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{A : Type} `{HasEquivs A} `{!HasMorExt A} {a b : A} : (b $<~> a) -> (opyon1 a $<~> opyon1 b). Proof. intro e. snrapply Build_NatEquiv. - intros c. exact (equiv_precompose_cat_equiv e). - rapply is1natural_opyoneda. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | natequiv_opyon_equiv | 7,800 |
{A : Type} `{Is1Cat A} (a : A) : A -> ZeroGpd := fun b => Build_ZeroGpd (a $-> b) _ _ _. Global Instance is0functor_hom_0gpd {A : Type} `{Is1Cat A} : Is0Functor (A:=A^op*A) (B:=ZeroGpd) (uncurry ( (A:=A))). Proof. nrapply Build_Is0Functor. intros [a1 a2] [b1 b2] [f1 f2]; unfold op in *; cbn in *. rapply (Build_Morphism_0Gpd ( a1 a2) ( b1 b2) (cat_postcomp b1 f2 o cat_precomp a2 f1)). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyon_0gpd | 7,801 |
{A : Type} `{Is1Cat A} (a : A) (F : A -> ZeroGpd) `{!Is0Functor F, !Is1Functor F} : F a -> (opyon_0gpd a $=> F). Proof. intros x b. refine (Build_Morphism_0Gpd (opyon_0gpd a b) (F b) (fun f => fmap F f x) _). rapply Build_Is0Functor. intros f1 f2 h. exact (fmap2 F h x). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyoneda_0gpd | 7,802 |
{A : Type} `{Is1Cat A} (a : A) (F : A -> ZeroGpd) {ff : Is0Functor F} : (opyon_0gpd a $=> F) -> F a := fun alpha => alpha a (Id a). Global Instance is1natural_opyoneda_0gpd {A : Type} `{Is1Cat A} (a : A) (F : A -> ZeroGpd) `{!Is0Functor F, !Is1Functor F} (x : F a) : Is1Natural (opyon_0gpd a) F (opyoneda_0gpd a F x). Proof. snrapply Build_Is1Natural. unfold opyon_0gpd, opyoneda_0gpd; intros b c f g; cbn in *. exact (fmap_comp F g f x). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | un_opyoneda_0gpd | 7,803 |
{A : Type} `{Is1Cat A} (a : A) (F : A -> ZeroGpd) `{!Is0Functor F, !Is1Functor F} (x x' : F a) (p : forall b : A, opyoneda_0gpd a F x b $== opyoneda_0gpd a F x' b) : x $== x'. Proof. refine ((fmap_id F a x)^$ $@ _ $@ fmap_id F a x'). cbn in p. exact (p a (Id a)). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyoneda_isinj_0gpd | 7,804 |
{A : Type} `{Is1Cat A} (a b : A) (f g : b $-> a) (p : forall (c : A) (h : a $-> c), h $o f $== h $o g) : f $== g := opyoneda_isinj_0gpd a _ f g p. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyon_faithful_0gpd | 7,805 |
{A : Type} `{Is1Cat A} (a : A) (F : A -> ZeroGpd) `{!Is0Functor F, !Is1Functor F} (x : F a) : un_opyoneda_0gpd a F (opyoneda_0gpd a F x) $== x := fmap_id F a x. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyoneda_issect_0gpd | 7,806 |
{A : Type} `{Is1Cat A} (a : A) (F : A -> ZeroGpd) `{!Is0Functor F, !Is1Functor F} (alpha : opyon_0gpd a $=> F) {alnat : Is1Natural (opyon_0gpd a) F alpha} (b : A) : opyoneda_0gpd a F (un_opyoneda_0gpd a F alpha) b $== alpha b. Proof. unfold opyoneda, un_opyoneda, opyon; intros f. refine ((isnat alpha f (Id a))^$ $@ _). cbn. apply (fmap (alpha b)). exact (cat_idr f). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyoneda_isretr_0gpd | 7,807 |
{A : Type} `{Is1Cat A} (a b : A) : (opyon_0gpd a $=> opyon_0gpd b) -> (b $-> a) := un_opyoneda_0gpd a (opyon_0gpd b). | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyon_cancel_0gpd | 7,808 |
{A : Type} `{Is1Cat A} (a : A) : Fun11 A ZeroGpd := Build_Fun11 _ _ (opyon_0gpd a). | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyon1_0gpd | 7,809 |
{A : Type} `{HasEquivs A} {a b : A} (f : opyon1_0gpd a $<~> opyon1_0gpd b) : b $<~> a. Proof. set (fa := (cate_fun f a) (Id a)). set (gb := (cate_fun f^-1$ b) (Id b)). srapply (cate_adjointify fa gb). - exact (opyoneda_isretr_0gpd _ _ f^-1$ a fa $@ cat_eissect (f a) (Id a)). - exact (opyoneda_isretr_0gpd _ _ f b gb $@ cat_eisretr (f b) (Id b)). Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | opyon_equiv_0gpd | 7,810 |
