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Build_Is1Cat' (A : Type) `{!IsGraph A, !Is2Graph A, !Is01Cat A} (is01cat_hom : forall a b : A, Is01Cat (a $-> b)) (is0gpd_hom : forall a b : A, Is0Gpd (a $-> b)) (is0functor_postcomp : forall (a b c : A) (g : b $-> c), Is0Functor (cat_postcomp a g)) (is0functor_precomp : forall (a b c : A) (f : a $-> b), Is0Functor (cat_precomp c f)) (cat_assoc : forall (a b c d : A) (f : a $-> b) (g : b $-> c) (h : c $-> d), h $o g $o f $== h $o (g $o f)) (cat_idl : forall (a b : A) (f : a $-> b), Id b $o f $== f) (cat_idr : forall (a b : A) (f : a $-> b), f $o Id a $== f) : Is1Cat A := Build_Is1Cat A _ _ _ is01cat_hom is0gpd_hom is0functor_postcomp is0functor_precomp cat_assoc (fun a b c d f g h => (cat_assoc a b c d f g h)^$) cat_idl cat_idr. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | Build_Is1Cat' | 7,400 |
{A} `{Is1Cat A} {a b c : A} {f g : a $-> b} (h : b $-> c) (p : f $== g) : h $o f $== h $o g := fmap (cat_postcomp a h) p. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | cat_postwhisker | 7,401 |
{A} `{Is1Cat A} {a b c : A} {f g : b $-> c} (p : f $== g) (h : a $-> b) : f $o h $== g $o h := fmap (cat_precomp c h) p. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | cat_prewhisker | 7,402 |
{A} `{Is1Cat A} {a b c : A} {f g : a $-> b} {h k : b $-> c} (p : f $== g) (q : h $== k ) : h $o f $== k $o g := (q $@R _) $@ (_ $@L p). | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | cat_comp2 | 7,403 |
{A} `{Is1Cat A} {b c: A} (f : b $-> c) := forall a (g h : a $-> b), f $o g $== f $o h -> g $== h. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | Monic | 7,404 |
{A} `{Is1Cat A} {a b : A} (f : a $-> b) := forall c (g h : b $-> c), g $o f $== h $o f -> g $== h. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | Epic | 7,405 |
{A} `{Is1Cat A} {a b : A} (f : a $-> b) := | Record | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | SectionOf | 7,406 |
{A} `{Is1Cat A} {a b : A} (f : a $-> b) := | Record | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | RetractionOf | 7,407 |
{A : Type} `{Is1Cat A} (x : A) := forall (y : A), {f : x $-> y & forall g, f $== g}. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | IsInitial | 7,408 |
{A : Type} `{Is1Cat A} (x y : A) {h : IsInitial x} : x $-> y := (h y). | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | mor_initial | 7,409 |
{A : Type} `{Is1Cat A} (x y : A) {h : IsInitial x} (f : x $-> y) : mor_initial x y $== f := (h y). | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | mor_initial_unique | 7,410 |
{A : Type} `{Is1Cat A} (y : A) := forall (x : A), {f : x $-> y & forall g, f $== g}. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | IsTerminal | 7,411 |
{A : Type} `{Is1Cat A} (x y : A) {h : IsTerminal y} : x $-> y := (h x). | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | mor_terminal | 7,412 |
{A : Type} `{Is1Cat A} (x y : A) {h : IsTerminal y} (f : x $-> y) : mor_terminal x y $== f := (h x). | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | mor_terminal_unique | 7,413 |
{A} `{HasMorExt A} {a b : A} {f g : a $-> b} (p : f $== g) : f = g := GpdHom_path^-1 p. Global Instance is1cat_strong_hasmorext {A : Type} `{HasMorExt A} : Is1Cat_Strong A. Proof. rapply Build_Is1Cat_Strong; hnf; intros; apply . + apply cat_assoc. + apply cat_assoc_opp. + apply cat_idl. + apply cat_idr. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | path_hom | 7,414 |
{A} `{Is1Gpd A} {a b c : A} (f : b $-> c) (g : a $-> b) : f^$ $o (f $o g) $== g := (cat_assoc _ _ _)^$ $@ (gpd_issect f $@R g) $@ cat_idl g. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_V_hh | 7,415 |
{A} `{Is1Gpd A} {a b c : A} (f : c $-> b) (g : a $-> b) : f $o (f^$ $o g) $== g := (cat_assoc _ _ _)^$ $@ (gpd_isretr f $@R g) $@ cat_idl g. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_h_Vh | 7,416 |
{A} `{Is1Gpd A} {a b c : A} (f : b $-> c) (g : a $-> b) : (f $o g) $o g^$ $== f := cat_assoc _ _ _ $@ (f $@L gpd_isretr g) $@ cat_idr f. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_hh_V | 7,417 |
{A} `{Is1Gpd A} {a b c : A} (f : b $-> c) (g : b $-> a) : (f $o g^$) $o g $== f := cat_assoc _ _ _ $@ (f $@L gpd_issect g) $@ cat_idr f. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_hV_h | 7,418 |
