Datasets:

phanerozoic

/

Coq-HoTT

like 2

forall {x1 x2 : C} {f : Category.Core.morphism C x2 x1} {p p0 : PreCategory} {f0 : Category.Core.morphism C x2 x1 -> Functor p0 p} {f1 : Functor p0 p0} {x0 : Category.Core.morphism (_ -> _) (f0 (f o 1)) (f0 f o f1)%functor} {H0' : IsIsomorphism x0} {x : Category.Core.morphism (_ -> _) f1 1%functor} {H1' : IsIsomorphism x} {fst_hyp : f o 1 = f} (rew_hyp : forall x3 : p0, (idtoiso (p0 -> p) (ap f0 fst_hyp) : Category.Core.morphism _ _ _) x3 = 1 o ((f0 f) _1 (x x3) o x0 x3)) {x3 : p} {x4 : p0} {f' : Category.Core.morphism p ((f0 f) x4) x3}, exist (fun f2 : Category.Core.morphism C x2 x1 => Category.Core.morphism p ((f0 f2) x4) x3) (f o 1) (f' o ((f0 f) _1 (x x4) o x0 x4)) = (f; f'). Proof. helper_t idtac. Qed.

Lemma

Require Import FunctorCategory.Morphisms. Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core Pseudofunctor.RewriteLaws. Require Import Category.Morphisms Cat.Morphisms. Require Import Functor.Composition.Core. Require Import Functor.Identity. Require Import FunctorCategory.Core. Require Import Basics Types HoTT.Tactics.

Categories\Grothendieck\PseudofunctorToCat.v

pseudofunctor_to_cat_right_identity_helper

PreCategory. Proof. refine (@Build_PreCategory Pair (fun s d => morphism s d) identity compose _ _ _ _); [ abstract ( intros ? ? ? ? [f ?] [g ?] [h ?]; exact (pseudofunctor_to_cat_assoc_helper (apD10 (ap components_of (p_composition_of_coherent_for_rewrite F _ _ _ _ f g h)))) ) | abstract ( intros ? ? [f ?]; exact (pseudofunctor_to_cat_left_identity_helper (apD10 (ap components_of (p_left_identity_of_coherent_for_rewrite F _ _ f)))) ) | abstract ( intros ? ? [f ?]; exact (pseudofunctor_to_cat_right_identity_helper (apD10 (ap components_of (p_right_identity_of_coherent_for_rewrite F _ _ f)))) ) ]. Defined.

Definition

Require Import FunctorCategory.Morphisms. Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core Pseudofunctor.RewriteLaws. Require Import Category.Morphisms Cat.Morphisms. Require Import Functor.Composition.Core. Require Import Functor.Identity. Require Import FunctorCategory.Core. Require Import Basics Types HoTT.Tactics.

Categories\Grothendieck\PseudofunctorToCat.v

category

Functor category C := Build_Functor category C c (fun s d => @ _ _) (fun _ _ _ _ _ => idpath) (fun _ => idpath).

Definition

Require Import FunctorCategory.Morphisms. Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core Pseudofunctor.RewriteLaws. Require Import Category.Morphisms Cat.Morphisms. Require Import Functor.Composition.Core. Require Import Functor.Identity. Require Import FunctorCategory.Core. Require Import Basics Types HoTT.Tactics.

Categories\Grothendieck\PseudofunctorToCat.v

pr1

PreCategory := (pseudofunctor_of_functor_to_cat F).

Definition

Require Import Category.Core Functor.Core. Require Import Pseudofunctor.FromFunctor. Require Import Cat.Core. Require Import Grothendieck.PseudofunctorToCat.

Categories\Grothendieck\ToCat.v

category

Functor category C := (pseudofunctor_of_functor_to_cat F).

Definition

Require Import Category.Core Functor.Core. Require Import Pseudofunctor.FromFunctor. Require Import Cat.Core. Require Import Grothendieck.PseudofunctorToCat.

Categories\Grothendieck\ToCat.v

pr1

Record

Require Import Category.Core Functor.Core. Require Import SetCategory.Core. Require Import Basics.Trunc Types.Sigma. Require Import Basics.Tactics.

Categories\Grothendieck\ToSet\Core.v

Pair

{ c : C | F c } <~> Pair. Proof. issig. Defined.

Definition

Require Import Category.Core Functor.Core. Require Import SetCategory.Core. Require Import Basics.Trunc Types.Sigma. Require Import Basics.Tactics.

Categories\Grothendieck\ToSet\Core.v

issig_pair

s d d' (m1 : morphism d d') (m2 : morphism s d) : (F _1 (m1 .1 o m2 .1)) s.(x) = d'.(x). Proof. etransitivity; [ | exact (m1.2) ]. etransitivity; [ | apply ap; exact (m2.2) ]. match goal with | [ |- ?f ?x = ?g (?h ?x) ] => change (f x = (g o h) x) end. match goal with | [ |- ?f ?x = ?g ?x ] => apply (@apD10 _ _ f g) end. apply (composition_of F). Defined.

Definition

Require Import Category.Core Functor.Core. Require Import SetCategory.Core. Require Import Basics.Trunc Types.Sigma. Require Import Basics.Tactics.

Categories\Grothendieck\ToSet\Core.v

compose_H

s d d' (m1 : morphism d d') (m2 : morphism s d) : morphism s d'. Proof. exists (m1.1 o m2.1). apply compose_H. Defined.

Definition

Require Import Category.Core Functor.Core. Require Import SetCategory.Core. Require Import Basics.Trunc Types.Sigma. Require Import Basics.Tactics.

Categories\Grothendieck\ToSet\Core.v

compose

s := apD10 (identity_of F s.

Definition

Require Import Category.Core Functor.Core. Require Import SetCategory.Core. Require Import Basics.Trunc Types.Sigma. Require Import Basics.Tactics.

Categories\Grothendieck\ToSet\Core.v

identity_H

s : morphism s s. Proof. exists 1. apply identity_H. Defined.

Definition

Require Import Category.Core Functor.Core. Require Import SetCategory.Core. Require Import Basics.Trunc Types.Sigma. Require Import Basics.Tactics.

Categories\Grothendieck\ToSet\Core.v

identity

PreCategory. Proof. refine (@Build_PreCategory Pair (fun s d => morphism s d) identity compose _ _ _ _); abstract ( repeat intro; apply path_sigma_uncurried; simpl; ((exists (associativity _ _ _ _ _ _ _ _)) || (exists (left_identity _ _ _ _)) || (exists (right_identity _ _ _ _))); exact (center _) ). Defined.

Definition

Require Import Category.Core Functor.Core. Require Import SetCategory.Core. Require Import Basics.Trunc Types.Sigma. Require Import Basics.Tactics.

Categories\Grothendieck\ToSet\Core.v

category

Functor category C := Build_Functor category C c (fun s d => @ _ _) (fun _ _ _ _ _ => idpath) (fun _ => idpath).

Definition

Require Import Category.Core Functor.Core. Require Import SetCategory.Core. Require Import Basics.Trunc Types.Sigma. Require Import Basics.Tactics.

Categories\Grothendieck\ToSet\Core.v

pr1

{s d : category F} : (s <~=~> d)%category <~> { e : (s.(c) <~=~> d.(c))%category | (F _1 e s.(x) = d.(x))%category }. Proof. simple refine (equiv_adjointify _ _ _ _). { intro m. simple refine (_; _). { exists (m : morphism _ _ _).1. exists (m^-1).1. { exact (ap proj1 (@left_inverse _ _ _ m _)). } { exact (ap proj1 (@right_inverse _ _ _ m _)). } } { exact (m : morphism _ _ _).2. } } { intro m. exists (m.1 : morphism _ _ _ ; m.2). eexists (m.1^-1; ((ap (F _1 (m.1)^-1) m.2)^) @ (ap10 ((((composition_of F _ _ _ _ _)^) @ (ap (fun m => F _1 m) (@left_inverse _ _ _ m.1 _)) @ (identity_of F _)) : (F _1 (m.1 : morphism _ _ _)^-1) o F _1 m.1 = idmap) s.(x))); apply path_sigma_hprop. - exact left_inverse. - exact right_inverse. } { intro x; apply path_sigma_hprop; apply path_isomorphic. reflexivity. } { intro x; apply path_isomorphic; reflexivity. } Defined.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Morphisms. Require Import SetCategory.Core. Require Import Grothendieck.ToSet.Core. Require Import HoTT.Basics HoTT.Types.

