Datasets:

phanerozoic

/

Coq-HoTT

like 2

braiding_op' {A : Type} `{HasEquivs A} {F : A -> A -> A} `{!Is0Bifunctor F, !Is1Bifunctor F, braid : !Braiding (A:=A^op) F} : Braiding F := braiding_op (A:=A^op) (braid := braid).

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

braiding_op'

a b : CatIsEquiv (braid a b) := catie_adjointify (braid a b) (braid b a) (braid_braid a b) (braid_braid b a).

Instance

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

catie_braid

a b : F a b $<~> F b a := Build_CatEquiv (braid a b).

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

braide

a b c f (g : c $-> _) : braid a b $o f $== g -> f $== braid b a $o g. Proof. intros p. apply (cate_monic_equiv (braide a b)). refine ((cate_buildequiv_fun _ $@R _) $@ p $@ _ $@ cat_assoc _ _ _). refine ((cat_idl _)^$ $@ (_^$ $@R _)). refine ((cate_buildequiv_fun _ $@R _) $@ _). apply braid_braid. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

moveL_braidL

a b c f (g : _ $-> c) : f $o braid a b $== g -> f $== g $o braid b a. Proof. intros p. apply (cate_epic_equiv (braide a b)). refine (_ $@ (cat_assoc _ _ _)^$). refine (_ $@ (_ $@L ((_ $@L cate_buildequiv_fun _) $@ _)^$)). 2: apply braid_braid. refine ((_ $@L _) $@ _ $@ (cat_idr _)^$). 1: apply cate_buildequiv_fun. exact p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

moveL_braidR

a b c f (g : c $-> _) : f $== braid b a $o g -> braid a b $o f $== g. Proof. intros p; symmetry; apply moveL_braidL; symmetry; exact p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

moveR_braidL

a b c f (g : _ $-> c) : f $== g $o braid b a -> f $o braid a b $== g. Proof. intros p; symmetry; apply moveL_braidR; symmetry; exact p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

moveR_braidR

a b c d f (g : d $-> _) : fmap01 F a (braid b c) $o f $== g -> f $== fmap01 F a (braid c b) $o g. Proof. intros p. apply (cate_monic_equiv (emap01 F a (braide b c))). refine (_ $@ cat_assoc _ _ _). refine (_ $@ (_ $@R _)). 2: { refine (_ $@ (_^$ $@R _)). 2: apply cate_buildequiv_fun. refine ((fmap_id _ _)^$ $@ fmap2 _ _ $@ fmap_comp _ _ _). refine ((_ $@R _) $@ _)^$. 1: apply cate_buildequiv_fun. apply braid_braid. } refine ((_ $@R _) $@ p $@ (cat_idl _)^$). refine (_ $@ fmap2 _ _); apply cate_buildequiv_fun. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

moveL_fmap01_braidL

a b c d f (g : _ $-> d) : f $o fmap01 F a (braid b c) $== g -> f $== g $o fmap01 F a (braid c b). Proof. intros p. apply (cate_epic_equiv (emap01 F a (braide b c))). refine (_ $@ (cat_assoc _ _ _)^$). refine (_ $@ (_ $@L _)). 2: { refine (_^$ $@ (_ $@L _^$)). 2: apply cate_buildequiv_fun. refine ((fmap_comp _ _ _)^$ $@ fmap2 _ _ $@ fmap_id _ _). refine ((_ $@L _) $@ _). 1: apply cate_buildequiv_fun. apply braid_braid. } refine ((_ $@L _) $@ p $@ (cat_idr _)^$). refine (_ $@ fmap2 _ _); apply cate_buildequiv_fun. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

moveL_fmap01_braidR

a b c d f (g : d $-> _) : f $== fmap01 F a (braid c b) $o g -> fmap01 F a (braid b c) $o f $== g. Proof. intros p; symmetry; apply moveL_fmap01_braidL; symmetry; exact p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

moveR_fmap01_braidL

a b c d f (g : _ $-> d) : f $== g $o fmap01 F a (braid c b) -> f $o fmap01 F a (braid b c) $== g. Proof. intros p; symmetry; apply moveL_fmap01_braidR; symmetry; exact p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

moveR_fmap01_braidR

a b c d e f (g : e $-> _) : fmap01 F a (fmap01 F b (braid c d)) $o f $== g -> f $== fmap01 F a (fmap01 F b (braid d c)) $o g. Proof. intros p. apply (cate_monic_equiv (emap01 F a (emap01 F b (braide c d)))). refine (_ $@ cat_assoc _ _ _). refine (_ $@ (_ $@R _)). 2: { refine (_ $@ (_^$ $@R _)). 2: apply cate_buildequiv_fun. refine ((fmap_id _ _)^$ $@ fmap2 _ _ $@ fmap_comp _ _ _). refine (_ $@ (_^$ $@R _)). 2: apply cate_buildequiv_fun. refine ((fmap_id _ _)^$ $@ fmap2 _ _ $@ fmap_comp _ _ _). refine ((_ $@R _) $@ _)^$. 1: apply cate_buildequiv_fun. apply braid_braid. } refine ((_ $@R _) $@ p $@ (cat_idl _)^$). refine (cate_buildequiv_fun _ $@ fmap02 _ _ _). refine (cate_buildequiv_fun _ $@ fmap02 _ _ _). apply cate_buildequiv_fun. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

moveL_fmap01_fmap01_braidL

a b c d e f (g : _ $-> e) : f $o fmap01 F a (fmap01 F b (braid c d)) $== g -> f $== g $o fmap01 F a (fmap01 F b (braid d c)). Proof. intros p. apply (cate_epic_equiv (emap01 F a (emap01 F b (braide c d)))). refine (_ $@ (cat_assoc _ _ _)^$). refine (_ $@ (_ $@L _)). 2: { refine (_^$ $@ (_ $@L _^$)). 2: apply cate_buildequiv_fun. refine ((fmap_comp _ _ _)^$ $@ fmap2 _ _ $@ fmap_id _ _). refine ((_ $@L _) $@ _). 1: apply cate_buildequiv_fun. refine ((fmap_comp _ _ _)^$ $@ fmap2 _ _ $@ fmap_id _ _). refine ((_ $@L _) $@ _). 1: apply cate_buildequiv_fun. apply braid_braid. } refine ((_ $@L _) $@ p $@ (cat_idr _)^$). refine (cate_buildequiv_fun _ $@ fmap02 _ _ _). refine (cate_buildequiv_fun _ $@ fmap02 _ _ _). apply cate_buildequiv_fun. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

moveL_fmap01_fmap01_braidR

a b c d e f (g : e $-> _) : f $== fmap01 F a (fmap01 F b (braid d c)) $o g -> fmap01 F a (fmap01 F b (braid c d)) $o f $== g. Proof. intros p; symmetry; apply moveL_fmap01_fmap01_braidL; symmetry; exact p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