{A} `{Is1Gpd A} {x y : A} {p q : x $-> y} (r : p $o q^$ $== Id _) : p $== q. Proof. refine ((cat_idr p)^$ $@ (p $@L (gpd_issect q)^$) $@ (cat_assoc _ _ _)^$ $@ _). refine ((r $@R q) $@ cat_idl q). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_moveL_1M | 7,419 |
{A} `{Is1Gpd A} {x y : A} {p : x $-> y} {q : y $-> x} (r : Id _ $== p $o q) : p^$ $== q. Proof. refine ((cat_idr p^$)^$ $@ (p^$ $@L r) $@ _). apply gpd_V_hh. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_moveR_V1 | 7,420 |
{A : Type} `{Is1Gpd A} {x y : A} {p q : x $-> y} (r : Id _ $== p^$ $o q) : p $== q. Proof. refine (_ $@ (cat_assoc _ _ _)^$ $@ ((gpd_isretr p) $@R q) $@ (cat_idl q)). exact ((cat_idr p)^$ $@ (p $@L r)). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_moveR_M1 | 7,421 |
{A : Type} `{Is1Gpd A} {x y : A} {p q : x $-> y} (r : Id _ $== q $o p^$) : p $== q. Proof. refine ((cat_idl p)^$ $@ _ $@ cat_idr q). refine (_ $@ cat_assoc _ _ _ $@ (q $@L (gpd_issect p)^$)^$). exact (r $@R p). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_moveR_1M | 7,422 |
{A : Type} `{Is1Gpd A} {x y : A} {p : x $-> y} {q : y $-> x} (r : p $o q $== Id _) : p $== q^$. Proof. refine (_ $@ (cat_idl q^$)). refine (_ $@ (r $@R q^$)). exact (gpd_hh_V _ _)^$. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_moveL_1V | 7,423 |
{A : Type} `{Is1Gpd A} {x y z : A} {p : y $-> z} {q : x $-> y} {r : x $-> z} (s : r $== p $o q) : r $o q^$ $== p := (s $@R q^$) $@ gpd_hh_V _ _. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_moveR_hV | 7,424 |
{A : Type} `{Is1Gpd A} {x y z : A} {p : y $-> z} {q : x $-> y} {r : x $-> z} (s : r $== p $o q) : p^$ $o r $== q := (p^$ $@L s) $@ gpd_V_hh _ _. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_moveR_Vh | 7,425 |
{A : Type} `{Is1Gpd A} {x y z : A} {p : y $-> z} {q : x $-> y} {r : x $-> z} (s : r $o q^$ $== p) : r $== p $o q := ((gpd_hV_h _ _)^$ $@ (s $@R _)). | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_moveL_hM | 7,426 |
{A : Type} `{Is1Gpd A} {x y z : A} {p : y $-> z} {q : x $-> y} {r : x $-> z} (s : p $o q $== r) : p $== r $o q^$ := (gpd_moveR_hV s^$)^$. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_moveL_hV | 7,427 |
{A : Type} `{Is1Gpd A} {x y z : A} {p : y $-> z} {q : x $-> y} {r : x $-> z} (s : p^$ $o r $== q) : r $== p $o q := ((gpd_h_Vh _ _)^$ $@ (p $@L s)). | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_moveL_Mh | 7,428 |
{A : Type} `{Is1Gpd A} {x y z : A} {p : y $-> z} {q : x $-> y} {r : x $-> z} (s : p $o q $== r) : q $== p^$ $o r := (gpd_moveR_Vh s^$)^$. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_moveL_Vh | 7,429 |
{A : Type} `{Is1Gpd A} {x y : A} {p q : x $-> y} (r : p $== q) : p^$ $== q^$. Proof. apply gpd_moveR_V1. apply gpd_moveL_hV. exact (cat_idl q $@ r^$). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_rev2 | 7,430 |
{A} `{Is1Gpd A} {a b c : A} (f : b $-> c) (g : a $-> b) : (f $o g)^$ $== g^$ $o f^$. Proof. apply gpd_moveR_V1. refine (_ $@ cat_assoc _ _ _). apply gpd_moveL_hV. refine (cat_idl _ $@ _). exact (gpd_hh_V _ _)^$. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_rev_pp | 7,431 |
{A} `{Is1Gpd A} {a : A} : (Id a)^$ $== Id a. Proof. refine ((gpd_rev2 (gpd_issect (Id a)))^$ $@ _). refine (gpd_rev_pp _ _ $@ _). apply gpd_isretr. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_rev_1 | 7,432 |
{A} `{Is1Gpd A} {a0 a1 : A} (g : a0 $== a1) : (g^$)^$ $== g. Proof. apply gpd_moveR_V1. exact (gpd_issect _)^$. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_rev_rev | 7,433 |
{A B} `{Is1Gpd A, Is1Gpd B} (F : A -> B) `{!Is0Functor F, !Is1Functor F} {a0 a1 : A} (f : a0 $== a1) : fmap F f^$ $== (fmap F f)^$. Proof. apply gpd_moveL_1V. refine ((fmap_comp _ _ _)^$ $@ _ $@ fmap_id _ _). rapply fmap2. apply gpd_issect. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_1functor_V | 7,434 |
{A} `{Is1Gpd A, !HasMorExt A} {a b c : A} (f : b $-> c) (g : a $-> b) : f^$ $o (f $o g) = g := path_hom (gpd_V_hh f g). | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_strong_V_hh | 7,435 |
{A} `{Is1Gpd A, !HasMorExt A} {a b c : A} (f : c $-> b) (g : a $-> b) : f $o (f^$ $o g) = g := path_hom (gpd_h_Vh f g). | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_strong_h_Vh | 7,436 |