Categories\Grothendieck\ToSet\Morphisms.v

isequiv_sigma_category_isomorphism

{s d} (a : c s = c d) : (transport (fun c : C => F c) a (x s) = x d) <~> (F _1 (idtoiso C a)) (x s) = x d. Proof. apply equiv_path. apply ap10, ap. destruct a; simpl. exact (ap10 (identity_of F _)^ _). Defined.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Univalent. Require Import Category.Morphisms. Require Import SetCategory.Core. Require Import Grothendieck.ToSet.Core Grothendieck.ToSet.Morphisms. Require Import HoTT.Basics.Equivalences HoTT.Basics.Trunc. Require Import HoTT.Types.Universe HoTT.Types.Sigma.

Categories\Grothendieck\ToSet\Univalent.v

category_isotoid_helper

{s d : category F} : s = d <~> (s <~=~> d)%category. Proof. refine (isequiv_sigma_category_isomorphism^-1 oE _ oE (equiv_ap' (issig_pair F)^-1 s d)). refine (_ oE (equiv_path_sigma _ _ _)^-1). simpl. simple refine (equiv_functor_sigma' _ _). { exists (@idtoiso C _ _). exact _. } { exact category_isotoid_helper. } Defined.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Univalent. Require Import Category.Morphisms. Require Import SetCategory.Core. Require Import Grothendieck.ToSet.Core Grothendieck.ToSet.Morphisms. Require Import HoTT.Basics.Equivalences HoTT.Basics.Trunc. Require Import HoTT.Types.Universe HoTT.Types.Sigma.

Categories\Grothendieck\ToSet\Univalent.v

category_isotoid

s d d' (m : morphism d d') (m' : morphism s d) : morphism s d' := transitivity m' m.

Definition

Require Import Category.Morphisms Category.Strict. Require Import Trunc PathGroupoids Basics.Tactics.

Categories\GroupoidCategory\Core.v

compose

x : morphism x x := reflexivity _.

Definition

Require Import Category.Morphisms Category.Strict. Require Import Trunc PathGroupoids Basics.Tactics.

Categories\GroupoidCategory\Core.v

identity

X `{IsTrunc 1 X} : PreCategory. Proof. refine (@Build_PreCategory X (@paths X) (@GroupoidCategoryInternals.identity X) (@GroupoidCategoryInternals.compose X) _ _ _ _); simpl; intros; path_induction; reflexivity. Defined.

Definition

Require Import Category.Morphisms Category.Strict. Require Import Trunc PathGroupoids Basics.Tactics.

Categories\GroupoidCategory\Core.v

groupoid_category

X `{IsHSet X} : IsStrictCategory (groupoid_category X). Proof. typeclasses eauto. Defined.

Lemma

Require Import Category.Morphisms Category.Strict. Require Import Trunc PathGroupoids Basics.Tactics.

Categories\GroupoidCategory\Core.v

isstrict_groupoid_category

`{Univalence} `{IsTrunc 1 X} : (groupoid_category X)^op = groupoid_category X. Proof. repeat match goal with | _ => intro | _ => progress cbn | _ => reflexivity | _ => apply path_forall | _ => apply (path_universe (symmetry _ _)) | _ => exact (center _) | _ => progress rewrite ?transport_path_universe, ?transport_path_universe_V | _ => progress path_category | _ => progress path_induction end. Qed.

Lemma

Require Import Category.Core GroupoidCategory.Core Category.Paths Category.Dual. Require Import HoTT.Types. Require Import Basics.Trunc Basics.Tactics.

Categories\GroupoidCategory\Dual.v

path_groupoid_dual

(s d : groupoid_category X) : s <~=~> d -> s = d := fun f => f : morphism _ _ _. Global Instance iscategory_groupoid_category : IsCategory (groupoid_category X). Proof. repeat intro. apply (isequiv_adjointify (@idtoiso (groupoid_category X) _ _) (@ _ _)); repeat intro; destruct_head @Isomorphic; destruct_head @IsIsomorphism; compute in *; path_induction_hammer. Qed.

Definition

Require Import Category.Core Category.Morphisms Category.Univalent GroupoidCategory.Core. Require Import Trunc Equivalences HoTT.Tactics.

Categories\GroupoidCategory\Morphisms.v

isotoid

`{@IsTerminalCategory one Hone Hone'} : Functor C one := Build_Functor C one (fun _ => center _) (fun _ _ _ => center _) (fun _ _ _ _ _ => contr _) (fun _ => contr _).

Definition

Require Import Category.Core Functor.Core Functor.Paths. Require Import InitialTerminalCategory.Core. Require Import NatCategory. Require Import HoTT.Basics.

Categories\InitialTerminalCategory\Functors.v

to_terminal

`{@IsTerminalCategory one Hone Hone'} (c : C) : Functor one C := Build_Functor one C (fun _ => c) (fun _ _ _ => identity c) (fun _ _ _ _ _ => symmetry _ _ (@identity_identity _ _)) (fun _ => idpath).

Definition

Require Import Category.Core Functor.Core Functor.Paths. Require Import InitialTerminalCategory.Core. Require Import NatCategory. Require Import HoTT.Basics.

Categories\InitialTerminalCategory\Functors.v

from_terminal

`{@IsInitialCategory zero} : Functor zero C := Build_Functor zero C (fun x => initial_category_ind _ x) (fun x _ _ => initial_category_ind _ x) (fun x _ _ _ _ => initial_category_ind _ x) (fun x => initial_category_ind _ x).

Definition

Require Import Category.Core Functor.Core Functor.Paths. Require Import InitialTerminalCategory.Core. Require Import NatCategory. Require Import HoTT.Basics.

Categories\InitialTerminalCategory\Functors.v

from_initial

C : Functor C 1 := Eval simpl in to_terminal C.

Definition

Require Import Category.Core Functor.Core Functor.Paths. Require Import InitialTerminalCategory.Core. Require Import NatCategory. Require Import HoTT.Basics.

Categories\InitialTerminalCategory\Functors.v

to_1

C c : Functor 1 C := Eval simpl in from_terminal C c.

Definition

Require Import Category.Core Functor.Core Functor.Paths. Require Import InitialTerminalCategory.Core. Require Import NatCategory. Require Import HoTT.Basics.

Categories\InitialTerminalCategory\Functors.v

from_1

C : Functor 0 C := Eval simpl in from_initial C. Local Notation "! x" := (@from_terminal _ terminal_category _ _ _ x) : functor_scope. Section unique. Context `{Funext}. Global Instance trunc_initial_category_function `{@IsInitialCategory zero} T : Contr (zero -> T). Proof. refine (Build_Contr _ (initial_category_ind _) _). intro y. apply path_forall; intro x. apply (initial_category_ind _ x). Defined.

Definition

Require Import Category.Core Functor.Core Functor.Paths. Require Import InitialTerminalCategory.Core. Require Import NatCategory. Require Import HoTT.Basics.

Categories\InitialTerminalCategory\Functors.v

from_0

`{@IsInitialCategory zero} (F : Functor C zero) : IsInitialCategory C := fun P x => initial_category_ind P (F x). Global Instance trunc_terminal_category `{@IsTerminalCategory one H1 H2} : Contr (Functor C one). Proof. refine (Build_Contr _ (to_terminal C) _). intros. exact (center _). Defined.