moveR_fmap01_fmap01_braidL

a b c d e f (g : _ $-> e) : f $== g $o fmap01 F a (fmap01 F b (braid d c)) -> f $o fmap01 F a (fmap01 F b (braid c d)) $== g. Proof. intros p; symmetry; apply moveL_fmap01_fmap01_braidR; symmetry; exact p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

moveR_fmap01_fmap01_braidR

{a b c d} (f : a $-> c) (g : b $-> d) : braid c d $o fmap11 F f g $== fmap11 F g f $o braid a b. Proof. exact (isnat braid_uncurried (a := (a, b)) (a' := (c, d)) (f, g)). Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

braid_nat

{a b c} (f : a $-> b) : braid b c $o fmap10 F f c $== fmap01 F c f $o braid a c. Proof. refine ((_ $@L (fmap10_is_fmap11 _ _ _)^$) $@ _ $@ (fmap01_is_fmap11 _ _ _ $@R _)). exact (isnat braid_uncurried (a := (a, c)) (a' := (b, c)) (f, Id _)). Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

braid_nat_l

{a b c} (g : b $-> c) : braid a c $o fmap01 F a g $== fmap10 F g a $o braid a b. Proof. refine ((_ $@L (fmap01_is_fmap11 _ _ _)^$) $@ _ $@ (fmap10_is_fmap11 _ _ _ $@R _)). exact (isnat braid_uncurried (a := (a, b)) (a' := (a, c)) (Id _ , g)). Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

braid_nat_r

symmetricbraiding_op' {A : Type} {F : A -> A -> A} `{HasEquivs A, !Is0Bifunctor F, !Is1Bifunctor F, H' : !SymmetricBraiding (A:=A^op) F} : SymmetricBraiding F := symmetricbraiding_op (A:=A^op) (F := F). Global Instance ismonoidal_op {A : Type} (tensor : A -> A -> A) (unit : A) `{IsMonoidal A tensor unit} : IsMonoidal A^op tensor unit. Proof. snrapply Build_IsMonoidal. 1-5: exact _. - intros a b; unfold op in a, b; simpl. refine (_^$ $@ _ $@ (_ $@L _)). 1,3: exact (emap_inv _ _ $@ cate_buildequiv_fun _). nrefine (cate_inv2 _ $@ cate_inv_compose' _ _). refine (cate_buildequiv_fun _ $@ _ $@ ((cate_buildequiv_fun _)^$ $@R _) $@ (cate_buildequiv_fun _)^$). rapply cat_tensor_triangle_identity. - intros a b c d; unfold op in a, b, c, d; simpl. refine (_ $@ ((_ $@L _) $@@ _)). 2,3: exact (emap_inv _ _ $@ cate_buildequiv_fun _). refine ((cate_inv_compose' _ _)^$ $@ cate_inv2 _ $@ cate_inv_compose' _ _ $@ (_ $@L cate_inv_compose' _ _)). refine (cate_buildequiv_fun _ $@ _ $@ ((cate_buildequiv_fun _)^$ $@R _) $@ (cate_buildequiv_fun _)^$). refine (_ $@ (cate_buildequiv_fun _ $@@ (cate_buildequiv_fun _ $@R _))^$). rapply cat_tensor_pentagon_identity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

symmetricbraiding_op'

ismonoidal_op' {A : Type} (tensor : A -> A -> A) (unit : A) `{HasEquivs A} `{!IsMonoidal A^op tensor unit} : IsMonoidal A tensor unit := ismonoidal_op (A:=A^op) tensor unit. Global Instance issymmetricmonoidal_op {A : Type} (tensor : A -> A -> A) (unit : A) `{IsSymmetricMonoidal A tensor unit} : IsSymmetricMonoidal A^op tensor unit. Proof. snrapply Build_IsSymmetricMonoidal. - rapply ismonoidal_op. - rapply symmetricbraiding_op. - intros a b c; unfold op in a, b, c; simpl. snrefine (_ $@ (_ $@L (_ $@R _))). 2: exact ((braide _ _)^-1$). 2: { nrapply cate_moveR_V1. symmetry. nrefine ((_ $@R _) $@ _). 1: nrapply cate_buildequiv_fun. rapply braid_braid. } snrefine ((_ $@R _) $@ _). { refine (emap _ _)^-1$. rapply braide. } { symmetry. refine (cate_inv_adjointify _ _ _ _ $@ fmap2 _ _). nrapply cate_inv_adjointify. } snrefine ((_ $@L (_ $@L _)) $@ _). { refine (emap (flip tensor c) _)^-1$. rapply braide. } { symmetry. refine (cate_inv_adjointify _ _ _ _ $@ fmap2 _ _). nrapply cate_inv_adjointify. } refine ((_ $@L _)^$ $@ _^$ $@ cate_inv2 _ $@ _ $@ (_ $@L _)). 1,2,4,5: rapply cate_inv_compose'. refine (_ $@ (_ $@@ _) $@ _ $@ (_ $@R _)^$ $@ _^$). 1-3,5-6: rapply cate_buildequiv_fun. refine ((fmap02 _ _ _ $@@ ((_ $@ fmap20 _ _ _) $@R _)) $@ cat_symm_tensor_hexagon a b c $@ ((_ $@L _^$) $@R _)). 1-4: nrapply cate_buildequiv_fun. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

ismonoidal_op'

issymmetricmonoidal_op' {A : Type} (tensor : A -> A -> A) (unit : A) `{HasEquivs A} `{H' : !IsSymmetricMonoidal A^op tensor unit} : IsSymmetricMonoidal A tensor unit := issymmetricmonoidal_op (A:=A^op) tensor unit.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

issymmetricmonoidal_op'

{A} (tensor : A -> A -> A) (unit : A) `{IsMonoidal A tensor unit} (x y : A) : (left_unitor (tensor x y) : _ $-> _) $== fmap10 tensor (left_unitor x) y $o associator unit x y. Proof. refine ((cate_moveR_eV _ _ _ (isnat_natequiv left_unitor _))^$ $@ ((_ $@L _) $@R _) $@ cate_moveR_eV _ _ _ (isnat_natequiv left_unitor _)). refine (_ $@ (fmap01_comp _ _ _ _)^$). refine (_ $@ (cate_moveR_Ve _ _ _ (associator_nat_m _ _ _)^$ $@R _)). nrefine (_ $@ cat_assoc_opp _ _ _). change (fmap (tensor ?x) ?f) with (fmap01 tensor x f). change (cate_fun' _ _ (cat_tensor_left_unitor ?x)) with (cate_fun (cat_tensor_left_unitor x)). apply cate_moveL_Ve. refine ((_ $@L triangle_identity _ _ _ _ _ _) $@ _). nrefine (cat_assoc_opp _ _ _ $@ _ $@ cat_assoc_opp _ _ _). refine (_ $@ ((fmap20 _ (triangle_identity _ _ _ _ _ _) _ $@ fmap10_comp _ _ _ _)^$ $@R _)). refine (_ $@ cat_assoc_opp _ _ _). refine (_ $@ (_ $@L (pentagon_identity _ _ _ _ _ _ $@ cat_assoc _ _ _))). refine ((_ $@R _) $@ cat_assoc _ _ _). exact (associator_nat_l _ _ _). Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

