concepts
* [[algebraic theory]] / [[Lawvere theory]] / [[(β,1)-algebraic theory]]
* [[monad]]
* **2-monad**/ [[doctrine]]
* [[Kleisli 2-category]]
* [[(β,1)-monad]]
* [[operad]] / [[(β,1)-operad]]
## References
* R. Blackwell, G. M. Kelly, and A. J. Power, _Two-dimensional monad theory_, Jour. Pure Appl. Algebra 59 (1989), 1--41
* F. Marmolejo, _Doctrines whose structure forms a fully faithful adjoint string_, Theory and Applications of Categories 3 (1997), 23--44. ([TAC](http://www.tac.mta.ca/tac/volumes/1997/n2/3-02abs.html))
* [[Stephen Lack]], _A coherent approach to pseudomonads_, Advances in Mathematics 152 (2000), 179--202. ([doi:10.1006/aima.1999.1881](https://doi.org/10.1006/aima.1999.1881))
* [[Max Kelly]] and [[Steve Lack]], _On property-like structures_, Theory and Applications of Categories 3 (1997) 213--250. ([TAC](http://www.tac.mta.ca/tac/volumes/1997/n9/3-09abs.html))
* [[Stephen Lack]], _Homotopy-theoretic aspects of 2-monads_ ([arXiv:math/0607646](http://arxiv.org/abs/math/0607646))
* [[Stephen Lack]], _Codescent objects and coherence_, JPAA 175 (2002), 223--241.
* I. J. Le Creurer, F. Marmolejo, E. M. Vitale, _Beck's theorem for pseudo-monads_, J. Pure Appl. Algebra 173 (2002), no. 3, 293--313.
* [[John Power]], *Unicity of enrichment over Cat or Gpd*, Appl. Categ. Struct. 2009, 1--7.
On the relation to [[symmetric operads]]:
* {#Kock12} [[Joachim Kock]], _Data types with symmetries and polynomial functors over groupoids_, 28th Conference on the Mathematical Foundations of Programming Semantics (Bath, June 2012); in Electronic Notes in Theoretical Computer Science. ([arXiv:1210.0828](http://arxiv.org/abs/1210.0828))
* {#Weber14} [[Mark Weber]], _Operads as polynomial 2-monads_ ([arXiv:1412.7599](http://arxiv.org/abs/1412.7599))
On the [[extension system]]-perspective (as for [[monads in computer science]]) generalized to pseudomonads:
Generalizing the perspective of [[extension systems]] to [[pseudomonads]]:
* [[Francisco Marmolejo]], [[Richard J. Wood]], *No-iteration pseudomonads*, [[TAC]] **28** 14 (2013) 371-402 [[tac:28-14](http://www.tac.mta.ca/tac/volumes/28/14/28-14abs.html)]
and relating to [[Kan extensions]]:
* [[Francisco Marmolejo]], [[Richard J. Wood]], *Kan extensions and lax idempotent pseudomonads*, [[TAC]] **26** 1 (2012) 1-29 [[26-01](http://www.tac.mta.ca/tac/volumes/26/1/26-01abs.html)]
[[!redirects strict 2-monad]]
[[!redirects weak 2-monad]]
[[!redirects pseudo monad]]
[[!redirects pseudo-monad]]
[[!redirects pseudomonad]]
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[[!redirects 2-comonads]]
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2-morphism | https://ncatlab.org/nlab/source/2-morphism |
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* [[object]]
* [[morphism]]
* **2-morphism**
* [[3-morphism]]
* [[k-morphism]]
***
#Contents#
* automatic table of contents goes here
{:toc}
## Definition
A **2-morphism** in an [[n-category]] is a [[k-morphism]] for $k = 2$: it is a higher morphism between ordinary 1-[[morphism]]s.
So in the hierarchy of $n$-categories, the first step where 2-morphisms appear is in a [[2-category]]. This includes cases such as [[bicategory]], [[2-groupoid]] or [[double category]].
