Datasets:

phanerozoic

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Coq-HoTT

like 2

s d (m : morphism (C1 -> (C2 -> D)) s d) : morphism (C1 * C2 -> D) (functor_object_of s) (functor_object_of d). Proof. simpl. refine (Build_NaturalTransformation (functor_object_of s) (functor_object_of d) (fun c => m (fst c) (snd c)) _); abstract ( repeat match goal with | [ |- context[components_of ?T ?x o components_of ?U ?x] ] => change (T x o U x) with ((compose (C := (_ -> _)) T U) x) | _ => f_ap | _ => rewrite !commutes | _ => do_exponential4 end ). Defined.

Definition

Require Import Category.Core Category.Prod FunctorCategory.Core Functor.Core NaturalTransformation.Core NaturalTransformation.Paths. Require Import Basics.Tactics.

Categories\ExponentialLaws\Law4\Functors.v

functor_morphism_of

Functor (C1 -> (C2 -> D)) (C1 * C2 -> D). Proof. refine (Build_Functor (C1 -> (C2 -> D)) (C1 * C2 -> D) functor_object_of functor_morphism_of _ _); abstract by path_natural_transformation. Defined.

Definition

Require Import Category.Core Category.Prod FunctorCategory.Core Functor.Core NaturalTransformation.Core NaturalTransformation.Paths. Require Import Basics.Tactics.

Categories\ExponentialLaws\Law4\Functors.v

functor

C1 -> (C2 -> D)%category. Proof. intro c1. refine (Build_Functor C2 D (fun c2 => F (c1, c2)) (fun s2 d2 m2 => F _1 ((identity c1, m2) : morphism (_ * _) (c1, s2) (c1, d2))) _ _); abstract do_exponential4_inverse. Defined.

Definition

Require Import Category.Core Category.Prod FunctorCategory.Core Functor.Core NaturalTransformation.Core NaturalTransformation.Paths. Require Import Basics.Tactics.

Categories\ExponentialLaws\Law4\Functors.v

inverse_object_of_object_of

s d (m : morphism C1 s d) : morphism (C2 -> D) (inverse_object_of_object_of s) (inverse_object_of_object_of d). Proof. refine (Build_NaturalTransformation (inverse_object_of_object_of s) (inverse_object_of_object_of d) (fun c => F _1 ((m, identity c) : morphism (_ * _) (s, c) (d, c))) _); abstract do_exponential4_inverse. Defined.

Definition

Require Import Category.Core Category.Prod FunctorCategory.Core Functor.Core NaturalTransformation.Core NaturalTransformation.Paths. Require Import Basics.Tactics.

Categories\ExponentialLaws\Law4\Functors.v

inverse_object_of_morphism_of

(C1 -> (C2 -> D))%category. Proof. refine (Build_Functor C1 (C2 -> D) inverse_object_of_object_of inverse_object_of_morphism_of _ _); abstract (path_natural_transformation; do_exponential4_inverse). Defined.

Definition

Require Import Category.Core Category.Prod FunctorCategory.Core Functor.Core NaturalTransformation.Core NaturalTransformation.Paths. Require Import Basics.Tactics.

Categories\ExponentialLaws\Law4\Functors.v

inverse_object_of

s d (m : morphism (C1 * C2 -> D) s d) : forall c, morphism (C2 -> D) ((inverse_object_of s) c) ((inverse_object_of d) c). Proof. intro c. refine (Build_NaturalTransformation ((inverse_object_of s) c) ((inverse_object_of d) c) (fun c' => m (c, c')) _). abstract do_exponential4_inverse. Defined.

Definition

Require Import Category.Core Category.Prod FunctorCategory.Core Functor.Core NaturalTransformation.Core NaturalTransformation.Paths. Require Import Basics.Tactics.

Categories\ExponentialLaws\Law4\Functors.v

inverse_morphism_of_components_of

s d (m : morphism (C1 * C2 -> D) s d) : morphism (C1 -> (C2 -> D)) (inverse_object_of s) (inverse_object_of d). Proof. refine (Build_NaturalTransformation (inverse_object_of s) (inverse_object_of d) (inverse_morphism_of_components_of m) _). abstract (path_natural_transformation; do_exponential4_inverse). Defined.

Definition

Require Import Category.Core Category.Prod FunctorCategory.Core Functor.Core NaturalTransformation.Core NaturalTransformation.Paths. Require Import Basics.Tactics.