{A : Type} `{HasEquivs A} {x y z : A} (f : y $<~> z) : opyon_0gpd x y $<~> opyon_0gpd x z := emap (opyon_0gpd x) f. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | equiv_postcompose_cat_equiv_0gpd | 7,811 |
{A : Type} `{HasEquivs A} {x y z : A} (f : x $<~> y) : opyon_0gpd y z $<~> opyon_0gpd x z := @equiv_postcompose_cat_equiv_0gpd A^op _ _ _ _ _ z y x f. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | equiv_precompose_cat_equiv_0gpd | 7,812 |
{A : Type} `{HasEquivs A} {a b : A} (e : b $<~> a) : opyon1_0gpd a $<~> opyon1_0gpd b. Proof. snrapply Build_NatEquiv. - intro c; exact (equiv_precompose_cat_equiv_0gpd e). - rapply is1natural_opyoneda_0gpd. Defined. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | natequiv_opyon_equiv_0gpd | 7,813 |
{A : Type} `{IsGraph A} (a : A) : A^op -> Type := opyon (A:=A^op) a. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yon | 7,814 |
{A : Type} `{Is01Cat A} (a : A) (F : A^op -> Type) `{!Is0Functor F} : F a -> (yon a $=> F) := @opyoneda (A^op) _ _ a F _. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yoneda | 7,815 |
{A : Type} `{Is01Cat A} (a : A) (F : A^op -> Type) `{!Is0Functor F} : (yon a $=> F) -> F a := un_opyoneda (A:=A^op) a F. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | un_yoneda | 7,816 |
{A : Type} `{Is1Cat A} (a : A) (F : A^op -> Type) `{!Is0Functor F, !Is1Functor F} (x x' : F a) (p : forall b, yoneda a F x b == yoneda a F x' b) : x = x' := opyoneda_isinj (A:=A^op) a F x x' p. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yoneda_isinj | 7,817 |
{A : Type} `{Is1Cat_Strong A} (a b : A) (f g : b $-> a) (p : forall (c : A) (h : c $-> b), f $o h = g $o h) : f = g := opyon_faithful (A:=A^op) b a f g p. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yon_faithful | 7,818 |
{A : Type} `{Is1Cat A} (a : A) (F : A^op -> Type) `{!Is0Functor F, !Is1Functor F} (x : F a) : un_yoneda a F (yoneda a F x) = x := opyoneda_issect (A:=A^op) a F x. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yoneda_issect | 7,819 |
{A : Type} `{Is1Cat_Strong A} (a : A) (F : A^op -> Type) `{!Is0Functor F} (* Without the hint here, Coq guesses to first project from [Is1Cat_Strong A] and then pass to opposites, whereas what we need is to first pass to opposites and then project. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yoneda_isretr | 7,820 |
{A : Type} `{Is01Cat A} (a b : A) : (yon a $=> yon b) -> (a $-> b) := un_yoneda a (yon b). | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yon_cancel | 7,821 |
{A : Type} `{Is01Cat A} (a : A) : Fun01 A^op Type := opyon1 (A:=A^op) a. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yon1 | 7,822 |
{A : Type} `{Is1Cat A} `{!HasMorExt A} (a : A) : Fun11 A^op Type := opyon11 (A:=A^op) a. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yon11 | 7,823 |
{A : Type} `{HasEquivs A} `{!Is1Cat_Strong A} (a b : A) : (yon1 a $<~> yon1 b) -> (a $<~> b) := opyon_equiv (A:=A^op). | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yon_equiv | 7,824 |
{A : Type} `{HasEquivs A} `{!HasMorExt A} (a b : A) : (a $<~> b) -> (yon1 a $<~> yon1 b) := natequiv_opyon_equiv (A:=A^op). | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | natequiv_yon_equiv | 7,825 |
{A : Type} `{Is1Cat A} (a : A) : A^op -> ZeroGpd := opyon_0gpd (A:=A^op) a. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yon_0gpd | 7,826 |
{A : Type} `{Is1Cat A} (a : A) (F : A^op -> ZeroGpd) `{!Is0Functor F, !Is1Functor F} : F a -> (yon_0gpd a $=> F) := opyoneda_0gpd (A:=A^op) a F. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yoneda_0gpd | 7,827 |