## Shapes
There are different [[geometric shapes for higher structures]]: [[globe]]s, [[simplex|simplices]], [[cube]]s, etc. Accordingly, 2-morphisms may appear in different guises:
A **globular** $2$-morphism looks like this:
$$
a\mathrlap{\begin{matrix}\begin{svg}
<svg width="76" height="37" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="79929">
<g>
<title>Layer 1</title>
<path marker-end="url(#se_marker_end_svg_79929_2)" id="svg_79929_2" d="m2,18.511721c31.272522,-14.782231 42.439789,-16.425501 71.625,-1.25" stroke="#000000" fill="none"/>
<path id="svg_79929_13" marker-end="url(#se_marker_end_svg_79929_2)" d="m2,24.511721c33.286949,14.464769 40.259941,16.4624 71.500008,1.75" stroke="#000000" fill="none"/>
</g>
<defs>
<marker refY="50" refX="50" markerHeight="5" markerWidth="5" viewBox="0 0 100 100" orient="auto" markerUnits="strokeWidth" id="se_marker_end_svg_79929_2">
<path stroke-width="10" stroke="#000000" fill="#000000" d="m100,50l-100,40l30,-40l-30,-40l100,40z" id="svg_79929_3"/>
</marker>
</defs>
</svg>
\end{svg}\includegraphics[width=56]{curvearrows}\end{matrix}}{\phantom{a}\space{0}{0}{13}\Downarrow\space{0}{0}{13}\phantom{a}} b
$$
A **simplicial** $2$-morphism looks like this:
$$
\begin{matrix}
&& b
\\
& \nearrow &\Downarrow& \searrow
\\
a &&\to&& c
\end{matrix}
$$
A **cubical** $2$-morphism looks like this:
$$
\begin{matrix}
& & b \\
& \nearrow & & \searrow \\
a & & \Downarrow & & d \\
& \searrow & & \nearrow \\
& & c
\end{matrix}
$$
Of course, using [[identity morphisms]] and [[composition]], we can turn one into the other; which is more fundamental depends on which shapes you prefer.
## Examples
* In the [[2-category]] [[Cat]], 2-morphisms are [[natural transformation]]s between functors.
* In a [[path 2-groupoid]] 2-morphisms are certain surfaces or images of surfaces in a space, going between paths in that space.
[[!redirects 2-morphism]]
[[!redirects 2-morphisms]]
[[!redirects 2-cell]]
[[!redirects 2-cells]] |
2-morphism - SVG | https://ncatlab.org/nlab/source/2-morphism+-+SVG | <svg width="100" height="75" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="47584">
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</marker>
</defs>
</svg>
category: svg
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2-morphisms > history | https://ncatlab.org/nlab/source/2-morphisms+%3E+history | < [[2-morphisms]]
[[!redirects 2-morphisms -- history]] |
2-periodic Morava K-theory | https://ncatlab.org/nlab/source/2-periodic+Morava+K-theory |
## Idea
The 2-[[periodic ring spectrum]]-version of [[Morava K-theory]].
Let $C$ be any [[elliptic curve]] over the [[prime field]] $\mathbb{F}_p$. This will have a [[formal group]] of either [[height of a formal group law|height]] $h=1$ ("ordinary curve") or height $h=2$ ("[[supersingular elliptic curve]]"). For any such $C$, there is a [[ring spectrum]] $K_C$ with coefficient ring $\pi_* K_C=\mathbb{F}_p[u,u^{-1}]$, with $u\in \pi_2$, which is [[complex orientable cohomology theory|complex orientable]], whose formal group is the formal group of $C$. This $K_C$ is the "2-periodic Morava $K$-theory" associated to the formal group.
(grapped from [this MO comment](http://mathoverflow.net/a/201436/381) by [[Charles Rezk]]).
[[!redirects periodic Morava K-theory]] |
2-periodic sphere spectrum | https://ncatlab.org/nlab/source/2-periodic+sphere+spectrum |
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#### Cohomology
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#Contents#
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## Idea
The 2-[[periodic ring spectrum]]-version of the [[sphere spectrum]]. It is formed by taking the [[Thom spectrum]] of the [[E2-operad|$E_2$]] map
$$
\mathbb{Z}
\;\simeq\;
\Omega^2 B U(1)
\longrightarrow
\Omega^2 B U
\longrightarrow
B U \times \mathbb{Z}
\,,
$$
where the last map is specified by [[Bott periodicity]] ([Lurie, Rotation Invariance, Remark 3.5.13](#LurieRotation)). It cannot be equipped with an [[E-infinity operad|$E_{\infty}$]] structure.
As a spectrum it is given by the [[direct sum]] ([[wedge sum]])
$\bigoplus_{n \in \mathbb{Z}} \mathbb{S}^{-2n}$
of all even-degree [[suspensions]] of the plain [[sphere spectrum]].