Categories\ExponentialLaws\Law4\Functors.v

inverse_morphism_of

Functor (C1 * C2 -> D) (C1 -> (C2 -> D)). Proof. refine (Build_Functor (C1 * C2 -> D) (C1 -> (C2 -> D)) inverse_object_of inverse_morphism_of _ _); abstract by path_natural_transformation. Defined.

Definition

Require Import Category.Core Category.Prod FunctorCategory.Core Functor.Core NaturalTransformation.Core NaturalTransformation.Paths. Require Import Basics.Tactics.

Categories\ExponentialLaws\Law4\Functors.v

inverse

c : functor C1 C2 D (inverse C1 C2 D c) = c. Proof. path_functor. abstract ( exp_laws_t; rewrite <- composition_of; exp_laws_t ). Defined.

Lemma

Require Import Category.Core Functor.Core. Require Import Functor.Paths. Require Import Functor.Identity Functor.Composition.Core. Require Import ExponentialLaws.Law4.Functors. Require Import ExponentialLaws.Tactics.

Categories\ExponentialLaws\Law4\Law.v

helper1

c x : inverse C1 C2 D (functor C1 C2 D c) x = c x. Proof. path_functor. abstract exp_laws_t. Defined.

Lemma

Require Import Category.Core Functor.Core. Require Import Functor.Paths. Require Import Functor.Identity Functor.Composition.Core. Require Import ExponentialLaws.Law4.Functors. Require Import ExponentialLaws.Tactics.

Categories\ExponentialLaws\Law4\Law.v

helper2_helper

c : inverse C1 C2 D (functor C1 C2 D c) = c. Proof. path_functor. exists (path_forall _ _ (helper2_helper c)). abstract (unfold helper2_helper; exp_laws_t). Defined.

Lemma

Require Import Category.Core Functor.Core. Require Import Functor.Paths. Require Import Functor.Identity Functor.Composition.Core. Require Import ExponentialLaws.Law4.Functors. Require Import ExponentialLaws.Tactics.

Categories\ExponentialLaws\Law4\Law.v

helper2

functor C1 C2 D o inverse C1 C2 D = 1 /\ inverse C1 C2 D o functor C1 C2 D = 1. Proof. split; path_functor; [ (exists (path_forall _ _ helper1)) | (exists (path_forall _ _ helper2)) ]; unfold helper1, helper2, helper2_helper; exp_laws_t. Qed.

Lemma

Require Import Category.Core Functor.Core. Require Import Functor.Paths. Require Import Functor.Identity Functor.Composition.Core. Require Import ExponentialLaws.Law4.Functors. Require Import ExponentialLaws.Tactics.

Categories\ExponentialLaws\Law4\Law.v

law

NaturalTransformation (hom_functor C) (hom_functor D o (F^op, F)). Proof. refine (Build_NaturalTransformation (hom_functor C) (hom_functor D o (F^op, F)) (fun (sd : object (C^op * C)) m => (F _1 m)%morphism) _ ). abstract ( repeat (intros [] || intro); simpl in *; repeat (apply path_forall; intro); simpl; rewrite !composition_of; reflexivity ). Defined.

Definition

Require Import Category.Core Functor.Core HomFunctor Category.Morphisms Category.Dual Functor.Dual Category.Prod Functor.Prod NaturalTransformation.Core SetCategory.Core Functor.Composition.Core. Require Import Basics.Trunc Types.Universe HIT.iso HoTT.Truncations.Core.

Categories\Functor\Attributes.v

induced_hom_natural_transformation

`{IsFull} `{IsFaithful} (H' : forall x y (m : morphism set_cat x y), IsEpimorphism m -> IsMonomorphism m -> IsIsomorphism m) : IsFullyFaithful. Proof. intros ? ?; hnf in * |- . apply H'; eauto. Qed.

Lemma

Require Import Category.Core Functor.Core HomFunctor Category.Morphisms Category.Dual Functor.Dual Category.Prod Functor.Prod NaturalTransformation.Core SetCategory.Core Functor.Composition.Core. Require Import Basics.Trunc Types.Universe HIT.iso HoTT.Truncations.Core.

Categories\Functor\Attributes.v

isfullyfaithful_isfull_isfaithful_helper

`{H' : IsEquiv _ _ m} : IsIsomorphism (m : morphism set_cat x y). Proof. exists (m^-1)%core; apply path_forall; intro; destruct H'; simpl in *; eauto. Qed.