{A} `{Is1Gpd A, !HasMorExt A} {a b c : A} (f : b $-> c) (g : a $-> b) : (f $o g) $o g^$ = f := path_hom (gpd_hh_V f g). | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_strong_hh_V | 7,437 |
{A} `{Is1Gpd A, !HasMorExt A} {a b c : A} (f : b $-> c) (g : b $-> a) : (f $o g^$) $o g = f := path_hom (gpd_hV_h f g). | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_strong_hV_h | 7,438 |
{A} `{Is1Gpd A, !HasMorExt A} {a b c : A} (f : b $-> c) (g : a $-> b) : (f $o g)^$ = g^$ $o f^$ := path_hom (gpd_rev_pp f g). | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_strong_rev_pp | 7,439 |
{A} `{Is1Gpd A, !HasMorExt A} {a : A} : (Id a)^$ = Id a := path_hom gpd_rev_1. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_strong_rev_1 | 7,440 |
{A} `{Is1Gpd A, !HasMorExt A} {a0 a1 : A} (g : a0 $== a1) : (g^$)^$ = g := path_hom (gpd_rev_rev g). | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_strong_rev_rev | 7,441 |
{A B} `{Is1Cat A, Is1Cat B, !HasMorExt B} (F : A -> B) `{!Is0Functor F, !Is1Functor F} (a : A) : fmap F (Id a) = Id (F a) := path_hom (fmap_id F a). | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | fmap_id_strong | 7,442 |
{A B} `{Is1Gpd A, Is1Gpd B, !HasMorExt B} (F : A -> B) `{!Is0Functor F, !Is1Functor F} {a0 a1 : A} (f : a0 $== a1) : fmap F f^$ = (fmap F f)^$ := path_hom (gpd_1functor_V F f). | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | gpd_strong_1functor_V | 7,443 |
{A B : Type} (F : A -> B) `{PreservesInitial A B F} (x y : A) (h : IsInitial x) : fmap F (mor_initial x y) $== mor_initial (F x) (F y). Proof. exact (mor_initial_unique _ _ _)^$. Defined. | Lemma | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | fmap_initial | 7,444 |
{A B : Type} (F : A -> B) `{PreservesTerminal A B F} (x y : A) (h : IsTerminal y) : fmap F (mor_terminal x y) $== mor_terminal (F x) (F y). Proof. exact (mor_terminal_unique _ _ _)^$. Defined. | Lemma | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | fmap_terminal | 7,445 |
(B C : Type) | Record | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | BasepointPreservingFunctor | 7,446 |
{B C D : Type} `{Is01Cat B, Is01Cat C, Is01Cat D} `{IsPointed B, IsPointed C, IsPointed D} (F : B -->* C) (G : C -->* D) : B -->* D. Proof. snrapply Build_BasepointPreservingFunctor. - exact (G o F). - exact _. - exact (bp_pointed G $o fmap G (bp_pointed F)). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. | WildCat\Core.v | basepointpreservingfunctor_compose | 7,447 |
{A : Type} {D : A -> Type} `{IsD01Cat A D} {a b c : A} {g : b $-> c} {a' : D a} {b' : D b} {c' : D c} (g' : DHom g b' c') : forall (f : a $-> b), DHom f a' b' -> DHom (g $o f) a' c' := fun _ f' => g' $o' f'. | Definition | Require Import Basics.Overture. Require Import Basics.PathGroupoids. Require Import Basics.Tactics. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Prod. | WildCat\Displayed.v | dcat_postcomp | 7,448 |
{A : Type} {D : A -> Type} `{IsD01Cat A D} {a b c : A} {f : a $-> b} {a' : D a} {b' : D b} {c' : D c} (f' : DHom f a' b') : forall (g : b $-> c), DHom g b' c' -> DHom (g $o f) a' c' := fun _ g' => g' $o' f'. | Definition | Require Import Basics.Overture. Require Import Basics.PathGroupoids. Require Import Basics.Tactics. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Prod. | WildCat\Displayed.v | dcat_precomp | 7,449 |
{A : Type} {D : A -> Type} `{IsD0Gpd A D} {a b : A} (f : a $== b) (a' : D a) (b' : D b) := DHom f a' b'. | Definition | Require Import Basics.Overture. Require Import Basics.PathGroupoids. Require Import Basics.Tactics. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Prod. | WildCat\Displayed.v | DGpdHom | 7,450 |
{A : Type} {D : A -> Type} `{IsD0Gpd A D} {a b c : A} {f : a $== b} {g : b $== c} {a' : D a} {b' : D b} {c' : D c} : DGpdHom f a' b' -> DGpdHom g b' c' -> DGpdHom (g $o f) a' c' := fun f' g' => g' $o' f'. | Definition | Require Import Basics.Overture. Require Import Basics.PathGroupoids. Require Import Basics.Tactics. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Prod. | WildCat\Displayed.v | dgpd_comp | 7,451 |
{A : Type} {D : A -> Type} `{IsD01Cat A D} {a b : A} (p : a = b) {a' : D a} {b': D b} (p' : transport D p a' = b') : DHom (Hom_path p) a' b'. Proof. destruct p, p'; apply DId. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.PathGroupoids. Require Import Basics.Tactics. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Prod. | WildCat\Displayed.v | DHom_path | 7,452 |