Definition

Require Import Category.Core Functor.Core Functor.Paths. Require Import InitialTerminalCategory.Core. Require Import NatCategory. Require Import HoTT.Basics.

Categories\InitialTerminalCategory\Functors.v

to_initial_category_empty

`{@IsInitialCategory zero} (F G : Functor zero C) : NaturalTransformation F G := Build_NaturalTransformation F G (fun x => initial_category_ind _ x) (fun x _ _ => initial_category_ind _ x). Global Instance trunc_from_initial `{Funext} `{@IsInitialCategory zero} (F G : Functor zero C) : Contr (NaturalTransformation F G). Proof. refine (Build_Contr _ ( F G) _). abstract ( intros; apply path_natural_transformation; intro x; exact (initial_category_ind _ x) ). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core NaturalTransformation.Paths. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Contractible.

Categories\InitialTerminalCategory\NaturalTransformations.v

from_initial

`{@IsTerminalCategory one H1 H2} (F G : Functor C one) : NaturalTransformation F G := Build_NaturalTransformation F G (fun x => center _) (fun _ _ _ => path_contr _ _). Global Instance trunc_to_terminal `{Funext} `{@IsTerminalCategory one H1 H2} (F G : Functor C one) : Contr (NaturalTransformation F G). Proof. refine (Build_Contr _ ( F G) _). abstract (path_natural_transformation; exact (contr _)). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core NaturalTransformation.Paths. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Contractible.

Categories\InitialTerminalCategory\NaturalTransformations.v

to_terminal

`{Funext} `{@IsTerminalCategory one Hone Hone'} (c : PreCategory) : Pseudofunctor one. Proof. refine (Build_Pseudofunctor one (fun _ => c) (fun _ _ _ => 1%functor) (fun _ _ _ _ _ => reflexivity _) (fun _ => reflexivity _) _ _ _); simpl; abstract ( intros; path_natural_transformation; rewrite ap_const; simpl; reflexivity ). Defined.

Definition

Require Import Category.Core Functor.Core. Require Import Functor.Identity. Require Import Pseudofunctor.Core. Require Import InitialTerminalCategory.Core. Require Import FunctorCategory.Morphisms. Require Import NaturalTransformation.Paths. Require Import NatCategory. Require Import PathGroupoids.

Categories\InitialTerminalCategory\Pseudofunctors.v

from_terminal

`{Funext} `{@IsInitialCategory zero} : Pseudofunctor zero := Build_Pseudofunctor zero (fun x => initial_category_ind _ x) (fun x _ _ => initial_category_ind _ x) (fun x _ _ _ _ => initial_category_ind _ x) (fun x => initial_category_ind _ x) (fun x => initial_category_ind _ x) (fun x => initial_category_ind _ x) (fun x => initial_category_ind _ x).

Definition

Require Import Category.Core Functor.Core. Require Import Functor.Identity. Require Import Pseudofunctor.Core. Require Import InitialTerminalCategory.Core. Require Import FunctorCategory.Morphisms. Require Import NaturalTransformation.Paths. Require Import NatCategory. Require Import PathGroupoids.

Categories\InitialTerminalCategory\Pseudofunctors.v

from_initial

`{Funext} c : Pseudofunctor 1 := Eval simpl in from_terminal c.

Definition

Require Import Category.Core Functor.Core. Require Import Functor.Identity. Require Import Pseudofunctor.Core. Require Import InitialTerminalCategory.Core. Require Import FunctorCategory.Morphisms. Require Import NaturalTransformation.Paths. Require Import NatCategory. Require Import PathGroupoids.

Categories\InitialTerminalCategory\Pseudofunctors.v

from_1

`{Funext} : Pseudofunctor 0 := Eval simpl in from_initial.

Definition

Require Import Category.Core Functor.Core. Require Import Functor.Identity. Require Import Pseudofunctor.Core. Require Import InitialTerminalCategory.Core. Require Import FunctorCategory.Morphisms. Require Import NaturalTransformation.Paths. Require Import NatCategory. Require Import PathGroupoids.

Categories\InitialTerminalCategory\Pseudofunctors.v

from_0

object ((C -> C') -> (C' -> D) -> (C -> D)) := Functor.

Definition

Require Import Category.Core Functor.Core. Require Import FunctorCategory.Core. Require Import Functor.Composition.Functorial.Core. Require Import UniversalProperties.

Categories\KanExtensions\Core.v

pullback_along_functor

(p : Functor C C') : object ((C' -> D) -> (C -> D)) := Eval hnf in pullback_along_functor p.

Definition

Require Import Category.Core Functor.Core. Require Import FunctorCategory.Core. Require Import Functor.Composition.Functorial.Core. Require Import UniversalProperties.

Categories\KanExtensions\Core.v

pullback_along

(p : Functor C C') (h : Functor C D) := @IsInitialMorphism (_ -> _) _ h (pullback_along p).

Definition

Require Import Category.Core Functor.Core. Require Import FunctorCategory.Core. Require Import Functor.Composition.Functorial.Core. Require Import UniversalProperties.

Categories\KanExtensions\Core.v

IsLeftKanExtensionAlong

(p : Functor C C') (h : Functor C D) := @IsTerminalMorphism _ (_ -> _) (pullback_along p) h.

Definition

Require Import Category.Core Functor.Core. Require Import FunctorCategory.Core. Require Import Functor.Composition.Functorial.Core. Require Import UniversalProperties.

Categories\KanExtensions\Core.v

IsRightKanExtensionAlong

Functor (C -> D) (C' -> D) := functor__of__initial_morphism has_left_kan_extensions.

Definition

Require Import Category.Core Functor.Core. Require Import KanExtensions.Core. Require Import Adjoint.UniversalMorphisms.Core. Require Import FunctorCategory.Core. Require Import Adjoint.Core.

Categories\KanExtensions\Functors.v

left_kan_extension_functor

left_kan_extension_functor -| pullback_along D p := adjunction__of__initial_morphism has_left_kan_extensions.

Definition

Require Import Category.Core Functor.Core. Require Import KanExtensions.Core. Require Import Adjoint.UniversalMorphisms.Core. Require Import FunctorCategory.Core. Require Import Adjoint.Core.

Categories\KanExtensions\Functors.v

left_kan_extension_adjunction

Functor (C -> D) (C' -> D) := functor__of__terminal_morphism has_right_kan_extensions.

Definition

Require Import Category.Core Functor.Core. Require Import KanExtensions.Core. Require Import Adjoint.UniversalMorphisms.Core. Require Import FunctorCategory.Core. Require Import Adjoint.Core.

Categories\KanExtensions\Functors.v

right_kan_extension_functor

pullback_along D p -| right_kan_extension_functor := adjunction__of__terminal_morphism has_right_kan_extensions.

Definition

Require Import Category.Core Functor.Core. Require Import KanExtensions.Core. Require Import Adjoint.UniversalMorphisms.Core. Require Import FunctorCategory.Core. Require Import Adjoint.Core.

Categories\KanExtensions\Functors.v

right_kan_extension_adjunction

`{Funext} A B (S : Pseudofunctor A) (T : Pseudofunctor B) `{forall a b, IsHSet (Functor (S a) (T b))} : PreCategory := @Build_PreCategory (@object _ _ _ S T) (@morphism _ _ _ S T) (@identity _ _ _ S T) (@compose _ _ _ S T) (@associativity _ _ _ S T) (@left_identity _ _ _ S T) (@right_identity _ _ _ S T) _.

Definition

Require Import Functor.Core. Require Import Category.Dual. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Pseudofunctors. Require Import Cat.Core. Require Import Category.Strict. Require Import NaturalTransformation.Paths. Require Import Pseudofunctor.Core. Import Functor.Identity.FunctorIdentityNotations. Import Pseudofunctor.Identity.PseudofunctorIdentityNotations. Import LaxComma.CoreLaws.LaxCommaCategory.