left_unitor_associator

{A} (tensor : A -> A -> A) (unit : A) `{IsMonoidal A tensor unit} (x y : A) : (fmap01 tensor x (right_unitor y) : _ $-> _) $== right_unitor (tensor x y) $o associator x y unit. Proof. refine ((cate_moveR_eV _ _ _ (isnat_natequiv right_unitor _))^$ $@ ((_ $@L _) $@R _) $@ cate_moveR_eV _ _ _ (isnat_natequiv right_unitor _)). refine (_ $@ (fmap10_comp tensor _ _ _)^$). refine ((cate_moveR_eV _ _ _ (associator_nat_m _ _ _))^$ $@ _). refine (_ $@ (cate_moveR_eV _ _ _ (triangle_identity _ _ _ _ _ _) $@R _)). apply cate_moveR_eV. refine ((_ $@L (fmap02 _ _ (cate_moveR_eV _ _ _ (triangle_identity _ _ _ _ _ _))^$ $@ fmap01_comp _ _ _ _)) $@ _). refine (cat_assoc_opp _ _ _ $@ _). nrefine ((associator_nat_r _ _ _ $@R _) $@ cat_assoc _ _ _ $@ _). do 2 nrefine (_ $@ cat_assoc_opp _ _ _). refine (_ $@L _). refine ((_ $@L (emap_inv' _ _)^$) $@ _). apply cate_moveR_eV. refine (_ $@ (_ $@L cate_buildequiv_fun _)^$). nrefine (_ $@ cat_assoc_opp _ _ _). apply cate_moveL_Ve. rapply pentagon_identity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

right_unitor_associator

{A} (tensor : A -> A -> A) (unit : A) `{IsMonoidal A tensor unit} : (left_unitor unit : tensor unit unit $-> _) $== right_unitor unit. Proof. refine ((cate_moveR_eV _ _ _ (isnat_natequiv left_unitor (left_unitor unit)))^$ $@ _). apply cate_moveR_eV. refine (_ $@ (_ $@L left_unitor_associator _ _ _ _)^$). nrefine (_ $@ (_ $@R _) $@ cat_assoc _ _ _). 2: rapply (isnat_natequiv right_unitor _)^$. nrefine ((_ $@L _) $@ cat_assoc_opp _ _ _). refine (triangle_identity _ _ _ _ _ _ $@ _). nrefine (_ $@R _). nrapply cate_monic_equiv. exact (isnat_natequiv right_unitor (right_unitor unit)). Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

left_unitor_unit_right_unitor_unit

{A B : Type} {tensorA : A -> A -> A} {tensorB : B -> B -> B} {IA : A} {IB : B} (F : A -> B) `{IsMonoidalFunctor A B tensorA tensorB IA IB F} {x x' y y': A} (f : x $-> x') (g : y $-> y') : fmap_tensor F (x', y') $o fmap11 tensorB (fmap F f) (fmap F g) $== fmap F (fmap11 tensorA f g) $o fmap_tensor F (x, y). Proof. destruct (fmap_tensor F) as [fmap_tensor_F nat]. exact (nat (x, y) (x', y') (f, g)). Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

fmap_tensor_nat

{A B : Type} {tensorA : A -> A -> A} {tensorB : B -> B -> B} {IA : A} {IB : B} (F : A -> B) `{IsMonoidalFunctor A B tensorA tensorB IA IB F} {x x' y : A} (f : x $-> x') : fmap_tensor F (x', y) $o fmap10 tensorB (fmap F f) (F y) $== fmap F (fmap10 tensorA f y) $o fmap_tensor F (x, y). Proof. refine ((_ $@L (fmap12 tensorB _ (fmap_id _ _) $@ fmap10_is_fmap11 _ _ _)^$) $@ _). refine (_ $@ (fmap2 F (fmap10_is_fmap11 _ _ _) $@R _)). snrapply fmap_tensor_nat. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

fmap_tensor_nat_l

{A B : Type} {tensorA : A -> A -> A} {tensorB : B -> B -> B} {IA : A} {IB : B} (F : A -> B) `{IsMonoidalFunctor A B tensorA tensorB IA IB F} {x y y' : A} (g : y $-> y') : fmap_tensor F (x, y') $o fmap01 tensorB (F x) (fmap F g) $== fmap F (fmap01 tensorA x g) $o fmap_tensor F (x, y). Proof. refine ((_ $@L (fmap21 tensorB (fmap_id _ _) _ $@ fmap01_is_fmap11 _ _ _)^$) $@ _). refine (_ $@ (fmap2 F (fmap01_is_fmap11 _ _ _) $@R _)). snrapply fmap_tensor_nat. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Opposite.

WildCat\Monoidal.v

fmap_tensor_nat_r

{a a'} (f : a $-> a') b c : twist a' b c $o fmap10 cat_tensor f (cat_tensor b c) $== fmap01 cat_tensor b (fmap10 cat_tensor f c) $o twist a b c. Proof. refine ((_ $@L _^$) $@ twist_nat a a' b b c c f (Id _) (Id _) $@ (_ $@R _)). - refine (fmap12 _ _ _ $@ fmap10_is_fmap11 _ _ _). rapply fmap11_id. - refine (fmap12 _ _ _ $@ fmap01_is_fmap11 _ _ _). rapply fmap10_is_fmap11. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

twist_nat_l

a {b b'} (g : b $-> b') c : twist a b' c $o fmap01 cat_tensor a (fmap10 cat_tensor g c) $== fmap10 cat_tensor g (cat_tensor a c) $o twist a b c. Proof. refine ((_ $@L _^$) $@ twist_nat a a b b' c c (Id _) g (Id _) $@ (_ $@R _)). - refine (fmap12 _ _ _ $@ fmap01_is_fmap11 _ _ _). rapply fmap10_is_fmap11. - refine (fmap12 _ _ _ $@ fmap10_is_fmap11 _ _ _). rapply fmap11_id. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

twist_nat_m

a b {c c'} (h : c $-> c') : twist a b c' $o fmap01 cat_tensor a (fmap01 cat_tensor b h) $== fmap01 cat_tensor b (fmap01 cat_tensor a h) $o twist a b c. Proof. refine ((_ $@L _^$) $@ twist_nat a a b b c c' (Id _) (Id _) h $@ (_ $@R _)). - refine (fmap12 _ _ _ $@ fmap01_is_fmap11 _ _ _). rapply fmap01_is_fmap11. - refine (fmap12 _ _ _ $@ fmap01_is_fmap11 _ _ _). rapply fmap01_is_fmap11. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

twist_nat_r

a b c d f (g : d $-> _) : twist a b c $o f $== g -> f $== twist b a c $o g. Proof. intros p. apply (cate_monic_equiv (twiste a b c)). nrefine ((cate_buildequiv_fun _ $@R _) $@ p $@ _ $@ cat_assoc _ _ _). refine ((cat_idl _)^$ $@ (_^$ $@R _)). refine ((cate_buildequiv_fun _ $@R _) $@ _). apply twist_twist. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