Lemma

Require Import Category.Core Functor.Core HomFunctor Category.Morphisms Category.Dual Functor.Dual Category.Prod Functor.Prod NaturalTransformation.Core SetCategory.Core Functor.Composition.Core. Require Import Basics.Trunc Types.Universe HIT.iso HoTT.Truncations.Core.

Categories\Functor\Attributes.v

isisomorphism_isequiv_set_cat

(Hepi : IsEpimorphism (m : morphism set_cat x y)) (Hmono : IsMonomorphism (m : morphism set_cat x y)) : @IsEquiv _ _ m (* NB: This depends on the (arguably accidental) fact that `ismono` and `isepi` from HoTT core are *definitionally* identical to the specialization of `IsMonomorphism` and `IsEpimorphism` to the category of sets.

Definition

Require Import Category.Core Functor.Core HomFunctor Category.Morphisms Category.Dual Functor.Dual Category.Prod Functor.Prod NaturalTransformation.Core SetCategory.Core Functor.Composition.Core. Require Import Basics.Trunc Types.Universe HIT.iso HoTT.Truncations.Core.

Categories\Functor\Attributes.v

isequiv_isepimorphism_ismonomorphism

Record

Require Import Category.Core.

Categories\Functor\Core.v

Functor

C D (F : Functor C D) : Functor C^op D^op := Build_Functor (C^op) (D^op) (object_of F) (fun s d (m : morphism C^op s d) => (F _1 m)%morphism) (fun d' d s m1 m2 => composition_of F s d d' m2 m1) (identity_of F).

Definition

Import Category.Dual.CategoryDualNotations. Require Import Category.Core Functor.Core.

Categories\Functor\Dual.v

opposite

C D (F : Functor C D) : (F^op)^op = F := idpath.

Definition

Import Category.Dual.CategoryDualNotations. Require Import Category.Core Functor.Core.

Categories\Functor\Dual.v

opposite_involutive

C : Functor C C := Build_Functor C C (fun x => x) (fun _ _ x => x) (fun _ _ _ _ _ => idpath) (fun _ => idpath).

Definition

Require Import Category.Core Functor.Core.

Categories\Functor\Identity.v

identity

functor_sig_T <~> Functor C D. Proof. issig. Defined.

Lemma

Require Import Category.Core Functor.Core. Require Import HoTT.Basics HoTT.Types HoTT.Tactics.

Categories\Functor\Paths.v

equiv_sig_functor

(F G : Functor C D) : path_functor'_T F G <~> (@equiv_inv _ _ _ equiv_sig_functor F = @equiv_inv _ _ _ equiv_sig_functor G). Proof. etransitivity; [ | by apply equiv_path_sigma ]. eapply @equiv_functor_sigma. repeat match goal with | [ |- context[(@equiv_inv ?A ?B ?f ?H ?F).1] ] => change ((@equiv_inv A B f H F).1) with (object_of F) end. Time exact (isequiv_idmap (object_of F = object_of G)). Abort. >> *)

Definition

Require Import Category.Core Functor.Core. Require Import HoTT.Basics HoTT.Types HoTT.Tactics.

Categories\Functor\Paths.v

equiv_path_functor_uncurried_sig

(F G : Functor C D) : path_functor'_T F G -> F = G. Proof. intros [? ?]. destruct F, G; simpl in *. path_induction; simpl. f_ap; eapply @center; abstract exact _. Defined.

Definition

Require Import Category.Core Functor.Core. Require Import HoTT.Basics HoTT.Types HoTT.Tactics.

Categories\Functor\Paths.v

path_functor_uncurried

F G HO HM : ap object_of (@path_functor_uncurried F G (HO; HM)) = HO. Proof. destruct F, G; simpl in *. path_induction_hammer. Qed.

Lemma

Require Import Category.Core Functor.Core. Require Import HoTT.Basics HoTT.Types HoTT.Tactics.

Categories\Functor\Paths.v

path_functor_uncurried_fst

F : @path_functor_uncurried F F (idpath; idpath) = idpath. Proof. destruct F; simpl in *. rewrite !(contr idpath). reflexivity. Qed.

Lemma

Require Import Category.Core Functor.Core. Require Import HoTT.Basics HoTT.Types HoTT.Tactics.

Categories\Functor\Paths.v

path_functor_uncurried_idpath

(F G : Functor C D) : F = G -> path_functor'_T F G := fun H' => (ap object_of H'; (transport_compose _ object_of _ _) ^ @ apD (@morphism_of _ _) H')%path.

Definition

Require Import Category.Core Functor.Core. Require Import HoTT.Basics HoTT.Types HoTT.Tactics.