Categories\LaxComma\Core.v

lax_comma_category

`{Funext} A B (S : Pseudofunctor A) (T : Pseudofunctor B) `{forall a b, IsHSet (Functor (S a) (T b))} : PreCategory := (lax_comma_category S T)^op. Global Instance isstrict_lax_comma_category `{Funext} A B (S : Pseudofunctor A) (T : Pseudofunctor B) `{IsStrictCategory A, IsStrictCategory B} `{forall a b, IsHSet (Functor (S a) (T b))} : IsStrictCategory (@lax_comma_category _ A B S T _). Proof. typeclasses eauto. Qed.

Definition

Require Import Functor.Core. Require Import Category.Dual. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Pseudofunctors. Require Import Cat.Core. Require Import Category.Strict. Require Import NaturalTransformation.Paths. Require Import Pseudofunctor.Core. Import Functor.Identity.FunctorIdentityNotations. Import Pseudofunctor.Identity.PseudofunctorIdentityNotations. Import LaxComma.CoreLaws.LaxCommaCategory.

Categories\LaxComma\Core.v

oplax_comma_category

(x y : comma_category) : x ≅ y -> x = y. Proof. intro i. destruct i as [i [i' ? ?]]. hnf in *. destruct i, i'. simpl in *. Global Instance comma_category_IsCategory `{IsCategory A, IsCategory B} : IsCategory comma_category. Proof. hnf. unfold IsStrictCategory in *. typeclasses eauto. Qed.

Definition

Require Import Functor.Core. Require Import Category.Dual. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Pseudofunctors. Require Import Cat.Core. Require Import Category.Strict. Require Import NaturalTransformation.Paths. Require Import Pseudofunctor.Core. Import Functor.Identity.FunctorIdentityNotations. Import Pseudofunctor.Identity.PseudofunctorIdentityNotations. Import LaxComma.CoreLaws.LaxCommaCategory.

Categories\LaxComma\Core.v

comma_category_isotoid

PreCategory := lax_comma_category S !a.

Definition

Require Import Functor.Core. Require Import Category.Dual. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Pseudofunctors. Require Import Cat.Core. Require Import Category.Strict. Require Import NaturalTransformation.Paths. Require Import Pseudofunctor.Core. Import Functor.Identity.FunctorIdentityNotations. Import Pseudofunctor.Identity.PseudofunctorIdentityNotations. Import LaxComma.CoreLaws.LaxCommaCategory.

Categories\LaxComma\Core.v

lax_slice_category

PreCategory := lax_comma_category !a S.

Definition

Require Import Functor.Core. Require Import Category.Dual. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Pseudofunctors. Require Import Cat.Core. Require Import Category.Strict. Require Import NaturalTransformation.Paths. Require Import Pseudofunctor.Core. Import Functor.Identity.FunctorIdentityNotations. Import Pseudofunctor.Identity.PseudofunctorIdentityNotations. Import LaxComma.CoreLaws.LaxCommaCategory.

Categories\LaxComma\Core.v

lax_coslice_category

PreCategory := oplax_comma_category S !a.

Definition

Require Import Functor.Core. Require Import Category.Dual. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Pseudofunctors. Require Import Cat.Core. Require Import Category.Strict. Require Import NaturalTransformation.Paths. Require Import Pseudofunctor.Core. Import Functor.Identity.FunctorIdentityNotations. Import Pseudofunctor.Identity.PseudofunctorIdentityNotations. Import LaxComma.CoreLaws.LaxCommaCategory.

Categories\LaxComma\Core.v

oplax_slice_category

PreCategory := oplax_comma_category !a S.

Definition

Require Import Functor.Core. Require Import Category.Dual. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Pseudofunctors. Require Import Cat.Core. Require Import Category.Strict. Require Import NaturalTransformation.Paths. Require Import Pseudofunctor.Core. Import Functor.Identity.FunctorIdentityNotations. Import Pseudofunctor.Identity.PseudofunctorIdentityNotations. Import LaxComma.CoreLaws.LaxCommaCategory.

Categories\LaxComma\Core.v

oplax_coslice_category

PreCategory := @lax_slice_category _ cat a (Pseudofunctor.

Definition

Require Import Functor.Core. Require Import Category.Dual. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Pseudofunctors. Require Import Cat.Core. Require Import Category.Strict. Require Import NaturalTransformation.Paths. Require Import Pseudofunctor.Core. Import Functor.Identity.FunctorIdentityNotations. Import Pseudofunctor.Identity.PseudofunctorIdentityNotations. Import LaxComma.CoreLaws.LaxCommaCategory.

Categories\LaxComma\Core.v

lax_slice_category_over

PreCategory := @lax_coslice_category _ cat a (Pseudofunctor.

Definition

Require Import Functor.Core. Require Import Category.Dual. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Pseudofunctors. Require Import Cat.Core. Require Import Category.Strict. Require Import NaturalTransformation.Paths. Require Import Pseudofunctor.Core. Import Functor.Identity.FunctorIdentityNotations. Import Pseudofunctor.Identity.PseudofunctorIdentityNotations. Import LaxComma.CoreLaws.LaxCommaCategory.

Categories\LaxComma\Core.v

lax_coslice_category_over

PreCategory := @oplax_slice_category _ cat a (Pseudofunctor.

Definition

Require Import Functor.Core. Require Import Category.Dual. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Pseudofunctors. Require Import Cat.Core. Require Import Category.Strict. Require Import NaturalTransformation.Paths. Require Import Pseudofunctor.Core. Import Functor.Identity.FunctorIdentityNotations. Import Pseudofunctor.Identity.PseudofunctorIdentityNotations. Import LaxComma.CoreLaws.LaxCommaCategory.

Categories\LaxComma\Core.v

oplax_slice_category_over

PreCategory := @oplax_coslice_category _ cat a (Pseudofunctor.

Definition

Require Import Functor.Core. Require Import Category.Dual. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Pseudofunctors. Require Import Cat.Core. Require Import Category.Strict. Require Import NaturalTransformation.Paths. Require Import Pseudofunctor.Core. Import Functor.Identity.FunctorIdentityNotations. Import Pseudofunctor.Identity.PseudofunctorIdentityNotations. Import LaxComma.CoreLaws.LaxCommaCategory.

Categories\LaxComma\Core.v

oplax_coslice_category_over

PreCategory := @lax_comma_category _ cat cat (Pseudofunctor.

Definition

Require Import Functor.Core. Require Import Category.Dual. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Pseudofunctors. Require Import Cat.Core. Require Import Category.Strict. Require Import NaturalTransformation.Paths. Require Import Pseudofunctor.Core. Import Functor.Identity.FunctorIdentityNotations. Import Pseudofunctor.Identity.PseudofunctorIdentityNotations. Import LaxComma.CoreLaws.LaxCommaCategory.

Categories\LaxComma\Core.v

lax_arrow_category

PreCategory := @oplax_comma_category _ cat cat (Pseudofunctor.

Definition

Require Import Functor.Core. Require Import Category.Dual. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Pseudofunctors. Require Import Cat.Core. Require Import Category.Strict. Require Import NaturalTransformation.Paths. Require Import Pseudofunctor.Core. Import Functor.Identity.FunctorIdentityNotations. Import Pseudofunctor.Identity.PseudofunctorIdentityNotations. Import LaxComma.CoreLaws.LaxCommaCategory.

Categories\LaxComma\Core.v

oplax_arrow_category

`{@LCC_Builder A B C x y z} : C := z.

Definition

Require Import Functor.Core. Require Import Category.Dual. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Pseudofunctors. Require Import Cat.Core. Require Import Category.Strict. Require Import NaturalTransformation.Paths. Require Import Pseudofunctor.Core. Import Functor.Identity.FunctorIdentityNotations. Import Pseudofunctor.Identity.PseudofunctorIdentityNotations. Import LaxComma.CoreLaws.LaxCommaCategory.

Categories\LaxComma\Core.v

get_LCC

`{@OLCC_Builder A B C x y z} : C := z.