{A : Type} `{Is1Cat A} (a : A) (F : A^op -> ZeroGpd) {ff : Is0Functor F} : (yon_0gpd a $=> F) -> F a := un_opyoneda_0gpd (A:=A^op) a F. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | un_yoneda_0gpd | 7,828 |
{A : Type} `{Is1Cat A} (a : A) (F : A^op -> ZeroGpd) `{!Is0Functor F, !Is1Functor F} (x x' : F a) (p : forall b : A, yoneda_0gpd a F x b $== yoneda_0gpd a F x' b) : x $== x' := opyoneda_isinj_0gpd (A:=A^op) a F x x' p. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yoneda_isinj_0gpd | 7,829 |
{A : Type} `{Is1Cat A} (a b : A) (f g : b $-> a) (p : forall (c : A) (h : c $-> b), f $o h $== g $o h) : f $== g := opyon_faithful_0gpd (A:=A^op) b a f g p. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yon_faithful_0gpd | 7,830 |
{A : Type} `{Is1Cat A} (a : A) (F : A^op -> ZeroGpd) `{!Is0Functor F, !Is1Functor F} (x : F a) : un_yoneda_0gpd a F (yoneda_0gpd a F x) $== x := opyoneda_issect_0gpd (A:=A^op) a F x. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yoneda_issect_0gpd | 7,831 |
{A : Type} `{Is1Cat A} (a : A) (F : A^op -> ZeroGpd) `{!Is0Functor F, !Is1Functor F} (alpha : yon_0gpd a $=> F) {alnat : Is1Natural (yon_0gpd a) F alpha} (b : A) : yoneda_0gpd a F (un_yoneda_0gpd a F alpha) b $== alpha b := opyoneda_isretr_0gpd (A:=A^op) a F alpha b. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yoneda_isretr_0gpd | 7,832 |
{A : Type} `{Is1Cat A} (a b : A) : (yon_0gpd a $=> yon_0gpd b) -> (a $-> b) := opyon_cancel_0gpd (A:=A^op) a b. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yon_cancel_0gpd | 7,833 |
{A : Type} `{Is1Cat A} (a : A) : Fun11 A^op ZeroGpd := opyon1_0gpd (A:=A^op) a. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yon1_0gpd | 7,834 |
{A : Type} `{HasEquivs A} {a b : A} (f : yon1_0gpd a $<~> yon1_0gpd b) : a $<~> b := opyon_equiv_0gpd (A:=A^op) f. | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | yon_equiv_0gpd | 7,835 |
{A : Type} `{HasEquivs A} {a b : A} (e : a $<~> b) : yon1_0gpd (A:=A) a $<~> yon1_0gpd b := natequiv_opyon_equiv_0gpd (A:=A^op) (e : CatEquiv (A:=A^op) b a). | Definition | Require Import Basics.Overture Basics.Tactics. Require Import WildCat.Core. Require Import WildCat.Equiv. Require Import WildCat.Universe. Require Import WildCat.Opposite. Require Import WildCat.FunctorCat. Require Import WildCat.NatTrans. Require Import WildCat.Prod. Require Import WildCat.Bifunctor. Require Import WildCat.ZeroGroupoid. | WildCat\Yoneda.v | natequiv_yon_equiv_0gpd | 7,836 |
{ carrier :> Type; isgraph_carrier : IsGraph carrier; is01cat_carrier : Is01Cat carrier; is0gpd_carrier : Is0Gpd carrier; }. | Record | Require Import Basics.Overture Basics.Tactics Require Import WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\ZeroGroupoid.v | ZeroGpd | 7,837 |
(G H : ZeroGpd) := { fun_0gpd :> carrier G -> carrier H; is0functor_fun_0gpd : Is0Functor fun_0gpd; }. | Record | Require Import Basics.Overture Basics.Tactics Require Import WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\ZeroGroupoid.v | Morphism_0Gpd | 7,838 |
{G H : ZeroGpd} (f : G $<~> H) : G -> H := fun_0gpd _ _ (cat_equiv_fun _ _ _ f). | Definition | Require Import Basics.Overture Basics.Tactics Require Import WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\ZeroGroupoid.v | equiv_fun_0gpd | 7,839 |
{G H : ZeroGpd} (f : G $<~> H) {x y : G} (h : equiv_fun_0gpd f x $== equiv_fun_0gpd f y) : x $== y. Proof. exact ((cat_eissect f x)^$ $@ fmap (equiv_fun_0gpd f^-1$) h $@ cat_eissect f y). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Require Import WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\ZeroGroupoid.v | isinj_equiv_0gpd | 7,840 |