moveL_twistL

a b c d f (g : _ $-> d) : f $o twist a b c $== g -> f $== g $o twist b a c. Proof. intros p. apply (cate_epic_equiv (twiste a b c)). refine (_ $@ (cat_assoc _ _ _)^$). refine (_ $@ (_ $@L ((_ $@L cate_buildequiv_fun _) $@ _)^$)). 2: apply twist_twist. refine ((_ $@L _) $@ _ $@ (cat_idr _)^$). 1: apply cate_buildequiv_fun. exact p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

moveL_twistR

a b c d f (g : d $-> _) : f $== twist b a c $o g -> twist a b c $o f $== g. Proof. intros p; symmetry; apply moveL_twistL; symmetry; exact p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

moveR_twistL

a b c d f (g : _ $-> d) : f $== g $o twist b a c -> f $o twist a b c $== g. Proof. intros p; symmetry; apply moveL_twistR; symmetry; exact p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

moveR_twistR

a b c d e f (g : e $-> _) : fmap01 cat_tensor a (twist b c d) $o f $== g -> f $== fmap01 cat_tensor a (twist c b d) $o g. Proof. intros p. apply (cate_monic_equiv (emap01 cat_tensor a (twiste b c d))). refine (_ $@ cat_assoc _ _ _). refine (_ $@ (_ $@R _)). 2: { refine (_ $@ (_^$ $@R _)). 2: apply cate_buildequiv_fun. refine ((fmap_id _ _)^$ $@ fmap2 _ _ $@ fmap_comp _ _ _). refine (_^$ $@ (_^$ $@R _)). 2: apply cate_buildequiv_fun. apply twist_twist. } refine ((_ $@R _) $@ p $@ (cat_idl _)^$). refine (cate_buildequiv_fun _ $@ fmap02 _ _ _). apply cate_buildequiv_fun. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

moveL_fmap01_twistL

a b c d e f (g : _ $-> e) : f $o fmap01 cat_tensor a (twist b c d) $== g -> f $== g $o fmap01 cat_tensor a (twist c b d). Proof. intros p. apply (cate_epic_equiv (emap01 cat_tensor a (twiste b c d))). refine (_ $@ (cat_assoc _ _ _)^$). refine (_ $@ (_ $@L _)). 2: { refine (_^$ $@ (_ $@L _^$)). 2: apply cate_buildequiv_fun. refine ((fmap_comp _ _ _)^$ $@ fmap2 _ _ $@ fmap_id _ _). refine ((_ $@L _) $@ _). 1: apply cate_buildequiv_fun. apply twist_twist. } refine ((_ $@L _) $@ p $@ (cat_idr _)^$). refine (cate_buildequiv_fun _ $@ fmap02 _ _ _). apply cate_buildequiv_fun. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

moveL_fmap01_twistR

a b c d e f (g : e $-> _) : f $== fmap01 cat_tensor a (twist c b d) $o g -> fmap01 cat_tensor a (twist b c d) $o f $== g. Proof. intros p; symmetry; apply moveL_fmap01_twistL; symmetry; exact p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

moveR_fmap01_twistL

a b c d e f (g : _ $-> e) : f $== g $o fmap01 cat_tensor a (twist c b d) -> f $o fmap01 cat_tensor a (twist b c d) $== g. Proof. intros p; symmetry; apply moveL_fmap01_twistR; symmetry; exact p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

moveR_fmap01_twistR

associator_twist' a b c : cat_tensor a (cat_tensor b c) $<~> cat_tensor (cat_tensor a b) c. Proof. refine (braide _ _ $oE _). nrefine (twiste _ _ _ $oE _). exact (emap01 cat_tensor a (braide _ _)). Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

associator_twist'

a b c : cate_fun (associator_twist' a b c) $== braid c (cat_tensor a b) $o (twist a c b $o fmap01 cat_tensor a (braid b c)). Proof. refine (cate_buildequiv_fun _ $@ (_ $@@ cate_buildequiv_fun _)). nrefine (cate_buildequiv_fun _ $@ (_ $@@ cate_buildequiv_fun _)). refine (cate_buildequiv_fun _ $@ fmap2 _ _). apply cate_buildequiv_fun. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

associator_twist'_unfold

Associator cat_tensor. Proof. snrapply Build_Associator. - exact '. - snrapply Build_Is1Natural. simpl; intros [[a b] c] [[a' b'] c'] [[f g] h]; simpl in f, g, h. change (?w $o ?x $== ?y $o ?z) with (Square z w x y). nrapply hconcatL. 1: apply '_unfold. nrapply hconcatR. 2: apply '_unfold. nrapply vconcat. 2: rapply braid_nat. nrapply vconcat. 2: apply twist_nat. nrapply hconcatL. 2: nrapply hconcatR. 1,3: symmetry; rapply fmap01_is_fmap11. rapply fmap11_square. 1: rapply vrefl. apply braid_nat. Defined.

Instance

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

associator_twist

LeftUnitor cat_tensor cat_tensor_unit. Proof. snrapply Build_NatEquiv'. - snrapply Build_NatTrans. + exact (fun a => right_unitor a $o braid cat_tensor_unit a). + snrapply Build_Is1Natural. intros a b f. change (?w $o ?x $== ?y $o ?z) with (Square z w x y). nrapply vconcat. 2: rapply (isnat right_unitor f). rapply braid_nat_r. - intros a. rapply compose_catie'. rapply catie_braid. Defined.

Instance

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

left_unitor_twist

TriangleIdentity cat_tensor cat_tensor_unit. Proof. intros a b. refine (_ $@ (_ $@L _)^$). 2: apply associator_twist'_unfold. refine (fmap02 _ a (cate_buildequiv_fun _) $@ _); cbn. refine (fmap01_comp _ _ _ _ $@ _). do 2 refine (_ $@ cat_assoc _ _ _). refine ((twist_unitor _ _ $@ (_ $@R _)) $@R _). apply braid_nat_r. Defined.