Definition

Require Import Functor.Core. Require Import Category.Dual. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Pseudofunctors. Require Import Cat.Core. Require Import Category.Strict. Require Import NaturalTransformation.Paths. Require Import Pseudofunctor.Core. Import Functor.Identity.FunctorIdentityNotations. Import Pseudofunctor.Identity.PseudofunctorIdentityNotations. Import LaxComma.CoreLaws.LaxCommaCategory.

Categories\LaxComma\Core.v

get_OLCC

x1 x2 x3 x4 (m1 : morphism x1 x2) (m2 : morphism x2 x3) (m3 : morphism x3 x4) : compose (compose m3 m2) m1 = compose m3 (compose m2 m1). Proof. refine (@path_morphism' _ _ (compose (compose m3 m2) m1) (compose m3 (compose m2 m1)) (Category.Core. _ _ _ _ _ _ _ _) (Category.Core. _ _ _ _ _ _ _ _) _). simpl in *. repeat match goal with | [ |- context[@morphism_inverse _ _ _ _ (@isisomorphism_isomorphic _ _ _ (Category.Morphisms.idtoiso ?C0 (ap (p_morphism_of ?F (s:=_) (d:=_)) (Category.Core. ?C ?x1 ?x2 ?x3 ?x4 ?m1 ?m2 ?m3))))] ] => generalize (@p_composition_of_coherent_inverse_for_rewrite _ C F x1 x2 x3 x4 m1 m2 m3); generalize (Category.Morphisms.idtoiso C0 (ap (p_morphism_of F (s:=_) (d:=_)) (Category.Core. C x1 x2 x3 x4 m1 m2 m3))) | [ |- context[Category.Morphisms.idtoiso ?C0 (ap (p_morphism_of ?F (s:=_) (d:=_)) (Category.Core. ?C ?x1 ?x2 ?x3 ?x4 ?m1 ?m2 ?m3))] ] => generalize (@p_composition_of_coherent_for_rewrite _ C F x1 x2 x3 x4 m1 m2 m3); generalize (Category.Morphisms.idtoiso C0 (ap (p_morphism_of F (s:=_) (d:=_)) (Category.Core. C x1 x2 x3 x4 m1 m2 m3))) end. simpl. destruct_head morphism. destruct_head object. simpl in *. repeat match goal with | [ |- context[p_composition_of ?F ?x ?y ?z ?m1 ?m2] ] => generalize dependent (p_composition_of F x y z m1 m2) | [ |- context[p_identity_of ?F ?x] ] => generalize dependent (p_identity_of F x) | [ |- context[p_morphism_of ?F ?x] ] => generalize dependent (p_morphism_of F x) | [ |- context[p_object_of ?F ?x] ] => generalize dependent (p_object_of F x) end. simpl. clear. repeat (let H := fresh "x" in intro H). repeat match goal with H : _ |- _ => revert H end. intro. >> *)

Lemma

Require Import Functor.Core NaturalTransformation.Core. Require Import Category.Strict. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core Pseudofunctor.RewriteLaws. Require Import NaturalTransformation.Composition.Laws. Require Import FunctorCategory.Morphisms. Require Import HoTT.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategory.

Categories\LaxComma\CoreLaws.v

associativity

{x x0 : PreCategory} {x1 : Functor x0 x} {x2 x3 : PreCategory} {x4 : Functor x3 x2} {x5 x6 : PreCategory} {x7 : Functor x6 x5} {x8 x9 : PreCategory} {x10 : Functor x9 x8} {x11 : Functor x9 x6} {x12 : Functor x9 x3} {x13 : Functor x0 x6} {x14 : Functor x9 x6} {x15 : Functor x8 x5} {x16 : Functor x x5} {x17 : Functor x9 x0} {x18 : Functor x8 x} {x19 : NaturalTransformation (x18 o x10) (x1 o x17)} {x20 : Functor x0 x3} {x21 : Functor x x2} {x22 : NaturalTransformation (x21 o x1) (x4 o x20)} {x23 : Functor x8 x2} {x24 : Functor x3 x6} {x25 : Functor x2 x5} {x26 : NaturalTransformation (x25 o x4) (x7 o x24)} {x27 : Functor x8 x5} {x28 : @Isomorphic (_ -> _) x27 (x25 o x23)%functor} {x29 : @Isomorphic (_ -> _) x23 (x21 o x18)%functor} {x30 : @Isomorphic (_ -> _) x16 (x25 o x21)%functor} {x31 : @Isomorphic (_ -> _) x15 (x16 o x18)%functor} {x32 : @Isomorphic (_ -> _) x14 (x13 o x17)%functor} {x33 : @Isomorphic (_ -> _) x13 (x24 o x20)%functor} {x34 : @Isomorphic (_ -> _) x12 (x20 o x17)%functor} {x35 : @Isomorphic (_ -> _) x11 (x24 o x12)%functor} {x36 : @Isomorphic (_ -> _) x14 x11} (x37 : (x36 : Category.Core.morphism _ _ _) = (x35 ^-1 o (x24 oL x34 ^-1 o (associator_1 x24 x20 x17 o ((x33 : Category.Core.morphism _ _ _) oR x17 o (x32 : Category.Core.morphism _ _ _)))))%natural_transformation) {x38 : @Isomorphic (_ -> _) x15 x27} (x39 : x38 ^-1 = (x31 ^-1 o (x30 ^-1 oR x18) o inverse (associator_1 x25 x21 x18) o (x25 oL (x29 : Category.Core.morphism _ _ _)) o (x28 : Category.Core.morphism _ _ _))%natural_transformation) : (x7 oL (x36 : Category.Core.morphism _ _ _) o (x7 oL x32 ^-1 o associator_1 x7 x13 x17 o (x7 oL x33 ^-1 o associator_1 x7 x24 x20 o (x26 oR x20) o associator_2 x25 x4 x20 o (x25 oL x22) o associator_1 x25 x21 x1 o ((x30 : Category.Core.morphism _ _ _) oR x1) oR x17) o associator_2 x16 x1 x17 o (x16 oL x19) o associator_1 x16 x18 x10 o ((x31 : Category.Core.morphism _ _ _) oR x10)) o (x38 ^-1 oR x10))%natural_transformation = (x7 oL x35 ^-1 o associator_1 x7 x24 x12 o (x26 oR x12) o associator_2 x25 x4 x12 o (x25 oL (x4 oL x34 ^-1 o associator_1 x4 x20 x17 o (x22 oR x17) o associator_2 x21 x1 x17 o (x21 oL x19) o associator_1 x21 x18 x10 o ((x29 : Category.Core.morphism _ _ _) oR x10))) o associator_1 x25 x23 x10 o ((x28 : Category.Core.morphism _ _ _) oR x10))%natural_transformation. Proof. t. Qed.

Lemma

Require Import Functor.Core NaturalTransformation.Core. Require Import Category.Strict. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core Pseudofunctor.RewriteLaws. Require Import NaturalTransformation.Composition.Laws. Require Import FunctorCategory.Morphisms. Require Import HoTT.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategory.

Categories\LaxComma\CoreLaws.v

associativity_helper

x1 x2 x3 x4 (m1 : morphism x1 x2) (m2 : morphism x2 x3) (m3 : morphism x3 x4) : compose (compose m3 m2) m1 = compose m3 (compose m2 m1). Proof. refine (@path_morphism' _ A B S T _ _ (compose (compose m3 m2) m1) (compose m3 (compose m2 m1)) (Category.Core. _ _ _ _ _ _ _ _) (Category.Core. _ _ _ _ _ _ _ _) _). simpl. apply associativity_helper. - exact (p_composition_of_coherent_for_rewrite _ _ _ _ _ _ _ _). - exact (p_composition_of_coherent_inverse_for_rewrite _ _ _ _ _ _ _ _). Defined.

Lemma

Require Import Functor.Core NaturalTransformation.Core. Require Import Category.Strict. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core Pseudofunctor.RewriteLaws. Require Import NaturalTransformation.Composition.Laws. Require Import FunctorCategory.Morphisms. Require Import HoTT.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategory.