## References
* {#LurieRotation} [[Jacob Lurie]], _Rotation Invariance in Algebraic K-Theory_, ([pdf](https://people.math.harvard.edu/~lurie/papers/Waldhaus.pdf))
* [[Mohammed Abouzaid]], [[Andrew J. Blumberg]], A.2.3 in: _Arnold Conjecture and Morava K-theory_ ([arXiv:2103.01507](https://arxiv.org/abs/2103.01507)).
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2-plectic geometry | https://ncatlab.org/nlab/source/2-plectic+geometry |
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#### Higher symplectic geometry
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[[!include symplectic geometry - contents]]
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#Contents#
* table of contents
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## Idea
_2-Plectic geometry_ is the higher generalization of [[symplectic geometry]], the special case of [[n-plectic geometry]] ([[multisymplectic geometry]]) for $n = 2$. This is the input for [[higher prequantum geometry]] in degree 2.
As symplectic geometry naturally describes [[classical mechanics]] and, via [[geometric quantization]], [[quantum mechanics]], hence 1-dimensional [[quantum field theory]], so 2-plectic geometry naturally describes 2-dimensional [[classical field theory]] and, via its [[higher geometric quantization]], 2-dimensional [[QFT]].
## Examples
* A [[semisimple Lie group]] is canonically a 2-plectic manifold, with the canonical 3-form $\langle -, [-,-]\rangle$ on the [[Lie algebra]] (the canonical [[Lie algebra cohomology|Lie algebra cocycle]]), extended to a [[left invariant differential form]].
* A [[G2-manifold]] is manifold of [[dimension]] 7 characterized by carrying a 2-plectic form.
* A (pre-)[[symplectic groupoid]] is a [[Lie groupoid]] equipped with a (pre-)2-plectic structure.
## Related concepts
[[!include geometric quantization extensions - table]]
[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]
* [[Nambu mechanics]]
## References
### General
Over [[smooth manifolds]], the general setup is discussed in
* [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis (2011) ([arXiv:1106.4068](http://arxiv.org/abs/1106.4068))
and considered in the general context of [[higher differential geometry]]/[[extended prequantum field theory]] in
* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]],
_[[schreiber:Higher geometric prequantum theory]]_,
_[[schreiber:L-β algebras of local observables from higher prequantum bundles]]_
### Application to the string $\sigma$-model
{#ReferencesStringSigmaModel}
Applications to the 2-dimensional [[string]] [[sigma-model]] (see at [[string Lie 2-algebra]]) are discussed in
* {#BaezRogers10} [[John Baez]], [[Chris Rogers]], _Categorified Symplectic Geometry and the String Lie 2-Algebra_, Homology Homotopy Appl. Volume 12, Number 1 (2010), 221-236. ([arXiv:0901.4721](http://arxiv.org/abs/0901.4721),[euclid](https://projecteuclid.org/euclid.hha/1296223828)).
* [[John Baez]], [[Alexander Hoffnung]], [[Chris Rogers]], _Categorified Symplectic Geometry and the Classical String_ ([arXiv:0808.0246](http://arxiv.org/abs/0808.0246/))
A survey of some (potential) applications of 2-plectic geometry in [[string theory]] and [[M2-brane]] models is in section 2 of
* [[Christian Saemann]], [[Richard Szabo]], _Groupoid quantization of loop spaces_ ([arXiv:1203.5921](http://arxiv.org/abs/1203.5921))
and in
* [[Christian Saemann]], [[Richard Szabo]], _Quantization of 2-Plectic Manifolds_ ([arXiv:1106.1890](http://arxiv.org/abs/1106.1890))
[[!redirects 2-plectic manifold]]
[[!redirects 2-plectic manifolds]]
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2-plethory | https://ncatlab.org/nlab/source/2-plethory |
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#### Algebra
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#### Combinatorics
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## Idea
A **2-plethory** ([Baez-Moeller-Trimble](#BMT21)) is a categorification of a rig-[[plethory]]. Just as a rig-plethory is defined as a monoid in $(\mathsf{Birig}, \odot)$, a 2-plethory is a [[pseudomonoid]] in $(2\mathsf{Birig}, \odot)$.