Instance

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

triangle_twist

PentagonIdentity cat_tensor. Proof. clear twist_unitor right_unitor cat_tensor_unit. intros a b c d. refine ((_ $@@ _) $@ _ $@ ((fmap02 _ _ _ $@ _)^$ $@@ (_ $@@ (fmap20 _ _ _ $@ _))^$)). 1,2,4,6,7: apply associator_twist'_unfold. 2: refine (fmap01_comp _ _ _ _ $@ (_ $@L (fmap01_comp _ _ _ _))). 2: refine (fmap10_comp _ _ _ _ $@ (_ $@L (fmap10_comp _ _ _ _))). refine (cat_assoc _ _ _ $@ _). refine (_ $@L (cat_assoc _ _ _) $@ _). do 4 refine ((cat_assoc _ _ _)^$ $@ _). refine (_ $@ (((cat_assoc _ _ _) $@R _) $@R _)). do 2 refine (_ $@ ((cat_assoc _ _ _) $@R _)). do 2 refine (_ $@ cat_assoc _ _ _). apply moveL_fmap01_fmap01_braidR. apply moveL_fmap01_twistR. refine (_ $@ (cat_assoc _ _ _)^$). refine (_ $@ ((_ $@L _) $@ cat_idr _)^$). 2: refine ((fmap01_comp _ _ _ _)^$ $@ fmap02 _ _ _ $@ fmap01_id _ _ _). 2: apply braid_braid. apply moveL_twistR. refine (_ $@ (cat_assoc _ _ _)^$). refine (_ $@ (_ $@L _)). 2: apply braid_nat_r. refine (_ $@ cat_assoc _ _ _). apply moveL_fmap01_fmap01_braidR. refine (_ $@ (cat_assoc _ _ _)^$). refine (_ $@ (_ $@L _)). 2: apply braid_nat_r. refine (_ $@ cat_assoc _ _ _). apply moveL_fmap01_twistR. refine (_ $@ _). 2: apply braid_nat_r. apply moveR_fmap01_twistR. apply moveR_fmap01_fmap01_braidR. apply moveR_twistR. apply moveR_fmap01_twistR. refine (cat_assoc _ _ _ $@ _). refine ((_ $@L _) $@ _). 1: refine ((fmap01_comp _ _ _ _)^$ $@ fmap02 _ _ _ $@ fmap01_comp _ _ _ _). 1: apply braid_nat_r. refine ((cat_assoc _ _ _)^$ $@ _). apply moveR_fmap01_braidR. refine (cat_assoc _ _ _ $@ _). refine ((_ $@L _) $@ _). 1: apply twist_nat_m. refine ((cat_assoc _ _ _)^$ $@ _). apply moveR_twistR. refine (cat_assoc _ _ _ $@ _). refine ((_ $@L _) $@ _). 1: apply braid_nat_l. refine ((cat_assoc _ _ _)^$ $@ _). apply moveR_braidR. refine (cat_assoc _ _ _ $@ _). refine ((_ $@L _) $@ cat_idr _ $@ _). 1: refine ((fmap01_comp _ _ _ _)^$ $@ fmap02 _ _ _ $@ fmap01_id _ _ _). 1: apply braid_braid. apply moveL_braidR. apply moveL_twistR. apply moveL_fmap01_braidR. do 4 refine (_ $@ (cat_assoc _ _ _)^$). do 3 refine (cat_assoc _ _ _ $@ _). refine (_ $@L _). apply moveR_twistL. do 4 refine (_ $@ cat_assoc _ _ _). refine ((cat_assoc _ _ _)^$ $@ _). apply moveL_fmap01_twistR. apply moveL_twistR. do 2 refine (_ $@ (cat_assoc _ _ _)^$). do 3 refine (cat_assoc _ _ _ $@ _). apply moveL_twistL. refine (_ $@ cat_assoc _ _ _). do 4 refine ((cat_assoc _ _ _)^$ $@ _). apply moveR_twistR. apply moveR_fmap01_twistR. do 3 refine (_ $@ (cat_assoc _ _ _)^$). do 2 refine (cat_assoc _ _ _ $@ _). apply moveL_fmap01_braidL. do 2 refine (_ $@ cat_assoc _ _ _). do 3 refine ((cat_assoc _ _ _)^$ $@ _). apply twist_9_gon. Defined.

Instance

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

pentagon_twist

HexagonIdentity cat_tensor. Proof. intros a b c; simpl. refine (((_ $@L _) $@R _) $@ _ $@ (_ $@@ (_ $@R _))^$). 1,3,4: apply associator_twist'_unfold. do 2 refine (((cat_assoc _ _ _)^$ $@R _) $@ _). refine (cat_assoc _ _ _ $@ (_ $@L _) $@ _). { refine ((fmap_comp _ _ _)^$ $@ fmap2 _ _ $@ fmap_id _ _). apply braid_braid. } refine (cat_idr _ $@ _). refine (_ $@ cat_assoc _ _ _). refine (_ $@ ((cat_assoc _ _ _)^$ $@R _)). refine (_ $@ (((cat_idr _)^$ $@ (_ $@L _^$)) $@R _)). 2: apply braid_braid. refine (((braid_nat_r _)^$ $@R _) $@ _). refine (cat_assoc _ _ _ $@ (_ $@L _) $@ (cat_assoc _ _ _)^$). refine (_ $@ cat_assoc _ _ _). apply moveL_fmap01_braidR. apply twist_hexagon. Defined.

Instance

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

hexagon_twist

IsMonoidal A cat_tensor cat_tensor_unit := {}.

Instance

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

ismonoidal_twist

IsSymmetricMonoidal A cat_tensor cat_tensor_unit := {}.

Instance

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

issymmetricmonoidal_twist

twist_hex' a b c d : fmap01 cat_tensor c (twist a b d) $o twist a c (cat_tensor b d) $o fmap01 cat_tensor a (twist b c d) $== twist b c (cat_tensor a d) $o fmap01 cat_tensor b (twist a c d) $o twist a b (cat_tensor c d). Proof. pose proof (twist_hexagon c a d $@ cat_assoc _ _ _) as p. apply moveR_twistL in p. apply moveR_fmap01_braidL in p. apply (fmap02 cat_tensor b) in p. refine (_ $@ ((_ $@L p) $@R _)); clear p. apply moveL_twistR. apply moveL_twistL. refine (_ $@ (fmap01_comp _ _ _ _)^$). apply moveR_twistL. refine (_ $@ cat_assoc _ _ _). Abort. End TwistConstruction.

Definition

Require Import Basics.Overture Basics.Tactics Types.Forall WildCat.Monoidal. Require Import WildCat.Core WildCat.Bifunctor WildCat.Prod WildCat.Equiv. Require Import WildCat.NatTrans WildCat.Square WildCat.Opposite.

WildCat\MonoidalTwistConstruction.v

twist_hex'

{A : Type} {B : A -> Type} `{forall x, IsGraph (B x)} (F G : forall (x : A), B x) := forall (a : A), F a $-> G a.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

Transformation

{A B : Type} `{Is01Cat B} (F : A -> B) : F $=> F := fun a => Id (F a).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

trans_id

{A B : Type} `{Is01Cat B} {F G K : A -> B} (gamma : G $=> K) (alpha : F $=> G) : F $=> K := fun a => gamma a $o alpha a.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

trans_comp

{A B : Type} {C : B -> Type} {F G : forall x, C x} `{Is01Cat B} `{!forall x, IsGraph (C x)} `{!forall x, Is01Cat (C x)} (gamma : F $=> G) (K : A -> B) : F o K $=> G o K := gamma o K.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

trans_prewhisker

{A B C : Type} {F G : A -> B} (K : B -> C) `{Is01Cat B, Is01Cat C, !Is0Functor K} (gamma : F $=> G) : K o F $=> K o G := fun a => fmap K (gamma a).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

trans_postwhisker

{A} {B} `{Is01Cat B} (F : A -> B) (G : A -> B) (alpha : F $=> G) : Transformation (A:=A^op) (B:=fun _ => B^op) G (F : A^op -> B^op) := alpha.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

trans_op

{A B : Type} `{IsGraph A} `{Is1Cat B} {F G : A -> B} `{!Is0Functor F, !Is0Functor G} (alpha : F $=> G) (isnat : forall a a' (f : a $-> a'), alpha a' $o fmap F f $== fmap G f $o alpha a) : Is1Natural F G alpha. Proof. snrapply '. - exact isnat. - intros a a' f. exact (isnat a a' f)^$. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