Categories\LaxComma\CoreLaws.v

associativity

(s d : object) (m : morphism s d) : compose (identity _) m = m. Proof. refine (@path_morphism' _ _ (compose (identity _) m) m (Category.Core. _ _ _ _) (Category.Core. _ _ _ _) _). simpl in *. repeat match goal with | [ |- context[@morphism_inverse _ _ _ _ (@isisomorphism_isomorphic _ _ _ (Category.Morphisms.idtoiso ?C0 (ap (p_morphism_of ?F (s:=_) (d:=_)) (Category.Core. ?C ?x ?y ?f))))] ] => generalize (@p_left_identity_of_coherent_inverse_for_rewrite _ C F x y f); generalize (Category.Morphisms.idtoiso C0 (ap (p_morphism_of F (s:=_) (d:=_)) (Category.Core. C x y f))) | [ |- context[Category.Morphisms.idtoiso ?C0 (ap (p_morphism_of ?F (s:=_) (d:=_)) (Category.Core. ?C ?x ?y ?f))] ] => generalize (@p_left_identity_of_coherent_for_rewrite _ C F x y f); generalize (Category.Morphisms.idtoiso C0 (ap (p_morphism_of F (s:=_) (d:=_)) (Category.Core. C x y f))) end. simpl. destruct_head morphism. destruct_head object. simpl in *. repeat match goal with | [ |- context[p_composition_of ?F ?x ?y ?z ?m1 ?m2] ] => generalize dependent (p_composition_of F x y z m1 m2) | [ |- context[p_identity_of ?F ?x] ] => generalize dependent (p_identity_of F x) | [ |- context[p_morphism_of ?F ?x] ] => generalize dependent (p_morphism_of F x) | [ |- context[p_object_of ?F ?x] ] => generalize dependent (p_object_of F x) end. simpl. clear. repeat (let H := fresh "x" in intro H). repeat match goal with H : _ |- _ => revert H end. intro. >> *)

Lemma

Require Import Functor.Core NaturalTransformation.Core. Require Import Category.Strict. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core Pseudofunctor.RewriteLaws. Require Import NaturalTransformation.Composition.Laws. Require Import FunctorCategory.Morphisms. Require Import HoTT.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategory.

Categories\LaxComma\CoreLaws.v

left_identity

{x x0 : PreCategory} {x1 : Functor x0 x} {x2 x3 : PreCategory} {x4 : Functor x3 x2} {x5 x6 : Functor x3 x0} {x7 : Functor x2 x} {x8 : NaturalTransformation (x7 o x4) (x1 o x6)} {x9 : Functor x2 x} {x10 : Functor x0 x0} {x11 : Functor x x} {x12 : @Isomorphic (_ -> _) x11 1%functor} {x13 : @Isomorphic (_ -> _) x10 1%functor} {x14 : @Isomorphic (_ -> _) x9 (x11 o x7)%functor} {x15 : @Isomorphic (_ -> _) x5 (x10 o x6)%functor} {x16 : @Isomorphic (_ -> _) x5 x6} {x17 : (x16 : Category.Core.morphism _ _ _) = (left_identity_natural_transformation_1 x6 o ((x13 : Category.Core.morphism _ _ _) oR x6 o (x15 : Category.Core.morphism _ _ _)))%natural_transformation} {x18 : @Isomorphic (_ -> _) x9 x7} {x19 : x18 ^-1 = (x14 ^-1 o (x12 ^-1 oR x7) o inverse (left_identity_natural_transformation_1 x7))%natural_transformation} : (x1 oL (x16 : Category.Core.morphism _ _ _) o (x1 oL x15 ^-1 o associator_1 x1 x10 x6 o (x1 oL x13 ^-1 o right_identity_natural_transformation_2 x1 o left_identity_natural_transformation_1 x1 o ((x12 : Category.Core.morphism _ _ _) oR x1) oR x6) o associator_2 x11 x1 x6 o (x11 oL x8) o associator_1 x11 x7 x4 o ((x14 : Category.Core.morphism _ _ _) oR x4)) o (x18 ^-1 oR x4))%natural_transformation = x8. Proof. t. Qed.

Lemma

Require Import Functor.Core NaturalTransformation.Core. Require Import Category.Strict. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core Pseudofunctor.RewriteLaws. Require Import NaturalTransformation.Composition.Laws. Require Import FunctorCategory.Morphisms. Require Import HoTT.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategory.

Categories\LaxComma\CoreLaws.v

left_identity_helper

(s d : object) (m : morphism s d) : compose (identity _) m = m. Proof. refine (@path_morphism' _ A B S T _ _ (compose (identity _) m) m (Category.Core. _ _ _ _) (Category.Core. _ _ _ _) _). simpl. refine left_identity_helper. - exact (p_left_identity_of_coherent_for_rewrite _ _ _ _). - exact (p_left_identity_of_coherent_inverse_for_rewrite _ _ _ _). Defined.

Lemma

Require Import Functor.Core NaturalTransformation.Core. Require Import Category.Strict. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core Pseudofunctor.RewriteLaws. Require Import NaturalTransformation.Composition.Laws. Require Import FunctorCategory.Morphisms. Require Import HoTT.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategory.

Categories\LaxComma\CoreLaws.v

left_identity

(s d : object) (m : morphism s d) : compose m (identity _) = m. Proof. refine (@path_morphism' _ _ (compose m (identity _)) m (Category.Core. _ _ _ _) (Category.Core. _ _ _ _) _). simpl in *. repeat match goal with | [ |- context[@morphism_inverse _ _ _ _ (@isisomorphism_isomorphic _ _ _ (Category.Morphisms.idtoiso ?C0 (ap (p_morphism_of ?F (s:=_) (d:=_)) (Category.Core. ?C ?x ?y ?f))))] ] => generalize (@p_right_identity_of_coherent_inverse_for_rewrite _ C F x y f); generalize (Category.Morphisms.idtoiso C0 (ap (p_morphism_of F (s:=_) (d:=_)) (Category.Core. C x y f))) | [ |- context[Category.Morphisms.idtoiso ?C0 (ap (p_morphism_of ?F (s:=_) (d:=_)) (Category.Core. ?C ?x ?y ?f))] ] => generalize (@p_right_identity_of_coherent_for_rewrite _ C F x y f); generalize (Category.Morphisms.idtoiso C0 (ap (p_morphism_of F (s:=_) (d:=_)) (Category.Core. C x y f))) end. simpl. destruct_head morphism. destruct_head object. simpl in *. repeat match goal with | [ |- context[p_composition_of ?F ?x ?y ?z ?m1 ?m2] ] => generalize dependent (p_composition_of F x y z m1 m2) | [ |- context[p_identity_of ?F ?x] ] => generalize dependent (p_identity_of F x) | [ |- context[p_morphism_of ?F ?x] ] => generalize dependent (p_morphism_of F x) | [ |- context[p_object_of ?F ?x] ] => generalize dependent (p_object_of F x) end. simpl. clear. repeat (let H := fresh "x" in intro H). repeat match goal with H : _ |- _ => revert H end. intro. >> *)

Lemma

Require Import Functor.Core NaturalTransformation.Core. Require Import Category.Strict. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core Pseudofunctor.RewriteLaws. Require Import NaturalTransformation.Composition.Laws. Require Import FunctorCategory.Morphisms. Require Import HoTT.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategory.