The property that polynomials in $\mathbb{N}[x]$ can be composed is a consequence of the fact that $\mathbb{N}[x]$ is the free rig in one generator, and hence represents the identity functor $\mathsf{Rig}\to \mathsf{Rig}$. Similarly, the category of [[Schur functor]]s $\mathsf{Schur}$ can be seen as a 2-rig (in the sense below) representing the identity 2-functor $2\mathsf{Rig}\to 2\mathsf{Rig}$. As a consequence, $\mathsf{Schur}$ has the structure of a 2-plethory. Taking isomorphism classes, one recovers the rig-[[plethory]] structure on the positive [[symmetric function]]s, justifying the term "categorified [[plethysm]]".
## Definitions
Let $k$ be a field of characteristic zero, and let $\mathsf{Rig}$ be the category of [[rig]]s and rig morphisms.
Let $U:\mathsf{Rig} \to \mathsf{Set}$ be the [[forgetful functor]].
A birig is equivalently defined as:
* A rig object in $\mathsf{Rig}^\text{op}$.
* A rig $B$ and a lift of the representable functor $\mathsf{Rig}(B,-):\mathsf{Rig}\to \mathsf{Set}$ to a functor $\Phi_B:\mathsf{Rig}\to \mathsf{Rig}$ such that $U \circ \Phi_B=\mathsf{Rig}(B,-)$. Observe that any two representable 2-functors $\Phi_B$, $\Phi_B'$ can be composed. Denote the resulting object by $B\odot B'$, so that $\Phi_B \circ \Phi_B'=\Phi_{B' \circ B}$.
* A functor $\Phi:\mathsf{Rig}\to \mathsf{Rig}$ that is a right adjoint.
Denote by $(\mathsf{Birig}, \odot)$ the monoidal category of birigs, viewed as a [[full subcategory]] of $\mathsf Rig$.
A rig-[[plethory]] is equivalently defined as:
* A monoid in $(\mathsf{Birig}, \odot)$.
* A birig $B$ such that $\Phi_B$ is a comonad.
The categorified versions of the definitions above are as follows.
The category $2\mathsf{Rig}$ has:
* Objects: [[symmetric monoidal]] $k$-linear categories for which the tensor product is bilinear on hom-spaces and which are [[Cauchy complete]].
* 1-morphisms: [[symmetric monoidal]] [[linear functor]]s.
* 2-morphisms: [[symmetric monoidal]] linear natural transformations.
Let $\mathsf{U}:\mathsf{2Rig} \to \mathsf{Cat}$ be the forgetful [[2-functor]].
A **2-birig** is a 2-rig $\mathsf B$ and a lift of the 2-functor $\mathsf{2Rig}(B,-):\mathsf{2Rig}\to \mathsf{Cat}$ to a 2-functor $\Phi_\mathsf{B}:\mathsf{2Rig}\to \mathsf{2Rig}$ such that $\mathsf U \circ \Phi_\mathsf{B}=\mathsf{2Rig}(\mathsf{B},-)$. Again, any two representable 2-functors $\Phi_\mathsf{B}$, $\Phi_\mathsf{B'}$ can be composed. Denote the resulting object by $\mathsf{B}\odot \mathsf{B'}$, so that $\Phi_\mathsf{B} \circ \Phi_\mathsf{B'}=\Phi_{\mathsf{B'}\odot \mathsf{B}}$.
Denote by $(\mathsf{2Birig}, \odot)$ the [[monoidal 2-category]] of 2-birigs, viewed as a full [[sub-2-category]] of $\mathsf{2Rig}$.
A **2-plethory** can be equivalently defined as
* A [[pseudomonoid]] in $(2\mathsf{Birig}, \odot)$.
* A 2-comonad $\Phi:\mathsf{2Rig} \to \mathsf{2Rig}$ whose underlying 2-functor is a right 2-adjoint.
## Example: Schur functors
For any 2-plethory $\Phi$ with left 2-adjoint $\Psi$, the composition $\mathsf{U}\Phi$ is representable, with representing object $\Psi(\overline{k\mathsf{S}})$, where $\overline{k\mathsf{S}}$ is the [[Cauchy completion]] of the $k$-linearization of the [[permutation groupoid]] $\mathsf{S}$. In turn, the 2-rig $\overline{k\mathsf{S}}$ is equivalent to $\mathsf{Schur}$ made into a 2-rig with the pointwise tensor product.