Categories\Functor\Paths.v

path_functor_uncurried_inv

(F G : Functor C D) (HO : object_of F = object_of G) (HM : transport (fun GO => forall s d, morphism C s d -> morphism D (GO s) (GO d)) HO (morphism_of F) = morphism_of G) : F = G := path_functor_uncurried F G (HO; HM).

Definition

Require Import Category.Core Functor.Core. Require Import HoTT.Basics HoTT.Types HoTT.Tactics.

Categories\Functor\Paths.v

path_functor

(F G : Functor C D) (HO : object_of F == object_of G) (HM : forall s d m, transport (fun GO => forall s d, morphism C s d -> morphism D (GO s) (GO d)) (path_forall _ _ HO) (morphism_of F) s d m = G _1 m) : F = G. Proof. refine (path_functor F G (path_forall _ _ HO) _). repeat (apply path_forall; intro); apply HM. Defined.

Definition

Require Import Category.Core Functor.Core. Require Import HoTT.Basics HoTT.Types HoTT.Tactics.

Categories\Functor\Paths.v

path_functor_pointwise

(F G : Functor C D) : path_functor'_T F G <~> F = G := Build_Equiv _ _ (@path_functor_uncurried F G) _.

Definition

Require Import Category.Core Functor.Core. Require Import HoTT.Basics HoTT.Types HoTT.Tactics.

Categories\Functor\Paths.v

equiv_path_functor_uncurried

(F G : Functor C D) (p q : F = G) : ap object_of p = ap object_of q -> p = q. Proof. refine ((ap (@path_functor_uncurried F G)^-1)^-1 o _). refine ((path_sigma_uncurried _ _ _) o _); simpl. refine (pr1^-1). Defined.

Definition

Require Import Category.Core Functor.Core. Require Import HoTT.Basics HoTT.Types HoTT.Tactics.

Categories\Functor\Paths.v

path_path_functor_uncurried

Functor C (C + D) := Build_Functor C (C + D) (@ _ _) (fun _ _ m => m) (fun _ _ _ _ _ => idpath) (fun _ => idpath).

Definition

Require Import Category.Sum Functor.Core Functor.Composition.Core Functor.Identity. Require Import Functor.Paths HoTT.Tactics Types.Forall. Require Import Basics.Tactics.

Categories\Functor\Sum.v

inl

Functor D (C + D) := Build_Functor D (C + D) (@ _ _) (fun _ _ m => m) (fun _ _ _ _ _ => idpath) (fun _ => idpath).

Definition

Require Import Category.Sum Functor.Core Functor.Composition.Core Functor.Identity. Require Import Functor.Paths HoTT.Tactics Types.Forall. Require Import Basics.Tactics.

Categories\Functor\Sum.v

inr

(F : Functor C D) (F' : Functor C' D) : Functor (C + C') D. Proof. refine (Build_Functor (C + C') D (fun cc' => match cc' with | type_inl c => F c | type_inr c' => F' c' end) (fun s d => match s, d with | type_inl cs, type_inl cd => fun m : morphism _ cs cd => F _1 m | type_inr c's, type_inr c'd => fun m : morphism _ c's c'd => F' _1 m | _, _ => fun m => match m with end end%morphism) _ _); abstract ( repeat (intros [] || intro); simpl in *; auto with functor ). Defined.

Definition

Require Import Category.Sum Functor.Core Functor.Composition.Core Functor.Identity. Require Import Functor.Paths HoTT.Tactics Types.Forall. Require Import Basics.Tactics.

Categories\Functor\Sum.v

sum

C D : Functor (C + D) (D + C) := sum (inr _ _) (inl _ _).

Definition

Require Import Category.Sum Functor.Core Functor.Composition.Core Functor.Identity. Require Import Functor.Paths HoTT.Tactics Types.Forall. Require Import Basics.Tactics.

Categories\Functor\Sum.v

swap

{C D} c : (swap C D) ((swap D C) c) = c := match c with type_inl _ => idpath | type_inr _ => idpath end.

Definition

Require Import Category.Sum Functor.Core Functor.Composition.Core Functor.Identity. Require Import Functor.Paths HoTT.Tactics Types.Forall. Require Import Basics.Tactics.