Categories\LaxComma\CoreLaws.v

right_identity

{x x0 : PreCategory} {x1 : Functor x0 x} {x2 x3 : PreCategory} {x4 : Functor x3 x2} {x5 x6 : Functor x3 x0} {x7 : Functor x2 x} {x8 : NaturalTransformation (x7 o x4) (x1 o x6)} {x9 : Functor x2 x} {x10 : Functor x3 x3} {x11 : Functor x2 x2} {x12 : @Isomorphic (_ -> _) x11 1%functor} {x13 : @Isomorphic (_ -> _) x10 1%functor} {x14 : @Isomorphic (_ -> _) x9 (x7 o x11)%functor} {x15 : @Isomorphic (_ -> _) x5 (x6 o x10)%functor} {x16 : @Isomorphic (_ -> _) x5 x6} {x17 : (x16 : Category.Core.morphism _ _ _) = (right_identity_natural_transformation_1 x6 o (x6 oL (x13 : Category.Core.morphism _ _ _) o (x15 : Category.Core.morphism _ _ _)))%natural_transformation} {x18 : @Isomorphic (_ -> _) x9 x7} {x19 : x18 ^-1 = (x14 ^-1 o (x7 oL x12 ^-1) o inverse (right_identity_natural_transformation_1 x7))%natural_transformation} : (x1 oL (x16 : Category.Core.morphism _ _ _) o (x1 oL x15 ^-1 o associator_1 x1 x6 x10 o (x8 oR x10) o associator_2 x7 x4 x10 o (x7 oL (x4 oL x13 ^-1 o right_identity_natural_transformation_2 x4 o left_identity_natural_transformation_1 x4 o ((x12 : Category.Core.morphism _ _ _) oR x4))) o associator_1 x7 x11 x4 o ((x14 : Category.Core.morphism _ _ _) oR x4)) o (x18 ^-1 oR x4))%natural_transformation = x8. Proof. t. Qed.

Lemma

Require Import Functor.Core NaturalTransformation.Core. Require Import Category.Strict. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core Pseudofunctor.RewriteLaws. Require Import NaturalTransformation.Composition.Laws. Require Import FunctorCategory.Morphisms. Require Import HoTT.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategory.

Categories\LaxComma\CoreLaws.v

right_identity_helper

(s d : object) (m : morphism s d) : compose m (identity _) = m. Proof. refine (@path_morphism' _ A B S T _ _ (compose m (identity _)) m (Category.Core. _ _ _ _) (Category.Core. _ _ _ _) _). simpl. refine right_identity_helper. - exact (p_right_identity_of_coherent_for_rewrite _ _ _ _). - exact (p_right_identity_of_coherent_inverse_for_rewrite _ _ _ _). Defined.

Lemma

Require Import Functor.Core NaturalTransformation.Core. Require Import Category.Strict. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core Pseudofunctor.RewriteLaws. Require Import NaturalTransformation.Composition.Laws. Require Import FunctorCategory.Morphisms. Require Import HoTT.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategory.

Categories\LaxComma\CoreLaws.v

right_identity

Record

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Trunc Types.Sigma. Require Import Basics.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategoryParts.

Categories\LaxComma\CoreParts.v

object

object_sig_T <~> object. Proof. issig. Defined.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Trunc Types.Sigma. Require Import Basics.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategoryParts.

Categories\LaxComma\CoreParts.v

issig_object

(x y : object) : forall (Ha : x.(a) = y.(a)) (Hb : x.(b) = y.(b)), match Ha in _ = X, Hb in _ = Y return Functor (S X) (T Y) with | idpath, idpath => x.(f) end = y.(f) -> x = y. Proof. destruct x, y; simpl. intros; path_induction; reflexivity. Defined.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Trunc Types.Sigma. Require Import Basics.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategoryParts.

Categories\LaxComma\CoreParts.v

path_object

x y (H : { HaHb : (x.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Trunc Types.Sigma. Require Import Basics.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategoryParts.

Categories\LaxComma\CoreParts.v

path_object_uncurried

x y Ha Hb Hf : ap (@a) (@path_object x y Ha Hb Hf) = Ha. Proof. destruct x, y; simpl in *. destruct Ha, Hb, Hf; simpl in *. reflexivity. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Trunc Types.Sigma. Require Import Basics.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategoryParts.

Categories\LaxComma\CoreParts.v

ap_a_path_object

x y Ha Hb Hf : ap (@b) (@path_object x y Ha Hb Hf) = Hb. Proof. destruct x, y; simpl in *. destruct Ha, Hb, Hf; simpl in *. reflexivity. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Trunc Types.Sigma. Require Import Basics.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategoryParts.

Categories\LaxComma\CoreParts.v

ap_b_path_object

(abf a'b'f' : object) :=

Record

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Trunc Types.Sigma. Require Import Basics.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategoryParts.

Categories\LaxComma\CoreParts.v

morphism

abf a'b'f' : (morphism_sig_T abf a'b'f') <~> morphism abf a'b'f'. Proof. issig. Defined.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Trunc Types.Sigma. Require Import Basics.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategoryParts.

Categories\LaxComma\CoreParts.v

issig_morphism

abf a'b'f' (gh g'h' : morphism abf a'b'f') : forall (Hg : gh.(g) = g'h'.(g)) (Hh : gh.(h) = g'h'.(h)), match Hg in _ = g, Hh in _ = h return NaturalTransformation (p_morphism_of T h o abf.(f)) (a'b'f'.(f) o p_morphism_of S g) with | idpath, idpath => gh.(p) end = g'h'.(p) -> gh = g'h'. Proof. intros Hg Hh Hp. destruct gh, g'h'; simpl in *. destruct Hg, Hh, Hp. reflexivity. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Trunc Types.Sigma. Require Import Basics.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategoryParts.

Categories\LaxComma\CoreParts.v

path_morphism

abf a'b'f' gh g'h' (H : { HgHh : (gh.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Trunc Types.Sigma. Require Import Basics.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategoryParts.

Categories\LaxComma\CoreParts.v

path_morphism_uncurried

abf a'b'f' (gh g'h' : morphism abf a'b'f') : forall (Hg : gh.(g) = g'h'.(g)) (Hh : gh.(h) = g'h'.(h)), ((_ oL (Category.Morphisms.idtoiso (_ -> _) (ap (@p_morphism_of _ _ S _ _) Hg) : Category.Core.morphism _ _ _)) o (gh.(p)) o ((Category.Morphisms.idtoiso (_ -> _) (ap (@p_morphism_of _ _ T _ _) Hh) : Category.Core.morphism _ _ _)^-1 oR _) = g'h'.(p))%natural_transformation -> match Hg in _ = g, Hh in _ = h return NaturalTransformation (p_morphism_of T h o abf.(f)) (a'b'f'.(f) o p_morphism_of S g) with | idpath, idpath => gh.(p) end = g'h'.(p). Proof. simpl; intros Hg Hh Hp. destruct g'h'; simpl in *. destruct Hg, Hh, Hp; simpl in *. path_natural_transformation. autorewrite with functor morphism. reflexivity. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Trunc Types.Sigma. Require Import Basics.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategoryParts.

Categories\LaxComma\CoreParts.v

path_morphism'_helper

path_morphism' abf a'b'f' (gh g'h' : morphism abf a'b'f') (Hg : gh.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Trunc Types.Sigma. Require Import Basics.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategoryParts.

Categories\LaxComma\CoreParts.v

path_morphism'

abf a'b'f' gh g'h' (H : { HgHh : (gh.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Trunc Types.Sigma. Require Import Basics.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategoryParts.

Categories\LaxComma\CoreParts.v

path_morphism'_uncurried

s d d' (gh : morphism d d') (g'h' : morphism s d) : morphism s d'. Proof. exists (gh.(g) o g'h'.(g)) (gh.(h) o g'h'.(h)). exact ((_ oL (p_composition_of S _ _ _ _ _)^-1) o (associator_1 _ _ _) o (gh.(p) oR _) o (associator_2 _ _ _) o (_ oL g'h'.(p)) o (associator_1 _ _ _) o ((p_composition_of T _ _ _ _ _ : Category.Core.morphism _ _ _) oR _))%natural_transformation. Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Trunc Types.Sigma. Require Import Basics.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategoryParts.