{A : Type} {D : A -> Type} `{IsD0Gpd A D} {a b : A} (p : a = b) {a' : D a} {b': D b} (p' : transport D p a' = b') : DGpdHom (GpdHom_path p) a' b' := DHom_path p p'. | Definition | Require Import Basics.Overture. Require Import Basics.PathGroupoids. Require Import Basics.Tactics. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Prod. | WildCat\Displayed.v | DGpdHom_path | 7,453 |
{A : Type} {D : A -> Type} `{IsD1Cat A D} {a b c : A} {f g : a $-> b} {h : b $-> c} {p : f $== g} {a' : D a} {b' : D b} {c' : D c} {f' : DHom f a' b'} {g' : DHom g a' b'} (h' : DHom h b' c') (p' : DHom p f' g') : DHom (h $@L p) (h' $o' f') (h' $o' g') := dfmap (cat_postcomp a h) (dcat_postcomp h') p'. | Definition | Require Import Basics.Overture. Require Import Basics.PathGroupoids. Require Import Basics.Tactics. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Prod. | WildCat\Displayed.v | dcat_postwhisker | 7,454 |
{A : Type} {D : A -> Type} `{IsD1Cat A D} {a b c : A} {f : a $-> b} {g h : b $-> c} {p : g $== h} {a' : D a} {b' : D b} {c' : D c} {g' : DHom g b' c'} {h' : DHom h b' c'} (p' : DHom p g' h') (f' : DHom f a' b') : DHom (p $@R f) (g' $o' f') (h' $o' f') := dfmap (cat_precomp c f) (dcat_precomp f') p'. | Definition | Require Import Basics.Overture. Require Import Basics.PathGroupoids. Require Import Basics.Tactics. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Prod. | WildCat\Displayed.v | dcat_prewhisker | 7,455 |
{A : Type} {D : A -> Type} `{IsD1Cat A D} {a b c : A} {f g : a $-> b} {h k : b $-> c} {p : f $== g} {q : h $== k} {a' : D a} {b' : D b} {c' : D c} {f' : DHom f a' b'} {g' : DHom g a' b'} {h' : DHom h b' c'} {k' : DHom k b' c'} (p' : DHom p f' g') (q' : DHom q h' k') : DHom (p $@@ q) (h' $o' f') (k' $o' g') := (k' $@L' p') $o' (q' $@R' f'). | Definition | Require Import Basics.Overture. Require Import Basics.PathGroupoids. Require Import Basics.Tactics. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Prod. | WildCat\Displayed.v | dcat_comp2 | 7,456 |
{A} {D : A -> Type} `{IsD1Cat A D} {b c : A} {f : b $-> c} {mon : Monic f} {b' : D b} {c' : D c} (f' : DHom f b' c') := forall (a : A) (g h : a $-> b) (p : f $o g $== f $o h) (a' : D a) (g' : DHom g a' b') (h' : DHom h a' b'), DGpdHom p (f' $o' g') (f' $o' h') -> DGpdHom (mon a g h p) g' h'. | Definition | Require Import Basics.Overture. Require Import Basics.PathGroupoids. Require Import Basics.Tactics. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Prod. | WildCat\Displayed.v | DMonic | 7,457 |
{A} {D : A -> Type} `{IsD1Cat A D} {a b : A} {f : a $-> b} {epi : Epic f} {a' : D a} {b' : D b} (f' : DHom f a' b') := forall (c : A) (g h : b $-> c) (p : g $o f $== h $o f) (c' : D c) (g' : DHom g b' c') (h' : DHom h b' c'), DGpdHom p (g' $o' f') (h' $o' f') -> DGpdHom (epi c g h p) g' h'. Global Instance isgraph_total {A : Type} (D : A -> Type) `{IsDGraph A D} : IsGraph (sig D). Proof. srapply Build_IsGraph. intros [a a'] [b b']. exact {f : a $-> b & DHom f a' b'}. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.PathGroupoids. Require Import Basics.Tactics. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Prod. | WildCat\Displayed.v | DEpic | 7,458 |
{A : Type} {D : A -> Type} `{IsD1Cat_Strong A D} {a b c d : A} {f : a $-> b} {g : b $-> c} {h : c $-> d} {a' : D a} {b' : D b} {c' : D c} {d' : D d} (f' : DHom f a' b') (g' : DHom g b' c') (h' : DHom h c' d') : (transport (fun k => DHom k a' d') (cat_assoc_opp_strong f g h) (h' $o' (g' $o' f'))) = (h' $o' g') $o' f'. Proof. apply (moveR_transport_V (fun k => DHom k a' d') (cat_assoc_strong f g h) _ _). exact ((dcat_assoc_strong f' g' h')^). Defined. | Definition | Require Import Basics.Overture. Require Import Basics.PathGroupoids. Require Import Basics.Tactics. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Prod. | WildCat\Displayed.v | dcat_assoc_opp_strong | 7,459 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {f : a $-> b} `{!CatIsEquiv f} {a' : D a} {b' : D b} (f' : DHom f a' b') {fe' : DCatIsEquiv f'} : DCatEquiv (Build_CatEquiv f) a' b' := dcate_buildequiv f' (fe':=fe'). | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | Build_DCatEquiv | 7,460 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {f : a $-> b} {g : b $-> a} {r : f $o g $== Id b} {s : g $o f $== Id a} {a'} {b'} (f' : DHom f a' b') (g' : DHom g b' a') (r' : DHom r (f' $o' g') (DId b')) (s' : DHom s (g' $o' f') (DId a')) : DCatEquiv (cate_adjointify f g r s) a' b' := Build_DCatEquiv f' (fe':=dcatie_adjointify f' g' r' s'). | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_adjointify | 7,461 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {f : a $<~> b} {a' : D a} {b' : D b} (f' : DCatEquiv f a' b') : DCatEquiv (f^-1$) b' a'. Proof. snrapply dcate_adjointify. - exact (' f'). - exact f'. - exact (dcate_issect' f'). - exact (dcate_isretr' f'). Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_inv | 7,462 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {f : a $<~> b} {a' : D a} {b' : D b} (f' : DCatEquiv f a' b') : DGpdHom (cate_issect f) (dcate_fun f'^-1$' $o' f') (DId a'). Proof. refine (_ $@' ' f'). refine (_ $@R' (dcate_fun f')). apply dcate_buildequiv_fun. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_issect | 7,463 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {f : a $<~> b} {a' : D a} {b' : D b} (f' : DCatEquiv f a' b') : DGpdHom (cate_isretr f) (dcate_fun f' $o' f'^-1$') (DId b'). Proof. refine (_ $@' ' f'). refine (dcate_fun f' $@L' _). apply dcate_buildequiv_fun. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_isretr | 7,464 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {f : a $<~> b} {g : b $-> a} {p : f $o g $== Id b} {a' : D a} {b' : D b} (f' : DCatEquiv f a' b') (g' : DHom g b' a') (p' : DGpdHom p (dcate_fun f' $o' g') (DId b')) : DGpdHom (cate_inverse_sect f g p) (dcate_fun f'^-1$') g'. Proof. refine ((dcat_idr _)^$' $@' _). refine ((_ $@L' p'^$') $@' _). 1: exact isd0gpd_hom. refine (dcat_assoc_opp _ _ _ $@' _). refine (dcate_issect f' $@R' _ $@' _). apply dcat_idl. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_inverse_sect | 7,465 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {f : a $<~> b} {g : b $-> a} {p : g $o f $== Id a} {a' : D a} {b' : D b} (f' : DCatEquiv f a' b') (g' : DHom g b' a') (p' : DGpdHom p (g' $o' f') (DId a')) : DGpdHom (cate_inverse_retr f g p) (dcate_fun f'^-1$') g'. Proof. refine ((dcat_idl _)^$' $@' _). refine ((p'^$' $@R' _) $@' _). 1: exact isd0gpd_hom. refine (dcat_assoc _ _ _ $@' _). refine (_ $@L' dcate_isretr f' $@' _). apply dcat_idr. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_inverse_retr | 7,466 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {f : a $-> b} {g : b $-> a} {r : f $o g $== Id b} {s : g $o f $== Id a} {a' : D a} {b' : D b} (f' : DHom f a' b') (g' : DHom g b' a') (r' : DGpdHom r (f' $o' g') (DId b')) (s' : DGpdHom s (g' $o' f') (DId a')) : DGpdHom (cate_inv_adjointify f g r s) (dcate_fun (dcate_adjointify f' g' r' s')^-1$') g'. Proof. apply dcate_inverse_sect. exact ((dcate_buildequiv_fun f' $@R' _) $@' r'). Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_inv_adjointify | 7,467 |
{A} {D : A -> Type} `{DHasEquivs A D} {a : A} (a' : D a) : DCatEquiv (id_cate a) a' a' := Build_DCatEquiv (DId a'). | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | did_cate | 7,468 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {f : a $-> b} `{!CatIsEquiv f} {g : a $-> b} {p : f $== g} {a' : D a} {b' : D b} (f' : DHom f a' b') `{fe' : !DCatIsEquiv f'} {g' : DHom g a' b'} (p' : DGpdHom p f' g') : DCatIsEquiv (fe:=catie_homotopic f p) g'. Proof. snrapply dcatie_adjointify. - exact (Build_DCatEquiv (fe':=fe') f')^-1$'. - refine (p'^$' $@R' _ $@' _). 1: exact isd0gpd_hom. refine ((dcate_buildequiv_fun f')^$' $@R' _ $@' _). 1: exact isd0gpd_hom. apply dcate_isretr. - refine (_ $@L' p'^$' $@' _). 1: exact isd0gpd_hom. refine (_ $@L' (dcate_buildequiv_fun f')^$' $@' _). 1: exact isd0gpd_hom. apply dcate_issect. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcatie_homotopic | 7,469 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b c : A} {g : b $<~> c} {f : a $<~> b} {a' : D a} {b' : D b} {c' : D c} (g' : DCatEquiv g b' c') (f' : DCatEquiv f a' b') : DCatEquiv (compose_cate g f) a' c' := Build_DCatEquiv (dcate_fun g' $o' f'). | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcompose_cate | 7,470 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b c : A} {g : b $<~> c} {f : a $<~> b} {a' : D a} {b' : D b} {c' : D c} (g' : DCatEquiv g b' c') (f' : DCatEquiv f a' b') : DGpdHom (compose_cate_fun g f) (dcate_fun (g' $oE' f')) (dcate_fun g' $o' f') := dcate_buildequiv_fun _. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcompose_cate_fun | 7,471 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b c : A} {g : b $<~> c} {f : a $<~> b} {a' : D a} {b' : D b} {c' : D c} (g' : DCatEquiv g b' c') (f' : DCatEquiv f a' b') : DGpdHom (compose_cate_funinv g f) (dcate_fun g' $o' f') (dcate_fun (g' $oE' f')). Proof. apply dgpd_rev. apply dcate_buildequiv_fun. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcompose_cate_funinv | 7,472 |
{A} {D : A -> Type} `{DHasEquivs A D} {a : A} (a' : D a) : DGpdHom (id_cate_fun a) (dcate_fun (did_cate a')) (DId a') := dcate_buildequiv_fun _. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | did_cate_fun | 7,473 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b c d : A} {f : a $<~> b} {g : b $<~> c} {h : c $<~> d} {a'} {b'} {c'} {d'} (f' : DCatEquiv f a' b') (g' : DCatEquiv g b' c') (h' : DCatEquiv h c' d') : DGpdHom (compose_cate_assoc f g h) (dcate_fun ((h' $oE' g') $oE' f')) (dcate_fun (h' $oE' (g' $oE' f'))). Proof. refine (dcompose_cate_fun _ f' $@' _ $@' dcat_assoc (dcate_fun f') g' h' $@' _ $@' dcompose_cate_funinv h' _). - apply (dcompose_cate_fun h' g' $@R' _). - apply (_ $@L' dcompose_cate_funinv g' f'). Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcompose_cate_assoc | 7,474 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {f : a $<~> b} {a' : D a} {b' : D b} (f' : DCatEquiv f a' b') : DGpdHom (compose_cate_idl f) (dcate_fun (did_cate b' $oE' f')) (dcate_fun f'). Proof. refine (dcompose_cate_fun _ f' $@' _ $@' dcat_idl (dcate_fun f')). apply (dcate_buildequiv_fun _ $@R' _). Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcompose_cate_idl | 7,475 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {f : a $<~> b} {a' : D a} {b' : D b} (f' : DCatEquiv f a' b') : DGpdHom (compose_cate_idr f) (dcate_fun (f' $oE' did_cate a')) (dcate_fun f'). Proof. refine (dcompose_cate_fun f' _ $@' _ $@' dcat_idr (dcate_fun f')). rapply (_ $@L' dcate_buildequiv_fun _). Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcompose_cate_idr | 7,476 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b c : A} {f : b $<~> c} {g : a $-> b} {a' : D a} {b' : D b} {c' : D c} (f' : DCatEquiv f b' c') (g' : DHom g a' b') : DGpdHom (compose_V_hh f g) (dcate_fun f'^-1$' $o' (dcate_fun f' $o' g')) g' := (dcat_assoc_opp _ _ _) $@' (dcate_issect f' $@R' g') $@' dcat_idl g'. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcompose_V_hh | 7,477 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b c : A} {f : c $<~> b} {g : a $-> b} {a' : D a} {b' : D b} {c' : D c} (f' : DCatEquiv f c' b') (g' : DHom g a' b') : DGpdHom (compose_h_Vh f g) (dcate_fun f' $o' (dcate_fun f'^-1$' $o' g')) g' := (dcat_assoc_opp _ _ _) $@' (dcate_isretr f' $@R' g') $@' dcat_idl g'. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcompose_h_Vh | 7,478 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b c : A} {f : b $-> c} {g : a $<~> b} {a' : D a} {b' : D b} {c' : D c} (f' : DHom f b' c') (g' : DCatEquiv g a' b') : DGpdHom (compose_hh_V f g) ((f' $o' g') $o' g'^-1$') f' := dcat_assoc _ _ _ $@' (f' $@L' dcate_isretr g') $@' dcat_idr f'. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcompose_hh_V | 7,479 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b c : A} {f : b $-> c} {g : b $<~> a} {a' : D a} {b' : D b} {c' : D c} (f' : DHom f b' c') (g' : DCatEquiv g b' a') : DGpdHom (compose_hV_h f g) ((f' $o' g'^-1$') $o' g') f' := dcat_assoc _ _ _ $@' (f' $@L' dcate_issect g') $@' dcat_idr f'. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcompose_hV_h | 7,480 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {e : a $<~> b} {a' : D a} {b' : D b} (e' : DCatEquiv e a' b') : DMonic (mon:=cate_monic_equiv e) (dcate_fun e'). Proof. intros c f g p c' f' g' p'. refine ((dcompose_V_hh e' _)^$' $@' _ $@' dcompose_V_hh e' _). 1: exact isd0gpd_hom. exact (_ $@L' p'). Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_monic_equiv | 7,481 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {e : a $<~> b} {a' : D a} {b' : D b} (e' : DCatEquiv e a' b') : DEpic (epi:=cate_epic_equiv e) (dcate_fun e'). Proof. intros c f g p c' f' g' p'. refine ((dcompose_hh_V _ e')^$' $@' _ $@' dcompose_hh_V _ e'). 1: exact isd0gpd_hom. exact (p' $@R' _). Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_epic_equiv | 7,482 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b c : A} {e : b $<~> a} {f : a $-> c} {g : b $-> c} {p : f $== g $o e^-1$} {a' : D a} {b' : D b} {c' : D c} (e' : DCatEquiv e b' a') (f' : DHom f a' c') (g' : DHom g b' c') (p' : DGpdHom p f' (g' $o' e'^-1$')) : DGpdHom (cate_moveR_eM e f g p) (f' $o' e') g'. Proof. apply (dcate_epic_equiv e'^-1$'). exact (dcompose_hh_V _ _ $@' p'). Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_moveR_eM | 7,483 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b c : A} {e : b $<~> c} {f : a $-> c} {g : a $-> b} {p : f $== e $o g} {a' : D a} {b' : D b} {c' : D c} (e' : DCatEquiv e b' c') (f' : DHom f a' c') (g' : DHom g a' b') (p' : DGpdHom p f' (dcate_fun e' $o' g')) : DGpdHom (cate_moveR_Ve e f g p) (dcate_fun e'^-1$' $o' f') g'. Proof. apply (dcate_monic_equiv e'). exact (dcompose_h_Vh _ _ $@' p'). Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_moveR_Ve | 7,484 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {e : a $<~> b} {f : b $-> a} {p : e $o f $== Id b} {a' : D a} {b' : D b} {e' : DCatEquiv e a' b'} (f' : DHom f b' a') (p' : DGpdHom p (dcate_fun e' $o' f') (DId b')) : DGpdHom (cate_moveL_V1 f p) f' (dcate_fun e'^-1$'). Proof. apply (dcate_monic_equiv e'). nrapply (p' $@' (dcate_isretr e')^$'). exact isd0gpd_hom. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_moveL_V1 | 7,485 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {e : a $<~> b} {f : b $-> a} {p : f $o e $== Id a} {a' : D a} {b' : D b} {e' : DCatEquiv e a' b'} (f' : DHom f b' a') (p' : DGpdHom p (f' $o' e') (DId a')) : DGpdHom (cate_moveL_1V f p) f' (dcate_fun e'^-1$'). Proof. apply (dcate_epic_equiv e'). nrapply (p' $@' (dcate_issect e')^$'). exact isd0gpd_hom. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_moveL_1V | 7,486 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {e : a $<~> b} {f : b $-> a} {p : Id b $== e $o f} {a' : D a} {b' : D b} {e' : DCatEquiv e a' b'} (f' : DHom f b' a') (p' : DGpdHom p (DId b') (dcate_fun e' $o' f')) : DGpdHom (cate_moveR_V1 f p) (dcate_fun e'^-1$') f'. Proof. apply (dcate_monic_equiv e'). exact (dcate_isretr e' $@' p'). Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_moveR_V1 | 7,487 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {e : a $<~> b} {f : b $-> a} {p : Id a $== f $o e} {a' : D a} {b' : D b} {e' : DCatEquiv e a' b'} (f' : DHom f b' a') (p' : DGpdHom p (DId a') (f' $o' e')) : DGpdHom (cate_moveR_1V f p) (dcate_fun e'^-1$') f'. Proof. apply (dcate_epic_equiv e'). exact (dcate_issect e' $@' p'). Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_moveR_1V | 7,488 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {e f : a $<~> b} {p : cate_fun e $== cate_fun f} {a' : D a} {b' : D b} {e' : DCatEquiv e a' b'} {f' : DCatEquiv f a' b'} (p' : DGpdHom p (dcate_fun e') (dcate_fun f')) : DGpdHom (cate_inv2 p) (dcate_fun e'^-1$') (dcate_fun f'^-1$'). Proof. apply dcate_moveL_V1. rapply ((p'^$' $@R' _) $@' dcate_isretr _). exact isd0gpd_hom. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_inv2 | 7,489 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b c : A} {e : a $<~> b} {f : b $<~> c} {a' : D a} {b' : D b} {c' : D c} (e' : DCatEquiv e a' b') (f' : DCatEquiv f b' c') : DGpdHom (cate_inv_compose e f) (dcate_fun (f' $oE' e')^-1$') (dcate_fun (e'^-1$' $oE' f'^-1$')). Proof. refine (_ $@' (dcompose_cate_fun e'^-1$' f'^-1$')^$'). - snrapply dcate_inv_adjointify. - exact isd0gpd_hom. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_inv_compose | 7,490 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} {e : a $<~> b} {a' : D a} {b' : D b} (e' : DCatEquiv e a' b') : DGpdHom (cate_inv_V e) (dcate_fun (e'^-1$')^-1$') (dcate_fun e'). Proof. apply dcate_moveR_V1. apply dgpd_rev. apply dcate_issect. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcate_inv_V | 7,491 |