Build_Is1Natural

{A B : Type} `{Is01Cat A} `{Is1Cat B} {F : A -> B} `{!Is0Functor F} {G : A -> B} `{!Is0Functor G} {alpha : F $=> G} (gamma : F $=> G) `{!Is1Natural F G gamma} (p : forall a, alpha a $== gamma a) : Is1Natural F G alpha. Proof. snrapply Build_Is1Natural. intros a b f. exact ((p b $@R _) $@ isnat gamma f $@ (_ $@L (p a)^$)). Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

is1natural_homotopic

{A B : Type} `{IsGraph A} `{Is1Cat B} {F G : A -> B}

Record

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

NatTrans

{A B : Type} `{IsGraph A} `{Is1Cat B} (F G : A -> B) {ff : Is0Functor F} {fg : Is0Functor G} : _ <~> NatTrans F G := ltac:(issig).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

issig_NatTrans

{A B : Type} (F : A -> B) `{IsGraph A, Is1Cat B, !Is0Functor F} : NatTrans F F := Build_NatTrans (trans_id F) _.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

nattrans_id

{A B : Type} {F G K : A -> B} `{IsGraph A, Is1Cat B, !Is0Functor F, !Is0Functor G, !Is0Functor K} : NatTrans G K -> NatTrans F G -> NatTrans F K := fun alpha beta => Build_NatTrans (trans_comp alpha beta) _.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

nattrans_comp

{A B C : Type} {F G : B -> C} `{IsGraph A, Is1Cat B, Is1Cat C, !Is0Functor F, !Is0Functor G} (alpha : NatTrans F G) (K : A -> B) `{!Is0Functor K} : NatTrans (F o K) (G o K) := Build_NatTrans (trans_prewhisker alpha K) _.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

nattrans_prewhisker

{A B C : Type} {F G : A -> B} (K : B -> C) `{IsGraph A, Is1Cat B, Is1Cat C, !Is0Functor F, !Is0Functor G, !Is0Functor K, !Is1Functor K} : NatTrans F G -> NatTrans (K o F) (K o G) := fun alpha => Build_NatTrans (trans_postwhisker K alpha) _.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

nattrans_postwhisker

{A B : Type} `{Is01Cat A} `{Is1Cat B} {F G : A -> B} `{!Is0Functor F, !Is0Functor G} : NatTrans F G -> NatTrans (A:=A^op) (B:=B^op) (G : A^op -> B^op) (F : A^op -> B^op) := fun alpha => Build_NatTrans (trans_op F G alpha) _.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

nattrans_op

{A B : Type} `{IsGraph A} `{HasEquivs B}

Record

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

NatEquiv

{A B : Type} `{IsGraph A} `{HasEquivs B} (F G : A -> B) `{!Is0Functor F, !Is0Functor G} : _ <~> NatEquiv F G := ltac:(issig).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

issig_NatEquiv

{A B : Type} `{IsGraph A} `{HasEquivs B} {F G : A -> B} `{!Is0Functor F, !Is0Functor G} : NatEquiv F G -> NatTrans F G. Proof. intros alpha. nrapply Build_NatTrans. rapply (is1natural_natequiv alpha). Defined.

Lemma

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

nattrans_natequiv

{A B : Type} `{IsGraph A} `{HasEquivs B} {F G : A -> B} `{!Is0Functor F, !Is0Functor G} (alpha : NatEquiv F G) {a a' : A} (f : a $-> a') := isnat (nattrans_natequiv alpha) f.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

isnat_natequiv

Build_NatEquiv' {A B : Type} `{IsGraph A} `{HasEquivs B} {F G : A -> B} `{!Is0Functor F, !Is0Functor G} (alpha : NatTrans F G) `{forall a, CatIsEquiv (alpha a)} : NatEquiv F G. Proof. snrapply Build_NatEquiv. - intro a. refine (Build_CatEquiv (alpha a)). - snrapply Build_Is1Natural'. + intros a a' f. refine ((cate_buildequiv_fun _ $@R _) $@ _ $@ (_ $@L cate_buildequiv_fun _)^$). apply (isnat alpha). + intros a a' f. refine ((_ $@L cate_buildequiv_fun _) $@ _ $@ (cate_buildequiv_fun _ $@R _)^$). apply (isnat_tr alpha). Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

Build_NatEquiv'

{A B : Type} `{IsGraph A} `{HasEquivs B} {F : A -> B} `{!Is0Functor F} : NatEquiv F F := Build_NatEquiv' (nattrans_id F).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

natequiv_id

{A B} {F G H : A -> B} `{IsGraph A} `{HasEquivs B} `{!Is0Functor F, !Is0Functor G, !Is0Functor H} (alpha : NatEquiv G H) (beta : NatEquiv F G) : NatEquiv F H := Build_NatEquiv' (nattrans_comp alpha beta).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

natequiv_compose

{A B C} {H K : B -> C} `{IsGraph A, HasEquivs B, HasEquivs C, !Is0Functor H, !Is0Functor K} (alpha : NatEquiv H K) (F : A -> B) `{!Is0Functor F} : NatEquiv (H o F) (K o F) := Build_NatEquiv' (nattrans_prewhisker alpha F).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

natequiv_prewhisker

{A B C} {F G : A -> B} `{IsGraph A, HasEquivs B, HasEquivs C, !Is0Functor F, !Is0Functor G} (K : B -> C) (alpha : NatEquiv F G) `{!Is0Functor K, !Is1Functor K} : NatEquiv (K o F) (K o G). Proof. srefine (Build_NatEquiv' (nattrans_postwhisker K alpha)). 2: unfold nattrans_postwhisker, trans_postwhisker; cbn. all: exact _. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

natequiv_postwhisker

{A B : Type} `{Is01Cat A} `{HasEquivs B} {F G : A -> B} `{!Is0Functor F, !Is0Functor G} : NatEquiv F G -> NatEquiv (G : A^op -> B^op) F. Proof. intros [a n]. snrapply Build_NatEquiv. 1: exact a. by rapply is1natural_op. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

natequiv_op

{A B : Type} `{IsGraph A} `{HasEquivs B} {F G : A -> B} `{!Is0Functor F, !Is0Functor G} : NatEquiv F G -> NatEquiv G F. Proof. intros [alpha I]. snrapply Build_NatEquiv. 1: exact (fun a => (alpha a)^-1$). snrapply Build_Is1Natural'. + intros X Y f. apply vinverse, I. + intros X Y f. apply hinverse, I. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

natequiv_inverse

{A B C D : Type} `{IsGraph A, HasEquivs B, HasEquivs C, HasEquivs D} (F : C -> D) (G : B -> C) (K : A -> B) `{!Is0Functor F, !Is0Functor G, !Is0Functor K} : NatEquiv ((F o G) o K) (F o (G o K)). Proof. snrapply Build_NatEquiv. 1: intro; reflexivity. snrapply Build_Is1Natural. intros X Y f. refine (cat_prewhisker (id_cate_fun _) _ $@ cat_idl _ $@ _^$). refine (cat_postwhisker _ (id_cate_fun _) $@ cat_idr _). Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

natequiv_functor_assoc_ff_f

{B C : Type} `{Is1Cat B, Is1Gpd C} `{IsPointed B, IsPointed C} (F G : B -->* C) := {eta : F $=> G & eta (point _) $== bp_pointed F $@ (bp_pointed G)^$}.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