Categories\Functor\Sum.v

swap_involutive_helper

`{Funext} C D : swap C D o swap D C = 1. Proof. path_functor. exists (path_forall _ _ swap_involutive_helper). repeat (apply (@path_forall _); intro). repeat match goal with | [ |- context[transport (fun x' => forall y, @?C x' y) ?p ?f ?x] ] => simpl rewrite (@transport_forall_constant _ _ C _ _ p f x) end. transport_path_forall_hammer. by repeat match goal with | [ H : Empty |- _ ] => destruct H | [ H : (_ + _) |- _ ] => destruct H | _ => progress hnf in * end. Qed.

Lemma

Require Import Category.Sum Functor.Core Functor.Composition.Core Functor.Identity. Require Import Functor.Paths HoTT.Tactics Types.Forall. Require Import Basics.Tactics.

Categories\Functor\Sum.v

swap_involutive

s d d' (m1 : morphism C s d) (m2 : morphism C d d') : c_morphism_of (m2 o m1) = c_morphism_of m2 o c_morphism_of m1 := transport (@paths _ (c_morphism_of (m2 o m1))) (composition_of G _ _ _ _ _) (ap (fun m => G _1 m) (composition_of F _ _ _ m1 m2)).

Definition

Require Import Category.Core Functor.Core.

Categories\Functor\Composition\Core.v

compose_composition_of

x : c_morphism_of (identity x) = identity (c_object_of x) := transport (@paths _ _) (identity_of G _) (ap (fun m => G _1 m) (identity_of F x)).

Definition

Require Import Category.Core Functor.Core.

Categories\Functor\Composition\Core.v

compose_identity_of

Functor C E := Build_Functor C E (fun c => G (F c)) (fun _ _ m => G _1 (F _1 m)) compose_composition_of compose_identity_of.

Definition

Require Import Category.Core Functor.Core.

Categories\Functor\Composition\Core.v

compose

(F : Functor C D) : 1 o F = F. Proof. by path_functor. Defined.

Lemma

Require Import Category.Core Functor.Core Functor.Composition.Core Functor.Identity. Require Import Functor.Paths. Require Import Basics.PathGroupoids Basics.Tactics.

Categories\Functor\Composition\Laws.v

left_identity

(F : Functor C D) : F o 1 = F. Proof. by path_functor. Defined.

Lemma

Require Import Category.Core Functor.Core Functor.Composition.Core Functor.Identity. Require Import Functor.Paths. Require Import Basics.PathGroupoids Basics.Tactics.

Categories\Functor\Composition\Laws.v

right_identity

F : ap object_of (left_identity F) = idpath := @path_functor_uncurried_fst _ _ _ (1 o F) F 1 1.

Definition

Require Import Category.Core Functor.Core Functor.Composition.Core Functor.Identity. Require Import Functor.Paths. Require Import Basics.PathGroupoids Basics.Tactics.

Categories\Functor\Composition\Laws.v

left_identity_fst

F : ap object_of (right_identity F) = idpath := @path_functor_uncurried_fst _ _ _ (F o 1) F 1 1.

Definition

Require Import Category.Core Functor.Core Functor.Composition.Core Functor.Identity. Require Import Functor.Paths. Require Import Basics.PathGroupoids Basics.Tactics.

Categories\Functor\Composition\Laws.v

right_identity_fst

(F : Functor B C) (G : Functor C D) (H : Functor D E) : (H o G) o F = H o (G o F). Proof. by path_functor. Defined.

Lemma

Require Import Category.Core Functor.Core Functor.Composition.Core Functor.Identity. Require Import Functor.Paths. Require Import Basics.PathGroupoids Basics.Tactics.

Categories\Functor\Composition\Laws.v

associativity

F G H : ap object_of (associativity F G H) = idpath := @path_functor_uncurried_fst _ _ _ ((H o G) o F) (H o (G o F)) 1%path 1%path.

Definition

Require Import Category.Core Functor.Core Functor.Composition.Core Functor.Identity. Require Import Functor.Paths. Require Import Basics.PathGroupoids Basics.Tactics.

Categories\Functor\Composition\Laws.v

associativity_fst

C D E (F : Functor C D) (G : Functor D E) : (associativity F 1 G @ ap (compose G) (left_identity F)) = (ap (fun G' : Functor D E => G' o F) (right_identity G)). Proof. coherence_t. Qed.

Lemma

Require Import Category.Core Functor.Core Functor.Composition.Core Functor.Identity. Require Import Functor.Paths. Require Import Basics.PathGroupoids Basics.Tactics.

Categories\Functor\Composition\Laws.v

triangle

A B C D E (F : Functor A B) (G : Functor B C) (H : Functor C D) (K : Functor D E) : (associativity F G (K o H) @ associativity (G o F) H K) = (ap (fun KHG => KHG o F) (associativity G H K) @ associativity F (H o G) K @ ap (compose K) (associativity F G H)). Proof. coherence_t. Qed.