In particular, the identity $1: \mathsf{2Rig} \to \mathsf{2Rig}$ is a 2-plethory with underlying 2-rig $\overline{k\mathsf{S}}$. The set isomorphism classes $J(\overline{k\mathsf{S}})$ has a commutative monoid structure coming from the decategorifications of $\oplus$ and $\otimes$. Then, $J(\overline{k\mathsf{S}})$ is isomorphic to the monoid of positive [[symmetric functions]] $\Lambda_+$. The 2-plethory structure induces the well-known rig-plethory structure on $\Lambda_+$. In turn, the [[Grothendieck group]] of $\mathsf{Schur}$ is $K(\overline{k\mathsf{S}})=\mathbb{Z}\otimes_\mathbb{N} J(\overline{k\mathsf{S}})$, and the rig-plethory structure on $J(\overline{k\mathsf{S}})$ extends to a ring-plethory structure on $K(\overline{k\mathsf{S}})$.
## Related concepts
* [[biring]]
* [[plethory]]
* [[plethysm]]
* [[Schur functor]]
##References
* {#TW70} D. Tall and [[Gavin Wraith|G. Wraith]], Representable functors and operations on rings, _Proc. London Math. Soc._ **3** (1970), 619--643.
* {#BW05} [[James Borger]], [[Ben Wieland]], Plethystic algebra, _Advances in Mathematics_ **194** (2005), 246--283. ([web](http://wwwmaths.anu.edu.au/~borger/papers/03/paper03.html))
* {#BMT21} [[John Baez]], [[Joe Moeller]], [[Todd Trimble]], Schur functors and categorified plethysm, [arXiv:2106.00190](https://arxiv.org/abs/2106.00190)
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2-polycategory | https://ncatlab.org/nlab/source/2-polycategory | A **2-polycategory** is a [[polycategory]] enriched over [[Cat]]. It can be identified with a [[double polycategory]] with only identity vertical arrows. |
2-poset | https://ncatlab.org/nlab/source/2-poset |
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# $2$-posets
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A **2-poset** is any of several concepts that generalize ([[categorify]]) the notion of [[partial order|posets]] one step in [[higher category theory]]. One does not usually hear about $2$-posets by themselves but instead as special cases of [[2-category|$2$-categories]], such as the [[locally posetal 2-category|locally posetal]] ones.
$2$-posets can also be called **(1,2)-categories**, being a special case of [[(n,r)-categories]]. The concept generalizes to [[n-poset|$n$-posets]].
## Definition
### Explicit definition
A **2-poset** is a [[category]] $C$ such that
1. For each object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A, B)$, $S:Hom(A, B)$, there is a binary relation $R \leq_{A, B} S$
2. For each object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A, B)$, $R \leq_{A, B} R$.
3. For each object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A, B)$, $S:Hom(A, B)$, $T:Hom(A, B)$, $R \leq_{A, B} S$ and $S \leq_{A, B} T$ implies $R \leq_{A, B} T$.
4. For each object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A, B)$, $S:Hom(A, B)$, $R \leq_{A, B} S$ and $S \leq_{A, B} R$ implies $R = S$.
$C$ is only a **2-proset** if $C$ only satisfies 1-3.
### From infinity-categories
Fix a meaning of $\infty$-[[infinity-category|category]], however weak or strict you wish. Then a __$2$-poset__ is an $\infty$-category such that all parallel pairs of $j$-morphisms are [[equivalence|equivalent]] for $j \geq 2$. Thus, up to [[equivalence of categories|equivalence]], there is no point in mentioning anything beyond $2$-morphisms, not even whether two given parallel $2$-morphisms are equivalent. This definition may give a concept more general than a locally posetal $2$-category for your preferred definition of $2$-category, but it will be equivalent if you ignore irrelevant data.
## Examples
* [[Pos]]
* [[Rel]]
* [[simplex category]]
* [[Lat]]
* [[DistLat]]
* [[Frm]]
* [[Locale]]
* [[HeytAlg]]
* [[BoolAlg]]
* [[allegory]]
* [[bicategory of relations]]
* [[first-order hyperdoctrine with equality]]
Just as the motivating example of a $2$-category is the $2$-category [[Cat]] of categories, so the motivating example of a $2$-poset is the $2$-poset [[Pos]] of posets.
## Related concepts
* [[dagger 2-poset]]
* [[n-poset]]
* [[poset]]
* **2-poset**
[[!redirects 2-posets]]
[[!redirects 2-proset]]
[[!redirects 2-prosets]]
[[!redirects (1,2)-category]]
[[!redirects (1,2)-categories]]
[[!redirects locally ordered 2-category]]
[[!redirects locally ordered 2-categories]]
[[!redirects locally ordered]]
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## Contents ##
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## Definition ##
Given a [[dagger 2-poset]] $A$, the **2-poset of partial maps** $Map_\bot(A)$ is the [[subobject|sub]]-[[2-poset]] whose [[objects]] are the objects of $A$ and whose [[morphisms]] are the [[functional morphism in a dagger 2-poset|functional morphisms]] of $A$.