PointedTransformation

{B C : Type} `{Is1Cat B, Is1Gpd C} `{IsPointed B, IsPointed C} (F G : B -->* C) : (F $=>* G) -> (G $=>* F). Proof. intros [h p]. exists (fun x => (h x)^$). refine (gpd_rev2 p $@ _). refine (gpd_rev_pp _ _ $@ _). refine (_ $@L _). apply gpd_rev_rev. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

ptransformation_inverse

{B C : Type} `{Is1Cat B, Is1Gpd C} `{IsPointed B, IsPointed C} {F0 F1 F2 : B -->* C} : (F0 $=>* F1) -> (F1 $=>* F2) -> (F0 $=>* F2). Proof. intros [h0 p0] [h1 p1]. exists (trans_comp h1 h0). refine ((p1 $@R _) $@ (_ $@L p0) $@ _); unfold gpd_comp; cbn. refine (cat_assoc _ _ _ $@ _). rapply (fmap _). apply gpd_h_Vh. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Square. Require Import WildCat.Opposite.

WildCat\NatTrans.v

ptransformation_compose

(A : Type) := A. Notation "A ^" := ( A). #[global] Typeclasses Opaque . Global Instance isgraph_op {A : Type} `{IsGraph A} : IsGraph A^. Proof. apply Build_IsGraph. unfold ; exact (fun a b => b $-> a). Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core.

WildCat\Opposite.v

op

(A : Type) : IsGraph A := {| Hom := paths |}.

Definition

Require Import Basics.Overture Basics.Tactics Basics.PathGroupoids. Require Import WildCat.Core WildCat.TwoOneCat WildCat.NatTrans.

WildCat\Paths.v

isgraph_paths

(A : Type) `{IsGraph A} : Is2Graph A := fun _ _ => isgraph_paths _.

Definition

Require Import Basics.Overture Basics.Tactics Basics.PathGroupoids. Require Import WildCat.Core WildCat.TwoOneCat WildCat.NatTrans.

WildCat\Paths.v

is2graph_paths

(A : Type) `{Is2Graph A} : Is3Graph A := fun _ _ => is2graph_paths _. Local Existing Instances isgraph_paths is2graph_paths | 10. Global Instance is01cat_paths (A : Type) : Is01Cat A := {| Id := @idpath _ ; cat_comp := fun _ _ _ x y => concat y x |}. Global Instance is0gpd_paths (A : Type) : Is0Gpd A := {| gpd_rev := @inverse _ |}. Global Instance is0functor_cat_postcomp_paths (A : Type) `{Is01Cat A} (a b c : A) (g : b $-> c) : Is0Functor (cat_postcomp a g). Proof. snrapply Build_Is0Functor. exact (@ap _ _ (cat_postcomp a g)). Defined.

Definition

Require Import Basics.Overture Basics.Tactics Basics.PathGroupoids. Require Import WildCat.Core WildCat.TwoOneCat WildCat.NatTrans.

WildCat\Paths.v

is3graph_paths

{A : Type} `{IsPointedCat A} {a b : A} : a $-> b := (mor_initial _ b) $o (mor_terminal a _).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core WildCat.Opposite. Require Import WildCat.Equiv.

WildCat\PointedCat.v

zero_morphism

(h : zero_object $-> a) : h $== zero_morphism := (mor_initial_unique _ _ _)^$ $@ (mor_initial_unique _ _ _).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core WildCat.Opposite. Require Import WildCat.Equiv.

WildCat\PointedCat.v

cat_zero_source

(h : a $-> zero_object) : h $== zero_morphism := (mor_terminal_unique _ _ _)^$ $@ (mor_terminal_unique _ _ _).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core WildCat.Opposite. Require Import WildCat.Equiv.

WildCat\PointedCat.v

cat_zero_target

zero_morphism b c $o f $== zero_morphism a c. Proof. refine (cat_assoc _ _ _ $@ (_ $@L _^$)). apply mor_terminal_unique. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core WildCat.Opposite. Require Import WildCat.Equiv.

WildCat\PointedCat.v

cat_zero_l

g $o zero_morphism a b $== zero_morphism a c. Proof. refine ((_ $@R _) $@ cat_assoc _ _ _)^$. apply mor_initial_unique. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core WildCat.Opposite. Require Import WildCat.Equiv.

WildCat\PointedCat.v

cat_zero_r

`{!HasEquivs A} (be : b $<~> zero_object) : g $o f $== zero_morphism a c. Proof. refine (_ $@L (compose_V_hh be f)^$ $@ _). refine (cat_assoc_opp _ _ _ $@ _). refine (_ $@L (mor_terminal_unique a _ _)^$ $@ _). exact ((mor_initial_unique _ _ _)^$ $@R _). Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core WildCat.Opposite. Require Import WildCat.Equiv.

WildCat\PointedCat.v

cat_zero_m

Build_IsPointedFunctor' {A B : Type} (F : A -> B) `{Is1Cat A, Is1Cat B, !Is0Functor F, !Is1Functor F} `{!IsPointedCat A, !IsPointedCat B, !HasEquivs A, !HasEquivs B} (p : F zero_object $<~> zero_object) : IsPointedFunctor F. Proof. apply Build_IsPointedFunctor. + intros x inx. rapply isinitial_cate. symmetry. refine (p $oE _). rapply (emap F _). rapply cate_isinitial. + intros x tex. rapply isterminal_cate. refine (p $oE _). rapply (emap F _). rapply cate_isterminal. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core WildCat.Opposite. Require Import WildCat.Equiv.

WildCat\PointedCat.v

Build_IsPointedFunctor'

{A B : Type} (F : A -> B) `{IsPointedCat A, IsPointedCat B, !HasEquivs B, !Is0Functor F, !Is1Functor F, !IsPointedFunctor F} : F zero_object $<~> zero_object. Proof. rapply cate_isinitial. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core WildCat.Opposite. Require Import WildCat.Equiv.