{G H : ZeroGpd} (f : G $<~> H) (x : H) (y : G) (p : x $== equiv_fun_0gpd f y) : equiv_fun_0gpd f^-1$ x $== y := fmap (equiv_fun_0gpd f^-1$) p $@ cat_eissect f y. | Definition | Require Import Basics.Overture Basics.Tactics Require Import WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\ZeroGroupoid.v | moveR_equiv_V_0gpd | 7,841 |
{G H : ZeroGpd} (f : G $<~> H) (x : H) (y : G) (p : equiv_fun_0gpd f y $== x) : y $== equiv_fun_0gpd f^-1$ x := (cat_eissect f y)^$ $@ fmap (equiv_fun_0gpd f^-1$) p. Global Instance issurjinj_equiv_0gpd {G H : ZeroGpd} (f : G $<~> H) : IsSurjInj (equiv_fun_0gpd f). Proof. econstructor. - intro y. exists (equiv_fun_0gpd f^-1$ y). rapply cat_eisretr. - apply isinj_equiv_0gpd. Defined. | Definition | Require Import Basics.Overture Basics.Tactics Require Import WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\ZeroGroupoid.v | moveL_equiv_V_0gpd | 7,842 |
{G H : ZeroGpd} (F : G $-> H) {e : IsSurjInj F} : Cat_IsBiInv F. Proof. destruct e as [e0 e1]; unfold SplEssSurj in e0. srapply catie_adjointify. - snrapply Build_Morphism_0Gpd. 1: exact (fun y => (e0 y).1). snrapply Build_Is0Functor; cbn beta. intros y1 y2 m. apply e1. exact ((e0 y1).2 $@ m $@ ((e0 y2).2)^$). - cbn. apply e0. - cbn. intro x. apply e1. apply e0. Defined. | Definition | Require Import Basics.Overture Basics.Tactics Require Import WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\ZeroGroupoid.v | isequiv_0gpd_issurjinj | 7,843 |
(I : Type) (G : I -> ZeroGpd) : ZeroGpd. Proof. rapply (Build_ZeroGpd (forall i, G i)). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Require Import WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\ZeroGroupoid.v | prod_0gpd | 7,844 |
{I : Type} {G : I -> ZeroGpd} : forall i, prod_0gpd I G $-> G i. Proof. intros i. snrapply Build_Morphism_0Gpd. 1: exact (fun f => f i). snrapply Build_Is0Functor; cbn beta. intros f g p. exact (p i). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Require Import WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\ZeroGroupoid.v | prod_0gpd_pr | 7,845 |
{I : Type} {G : ZeroGpd} {H : I -> ZeroGpd} : (forall i, G $-> H i) <~> (G $-> prod_0gpd I H). Proof. snrapply Build_Equiv. { intro f. snrapply Build_Morphism_0Gpd. 1: exact (fun x i => f i x). snrapply Build_Is0Functor; cbn beta. intros x y p i; simpl. exact (fmap (f i) p). } snrapply Build_IsEquiv. - intro f. intros i. exact (prod_0gpd_pr i $o f). - intro f. reflexivity. - intro f. reflexivity. - reflexivity. Defined. | Definition | Require Import Basics.Overture Basics.Tactics Require Import WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\ZeroGroupoid.v | equiv_prod_0gpd_corec | 7,846 |
{I J : Type} (ie : I <~> J) (G : I -> ZeroGpd) (H : J -> ZeroGpd) (f : forall (i : I), G i $<~> H (ie i)) : prod_0gpd I G $<~> prod_0gpd J H. Proof. snrapply cate_adjointify. - snrapply Build_Morphism_0Gpd. + intros h j. exact (transport H (eisretr ie j) (cate_fun (f (ie^-1 j)) (h _))). + nrapply Build_Is0Functor. intros g h p j. destruct (eisretr ie j). refine (_ $o Hom_path (transport_1 _ _)). apply Build_Morphism_0Gpd. exact (p _). - exact (equiv_prod_0gpd_corec (fun i => (f i)^-1$ $o prod_0gpd_pr (ie i))). - intros h j. cbn. destruct (eisretr ie j). exact (cate_isretr (f _) _). - intros g i. cbn. refine (_ $o Hom_path (ap (cate_fun (f i)^-1$) (transport2 _ (eisadj ie i) _))). destruct (eissect ie i). exact (cate_issect (f _) _). Defined. | Definition | Require Import Basics.Overture Basics.Tactics Require Import WildCat.Core WildCat.Equiv WildCat.EquivGpd | WildCat\ZeroGroupoid.v | cate_prod_0gpd | 7,847 |