{A B : Type} {DA : A -> Type} `{DHasEquivs A DA} {DB : B -> Type} `{DHasEquivs B DB} (F : A -> B) `{!Is0Functor F, !Is1Functor F} (F' : forall (a : A), DA a -> DB (F a)) `{!IsD0Functor F F', !IsD1Functor F F'} {a b : A} {f : a $<~> b} {a' : DA a} {b' : DA b} (f' : DCatEquiv f a' b') : DCatEquiv (emap F f) (F' a a') (F' b b') := Build_DCatEquiv (dfmap F F' (dcate_fun f')). | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | demap | 7,492 |
{A B : Type} {DA : A -> Type} `{DHasEquivs A DA} {DB : B -> Type} `{DHasEquivs B DB} (F : A -> B) `{!Is0Functor F, !Is1Functor F} (F' : forall (a : A), DA a -> DB (F a)) `{!IsD0Functor F F', !IsD1Functor F F'} {a : A} {a' : DA a} : DGpdHom (emap_id F) (dcate_fun (demap F F' (did_cate a'))) (dcate_fun (did_cate (F' a a'))). Proof. refine (dcate_buildequiv_fun _ $@' _). refine (dfmap2 F F' (did_cate_fun a') $@' _ $@' _). - rapply dfmap_id. - apply dgpd_rev. exact (did_cate_fun (F' a a')). Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | demap_id | 7,493 |
{A B : Type} {DA : A -> Type} `{DHasEquivs A DA} {DB : B -> Type} `{DHasEquivs B DB} (F : A -> B) `{!Is0Functor F, !Is1Functor F} (F' : forall (a : A), DA a -> DB (F a)) `{!IsD0Functor F F', isd1f : !IsD1Functor F F'} {a b c : A} {f : a $<~> b} {g : b $<~> c} {a' : DA a} {b' : DA b} {c' : DA c} (f' : DCatEquiv f a' b') (g' : DCatEquiv g b' c') : DGpdHom (emap_compose F f g) (dcate_fun (demap F F' (g' $oE' f'))) (dfmap F F' (dcate_fun g') $o' dfmap F F' (dcate_fun f')). Proof. refine (dcate_buildequiv_fun _ $@' _). refine (dfmap2 F F' (dcompose_cate_fun _ _) $@' _). nrapply dfmap_comp; exact isd1f. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | demap_compose | 7,494 |
demap_compose' {A B : Type} {DA : A -> Type} `{DHasEquivs A DA} {DB : B -> Type} `{DHasEquivs B DB} (F : A -> B) `{!Is0Functor F, !Is1Functor F} (F' : forall (a : A), DA a -> DB (F a)) `{!IsD0Functor F F', !IsD1Functor F F'} {a b c : A} {f : a $<~> b} {g : b $<~> c} {a' : DA a} {b' : DA b} {c' : DA c} (f' : DCatEquiv f a' b') (g' : DCatEquiv g b' c') : DGpdHom (emap_compose' F f g) (dcate_fun (demap F F' (g' $oE' f'))) (dcate_fun ((demap F F' g') $oE' (demap F F' f'))). Proof. refine (demap_compose F F' f' g' $@' _). apply dgpd_rev. refine (dcompose_cate_fun _ _ $@' _). exact (dcate_buildequiv_fun _ $@@' dcate_buildequiv_fun _). Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | demap_compose' | 7,495 |
{A B : Type} {DA : A -> Type} `{DHasEquivs A DA} {DB : B -> Type} `{DHasEquivs B DB} (F : A -> B) `{!Is0Functor F, !Is1Functor F} (F' : forall (a : A), DA a -> DB (F a)) `{!IsD0Functor F F', !IsD1Functor F F'} {a b : A} {e : a $<~> b} {a' : DA a} {b' : DA b} (e' : DCatEquiv e a' b') : DGpdHom (emap_inv F e) (dcate_fun (demap F F' e')^-1$') (dcate_fun (demap F F' e'^-1$')). Proof. refine (dcate_inv_adjointify _ _ _ _ $@' _). apply dgpd_rev. exact (dcate_buildequiv_fun _). Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | demap_inv | 7,496 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} (p : a = b) (a' : D a) (b' : D b) : transport D p a' = b' -> DCatEquiv (cat_equiv_path a b p) a' b'. Proof. intro p'. destruct p, p'. reflexivity. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcat_equiv_path | 7,497 |
{A} {D : A -> Type} `{IsDUnivalent1Cat A D} {a b : A} (p : a = b) (a' : D a) (b' : D b) : DCatEquiv (cat_equiv_path a b p) a' b' -> transport D p a' = b' := (dcat_equiv_path p a' b')^-1. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcat_path_equiv | 7,498 |
{A} {D : A -> Type} `{DHasEquivs A D} {a b : A} (a' : D a) (b' : D b) : {p : a = b & p # a' = b'} -> {e : a $<~> b & DCatEquiv e a' b'} := functor_sigma (cat_equiv_path a b) (fun p => dcat_equiv_path p a' b'). Global Instance isunivalent1cat_total {A} `{IsUnivalent1Cat A} (D : A -> Type) `{!IsDGraph D, !IsD2Graph D, !IsD01Cat D, !IsD1Cat D, !DHasEquivs D} `{!IsDUnivalent1Cat D} : IsUnivalent1Cat (sig D). Proof. snrapply Build_IsUnivalent1Cat. intros aa' bb'. apply (isequiv_homotopic ( _ _ o (path_sigma_uncurried D aa' bb')^-1)). intros []; reflexivity. Defined. | Definition | Require Import Basics.Overture. Require Import Basics.Tactics. Require Import Basics.Equivalences. Require Import Types.Sigma. Require Import WildCat.Core. Require Import WildCat.Displayed. Require Import WildCat.Equiv. | WildCat\DisplayedEquiv.v | dcat_equiv_path_total | 7,499 |