WildCat\PointedCat.v

pfunctor_zero

{A B : Type} (F : A -> B) `{IsPointedCat A, IsPointedCat B, !HasEquivs B, !Is0Functor F, !Is1Functor F, !IsPointedFunctor F} {a b : A} : fmap F (zero_morphism a b) $== zero_morphism (F a) (F b). Proof. refine (fmap_comp F _ _ $@ _). refine (_ $@R _ $@ _). 1: nrapply fmap_initial; [exact _]. refine (_ $@L _ $@ _). 1: nrapply fmap_terminal; [exact _]. rapply cat_zero_m. rapply pfunctor_zero. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core WildCat.Opposite. Require Import WildCat.Equiv.

WildCat\PointedCat.v

fmap_zero_morphism

{A B C : Type} `{IsGraph A, IsGraph B, Is01Cat C} (F : A * B -> C) `{!forall a, Is0Functor (fun b => F (a,b)), !forall b, Is0Functor (fun a => F (a,b))} : Is0Functor F. Proof. snrapply Build_Is0Functor. intros [a b] [a' b'] [f g]. exact (fmap (fun a0 => F (a0,b')) f $o fmap (fun b0 => F (a,b0)) g). Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import Types.Prod.

WildCat\Prod.v

is0functor_prod_is0functor

{A B C : Type} `{Is1Cat A, Is1Cat B, Is1Cat C} (F : A * B -> C) `{!forall a, Is0Functor (fun b => F (a,b)), !forall b, Is0Functor (fun a => F (a,b))} `{!forall a, Is1Functor (fun b => F (a,b)), !forall b, Is1Functor (fun a => F (a,b))} (bifunctor_coh : forall a0 a1 (f : a0 $-> a1) b0 b1 (g : b0 $-> b1), fmap (fun b => F (a1,b)) g $o fmap (fun a => F (a,b0)) f $== fmap (fun a => F(a,b1)) f $o fmap (fun b => F (a0,b)) g) : Is1Functor F. Proof. snrapply Build_Is1Functor. - intros [a b] [a' b'] [f g] [f' g'] [p p']; unfold fst, snd in * |- . exact (fmap2 (fun b0 => F (a,b0)) p' $@@ fmap2 (fun a0 => F (a0,b')) p). - intros [a b]. exact ((fmap_id (fun b0 => F (a,b0)) b $@@ fmap_id (fun a0 => F (a0,b)) _) $@ cat_idr _). - intros [a b] [a' b'] [a'' b''] [f g] [f' g']; unfold fst, snd in * |- . refine ((fmap_comp (fun b0 => F (a,b0)) g g' $@@ fmap_comp (fun a0 => F (a0,b'')) f f') $@ _). nrefine (cat_assoc_opp _ _ _ $@ (_ $@R _) $@ cat_assoc _ _ _). refine (cat_assoc _ _ _ $@ (_ $@L _^$) $@ cat_assoc_opp _ _ _). nrapply bifunctor_coh. Defined.

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import Types.Prod.

WildCat\Prod.v

is1functor_prod_is1functor

{A B C : Type} `{IsGraph A, IsGraph B, IsGraph C} (F : A -> B -> C) {H2 : Is0Functor (uncurry F)} {a0 a1 : A} (f : a0 $-> a1) {b0 b1 : B} (g : b0 $-> b1) : F a0 b0 $-> F a1 b1 := @fmap _ _ _ _ (uncurry F) H2 (a0, b0) (a1, b1) (f, g).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import Types.Prod.

WildCat\Prod.v

fmap11_uncurry

{A B C : Type} `{IsGraph A, IsGraph B, IsGraph C} (F : A * B -> C) `{!Is0Functor F} {a0 a1 : A} (f : a0 $-> a1) {b0 b1 : B} (g : b0 $-> b1) : F (a0, b0) $-> F (a1, b1) := fmap (a := (a0, b0)) (b := (a1, b1)) F (f, g).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import Types.Prod.

WildCat\Prod.v

fmap_pair

{A B C : Type} `{Is1Cat A, Is1Cat B, Is1Cat C} (F : A * B -> C) `{!Is0Functor F, !Is1Functor F} {a0 a1 a2 : A} {b0 b1 b2 : B} (f : a0 $-> a1) (h : b0 $-> b1) (g : a1 $-> a2) (i : b1 $-> b2) : fmap_pair F (g $o f) (i $o h) $== fmap_pair F g i $o fmap_pair F f h := fmap_comp (a := (a0, b0)) (b := (a1, b1)) (c := (a2, b2)) F (f, h) (g, i).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import Types.Prod.

WildCat\Prod.v

fmap_pair_comp

{A B C : Type} `{Is1Cat A, Is1Cat B, Is1Cat C} (F : A * B -> C) `{!Is0Functor F, !Is1Functor F} {a0 a1 : A} {f f' : a0 $-> a1} (p : f $== f') {b0 b1 : B} {g g' : b0 $-> b1} (q : g $== g') : fmap_pair F f g $== fmap_pair F f' g' := fmap2 F (a := (a0, b0)) (b := (a1, b1)) (f := (f, g)) (g := (f' ,g')) (p, q).

Definition

Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import Types.Prod.

WildCat\Prod.v

fmap2_pair

{I A : Type} `{Is1Cat A} (prod : A) (x : I -> A) (z : A) (pr : forall i, prod $-> x i) : yon_0gpd prod z $-> prod_0gpd I (fun i => yon_0gpd (x i) z). Proof. snrapply equiv_prod_0gpd_corec. intros i. exact (fmap (fun x => yon_0gpd x z) (pr i)). Defined.

Definition

Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd

WildCat\Products.v

cat_prod_corec_inv

(I : Type) {A : Type} `{Is1Cat A} {x : I -> A} (cat_prod : A) (cat_pr : forall i : I, cat_prod $-> x i) (cat_prod_corec : forall z : A, (forall i : I, z $-> x i) -> (z $-> cat_prod)) (cat_prod_beta_pr : forall (z : A) (f : forall i, z $-> x i) (i : I), cat_pr i $o cat_prod_corec z f $== f i) (cat_prod_eta_pr : forall (z : A) (f g : z $-> cat_prod), (forall i : I, cat_pr i $o f $== cat_pr i $o g) -> f $== g) : Product I x. Proof. snrapply (' I A _ _ _ _ _ cat_prod cat_pr). intros z. nrapply isequiv_0gpd_issurjinj. nrapply Build_IsSurjInj. - intros f. exists (cat_prod_corec z f). intros i. nrapply cat_prod_beta_pr. - intros f g p. by nrapply cat_prod_eta_pr. Defined.

Definition

Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd

WildCat\Products.v

Build_Product

{z : A} : (yon_0gpd (cat_prod I x) z) $<~> prod_0gpd I (fun i => yon_0gpd (x i) z) := Build_CatEquiv (cat_prod_corec_inv (cat_prod I x) x z cat_pr).

Definition

Require Import Basics.Equivalences Basics.Overture Basics.Tactics. Require Import Types.Bool Types.Prod Types.Forall. Require Import WildCat.Bifunctor WildCat.Core WildCat.Equiv WildCat.EquivGpd

WildCat\Products.v

cate_cat_prod_corec_inv