## Examples ##
* For the dagger 2-poset [[Rel]] of [[sets]] and [[relations]], the [[2-poset]] of partial maps $Map_\bot(Rel)$ is equivalent to the [[category]] of sets and [[partial functions]] $Set_\bot$.
## See also ##
* [[partial map]]
* [[category of maps]]
* [[category of monic maps]]
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2-presheaf | https://ncatlab.org/nlab/source/2-presheaf |
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## Idea
The generalization of the notion of _[[presheaf]]_ from [[category theory]] to [[2-category theory]] is an [[indexed category]].
[[!redirects 2-presheaves]]
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2-pretopos | https://ncatlab.org/nlab/source/2-pretopos |
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#Contents#
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## Idea
The generalization of the notion of _[[pretopos]]_ from [[category theory]] to [[2-category theory]].
## Definition
Let $n$ be 2, (2,1), (1,2), or 1.
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###### Definition
An **$n$-pretopos** is an $n$-[[exact 2-category|exact]] [[(n,r)-category|n-category]] which is also [[extensive 2-category|extensive]]. An **infinitary $n$-pretopos** is an $n$-pretopos which is infinitary-extensive.
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###### Remark
As remarked [[coherent 2-category|here]], regularity plus extensivity implies coherency. Thus an $n$-pretopos is, in particular, a [[coherent 2-category|coherent]] $n$-category. Conversely, we have:
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###### Theorem
An $n$-category is an $n$-pretopos if and only if it is coherent and every (finitary) $n$-[[2-polycongruence|polycongruence]] is a kernel.
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## Examples
* [[Cat]] is a 2-pretopos. Likewise, [[Gpd]] is a (2,1)-pretopos and [[Pos]] is a (1,2)-pretopos.
* A 1-category is a 1-pretopos precisely when it is a [[pretopos]] in the usual sense. Note that, as remarked for [[exact 2-category|exactness]], a 1-category is unlikely to be an $n$-pretopos for any $n\gt 1$.
* Since no nontrivial [[(0,1)-categories]] are extensive, the definition as phrased above is not reasonable for $n=(0,1)$. However, for some purposes (such as the [[2-Giraud theorem|n-Giraud theorem]]), it is convenient to define an (infinitary) **(0,1)-pretopos** to simply be an (infinitary) coherent (0,1)-category (exactness being automatic).
## Properties
### Colimits
An $n$-pretopos has [[2-coproducts]] and [[2-quotients]] of $n$-congruences, which are an important class of colimits. However, it can fail to admit all finite colimits, for essentially the same reason as when $n=1$: namely, some ostensibly "finite" colimits secretly involve infinitary processes. In a 1-category, this manifests in the construction of arbitrary coequalizers and pushouts, where we must first _generate_ an equivalence relation by an infinitary process and then take its quotient.
For 2-categories it is even easier to find counterexamples: the 1-pretopos $FinSet$ does in fact have all finite colimits, but the 2-pretopos [[FinCat]] of finite categories (that is, finitely many objects and finitely many morphisms) does not have coinserters, coinverters, or coequifiers. (The category $FPCat$ of finitely _presented_ categories does have finite colimits, but fails to have finite limits.)
However, it is natural to conjecture that just as in the case $n=1$, once an $n$-pretopos is also countably-coherent, it does become finitely cocomplete. See [[colimits in an n-pretopos]].
## References
This is due to
* [[Mike Shulman]], _[[michaelshulman:2-pretopos]]_
based on
* [[Ross Street]], _[[StreetCBS]]_
[[!redirects 2-pretopoi]]
[[!redirects 2-pretoposes]]
[[!redirects n-pretopos]]
[[!redirects n-pretopoi]]
[[!redirects n-pretoposes]]
[[!redirects infinitary 2-pretopos]]
[[!redirects infinitary 2-pretopoi]]
[[!redirects infinitary 2-pretoposes]]
[[!redirects infinitary n-pretopos]]
[[!redirects infinitary n-pretopoi]]
[[!redirects infinitary n-pretoposes]]
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