Categories\LaxComma\CoreParts.v

compose

x : morphism x x. Proof. exists ( (x.(a))) ( (x.(b))). exact ((_ oL (p_identity_of S _ : Category.Core.morphism _ _ _)^-1) o (right_identity_natural_transformation_2 _) o (left_identity_natural_transformation_1 _) o ((p_identity_of T _ : Category.Core.morphism _ _ _) oR _))%natural_transformation. Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Category.Morphisms FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Trunc Types.Sigma. Require Import Basics.Tactics. Import Functor.Identity.FunctorIdentityNotations. Module Import LaxCommaCategoryParts.

Categories\LaxComma\CoreParts.v

identity

Functor (1 -> C) (D -> C) := @pullback_along _ D 1 C (Functors.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import ExponentialLaws.Law1.Functors FunctorCategory.Core. Require Import KanExtensions.Core InitialTerminalCategory.Core NatCategory. Require Import Functor.Paths. Import Comma.Core.

Categories\Limits\Core.v

diagonal_functor

diagonal_functor' : Functor C (D -> C) := diagonal_functor o ExponentialLaws.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import ExponentialLaws.Law1.Functors FunctorCategory.Core. Require Import KanExtensions.Core InitialTerminalCategory.Core NatCategory. Require Import Functor.Paths. Import Comma.Core.

Categories\Limits\Core.v

diagonal_functor'

diagonal_functor_diagonal_functor' X : diagonal_functor X = diagonal_functor' (X (center _)). Proof. path_functor. simpl. repeat (apply path_forall || intro). apply identity_of. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import ExponentialLaws.Law1.Functors FunctorCategory.Core. Require Import KanExtensions.Core InitialTerminalCategory.Core NatCategory. Require Import Functor.Paths. Import Comma.Core.

Categories\Limits\Core.v

diagonal_functor_diagonal_functor'

x (F : Functor D' D) : diagonal_functor C D x o F = diagonal_functor _ _ x. Proof. path_functor. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import ExponentialLaws.Law1.Functors FunctorCategory.Core. Require Import KanExtensions.Core InitialTerminalCategory.Core NatCategory. Require Import Functor.Paths. Import Comma.Core.

Categories\Limits\Core.v

compose_diagonal_functor

x (F : Functor D' D) : NaturalTransformation (diagonal_functor C D x o F) (diagonal_functor _ _ x) := Build_NaturalTransformation (diagonal_functor C D x o F) (diagonal_functor _ _ x) (fun z => identity _) (fun _ _ _ => transitivity (left_identity _ _ _ _) (symmetry _ _ (right_identity _ _ _ _))).

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import ExponentialLaws.Law1.Functors FunctorCategory.Core. Require Import KanExtensions.Core InitialTerminalCategory.Core NatCategory. Require Import Functor.Paths. Import Comma.Core.

Categories\Limits\Core.v

compose_diagonal_functor_natural_transformation

@IsRightKanExtensionAlong _ D 1 C (Functors.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import ExponentialLaws.Law1.Functors FunctorCategory.Core. Require Import KanExtensions.Core InitialTerminalCategory.Core NatCategory. Require Import Functor.Paths. Import Comma.Core.

Categories\Limits\Core.v

IsLimit

@IsLeftKanExtensionAlong _ D 1 C (Functors.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import ExponentialLaws.Law1.Functors FunctorCategory.Core. Require Import KanExtensions.Core InitialTerminalCategory.Core NatCategory. Require Import Functor.Paths. Import Comma.Core.

Categories\Limits\Core.v

IsColimit

(lim_obj : C) (lim_mor : morphism (D -> C) (diagonal_functor' C D lim_obj) F) (lim := CommaCategory.Build_object (diagonal_functor C D) !(F : object (_ -> _)) !lim_obj (center _) lim_mor) (UniversalProperty : forall (lim_obj' : C) (lim_mor' : morphism (D -> C) (diagonal_functor' C D lim_obj') F), Contr { m : morphism C lim_obj' lim_obj | lim_mor o morphism_of (diagonal_functor' C D) m = lim_mor' }) : IsTerminalMorphism lim. Proof. apply Build_IsTerminalMorphism. intros A' p'. specialize (UniversalProperty (A' (center _))).*) End Limit.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core. Require Import ExponentialLaws.Law1.Functors FunctorCategory.Core. Require Import KanExtensions.Core InitialTerminalCategory.Core NatCategory. Require Import Functor.Paths. Import Comma.Core.

Categories\Limits\Core.v

Build_IsLimit

Functor (D -> C) (1 -> C) := left_kan_extension_functor has_colimits.

Definition

Require Import Category.Core Functor.Core. Require Import KanExtensions.Functors. Require Import Limits.Core. Require Import FunctorCategory.Core. Require Import Adjoint.Core. Require Import NatCategory.

Categories\Limits\Functors.v

colimit_functor

colimit_functor -| diagonal_functor _ _ := left_kan_extension_adjunction has_colimits.

Definition

Require Import Category.Core Functor.Core. Require Import KanExtensions.Functors. Require Import Limits.Core. Require Import FunctorCategory.Core. Require Import Adjoint.Core. Require Import NatCategory.

Categories\Limits\Functors.v

colimit_adjunction

Functor (D -> C) (1 -> C) := right_kan_extension_functor has_limits.

Definition

Require Import Category.Core Functor.Core. Require Import KanExtensions.Functors. Require Import Limits.Core. Require Import FunctorCategory.Core. Require Import Adjoint.Core. Require Import NatCategory.

Categories\Limits\Functors.v

limit_functor

diagonal_functor _ _ -| limit_functor := right_kan_extension_adjunction has_limits.

Definition

Require Import Category.Core Functor.Core. Require Import KanExtensions.Functors. Require Import Limits.Core. Require Import FunctorCategory.Core. Require Import Adjoint.Core. Require Import NatCategory.

Categories\Limits\Functors.v

limit_adjunction

(tensor ∘ (Functor.

Definition

Require Import Basics.Utf8. Require Import Category.Core Category.Morphisms. Require Import Functor.Core Require Import NaturalTransformation.Core. Require Import FunctorCategory.Core FunctorCategory.Morphisms. Require Import ProductLaws.

Categories\Monoidal\MonoidalCategory.v

right_assoc

tensor ∘ (Functor.

Definition

Require Import Basics.Utf8. Require Import Category.Core Category.Morphisms. Require Import Functor.Core Require Import NaturalTransformation.Core. Require Import FunctorCategory.Core FunctorCategory.Morphisms. Require Import ProductLaws.

Categories\Monoidal\MonoidalCategory.v

left_assoc

NaturalIsomorphism right_assoc left_assoc.

Definition

Require Import Basics.Utf8. Require Import Category.Core Category.Morphisms. Require Import Functor.Core Require Import NaturalTransformation.Core. Require Import FunctorCategory.Core FunctorCategory.Morphisms. Require Import ProductLaws.

Categories\Monoidal\MonoidalCategory.v

associator

(A : C) := Core.

Definition

Require Import Basics.Utf8. Require Import Category.Core Category.Morphisms. Require Import Functor.Core Require Import NaturalTransformation.Core. Require Import FunctorCategory.Core FunctorCategory.Morphisms. Require Import ProductLaws.

Categories\Monoidal\MonoidalCategory.v

pretensor

pretensor I.

Definition

Require Import Basics.Utf8. Require Import Category.Core Category.Morphisms. Require Import Functor.Core Require Import NaturalTransformation.Core. Require Import FunctorCategory.Core FunctorCategory.Morphisms. Require Import ProductLaws.

Categories\Monoidal\MonoidalCategory.v

I_pretensor

(A : C) := Core.

Definition

Require Import Basics.Utf8. Require Import Category.Core Category.Morphisms. Require Import Functor.Core Require Import NaturalTransformation.Core. Require Import FunctorCategory.Core FunctorCategory.Morphisms. Require Import ProductLaws.

Categories\Monoidal\MonoidalCategory.v

posttensor