Lemma

Require Import Category.Core Functor.Core Functor.Composition.Core Functor.Identity. Require Import Functor.Paths. Require Import Basics.PathGroupoids Basics.Tactics.

Categories\Functor\Composition\Laws.v

pentagon

(F G : Functor B C) (T U : NaturalTransformation F G) (a : A) (b : B) (f : H a <~=~> b) (H' : T oR H = U oR H) : T b = U b. Proof. apply (ap components_of) in H'. apply apD10 in H'; hnf in H'; simpl in H'. rewrite <- !(path_components_of_isomorphic' f). rewrite H'. reflexivity. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Functorial.Core. Require Import NaturalTransformation.Composition.Core. Require Import NaturalTransformation.Isomorphisms. Require Import Functor.Attributes. Require Import FunctorCategory.Core. Require Import Category.Morphisms. Require Import NaturalTransformation.Paths. Require Import HoTT.Truncations.Core.

Categories\Functor\Composition\Functorial\Attributes.v

isfaithful_precomposition_essentially_surjective_helper

s d (m : morphism (C -> D) s d) : morphism ((D -> E) -> (C -> E)) (whiskerR_functor _ s) (whiskerR_functor _ d) := Build_NaturalTransformation (whiskerR_functor E s) (whiskerR_functor E d) (fun x => x oL m) (fun _ _ _ => exchange_whisker _ _).

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import Category.Prod FunctorCategory.Core NaturalTransformation.Composition.Functorial NaturalTransformation.Composition.Laws ExponentialLaws.Law4.Functors. Require Import NaturalTransformation.Paths.

Categories\Functor\Composition\Functorial\Core.v

compose_functor_morphism_of

object ((C -> D) -> ((D -> E) -> (C -> E))). Proof. refine (Build_Functor (C -> D) ((D -> E) -> (C -> E)) (@whiskerR_functor _ _ _ _) compose_functor_morphism_of _ _); abstract ( path_natural_transformation; rewrite ?composition_of, ?identity_of; reflexivity ). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import Category.Prod FunctorCategory.Core NaturalTransformation.Composition.Functorial NaturalTransformation.Composition.Laws ExponentialLaws.Law4.Functors. Require Import NaturalTransformation.Paths.

Categories\Functor\Composition\Functorial\Core.v

compose_functor

object ((C -> D) * (D -> E) -> (C -> E)) := ExponentialLaws.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import Category.Prod FunctorCategory.Core NaturalTransformation.Composition.Functorial NaturalTransformation.Composition.Laws ExponentialLaws.Law4.Functors. Require Import NaturalTransformation.Paths.

Categories\Functor\Composition\Functorial\Core.v

compose_functor_uncurried

compose_functor' : object ((D -> E) -> ((C -> D) -> (C -> E))) := ExponentialLaws.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import Category.Prod FunctorCategory.Core NaturalTransformation.Composition.Functorial NaturalTransformation.Composition.Laws ExponentialLaws.Law4.Functors. Require Import NaturalTransformation.Paths.

Categories\Functor\Composition\Functorial\Core.v

compose_functor'

s d (m : morphism (C -> D) s d) : morphism (C' -> D') (pointwise_object_of s) (pointwise_object_of d) := Eval simpl in G oL m oR F.

Definition

Require Import Category.Core Functor.Core FunctorCategory.Core NaturalTransformation.Paths Functor.Composition.Core NaturalTransformation.Composition.Core.

Categories\Functor\Pointwise\Core.v

pointwise_morphism_of

s d (m : morphism (C -> D) s d) : morphism (C -> D') (pointwise_whiskerL_object_of s) (pointwise_whiskerL_object_of d) := Eval simpl in G oL m.

Definition

Require Import Category.Core Functor.Core FunctorCategory.Core NaturalTransformation.Paths Functor.Composition.Core NaturalTransformation.Composition.Core.

Categories\Functor\Pointwise\Core.v

pointwise_whiskerL_morphism_of

s d (m : morphism (C -> D) s d) : morphism (C' -> D) (pointwise_whiskerR_object_of s) (pointwise_whiskerR_object_of d) := Eval simpl in m oR F.

Definition

Require Import Category.Core Functor.Core FunctorCategory.Core NaturalTransformation.Paths Functor.Composition.Core NaturalTransformation.Composition.Core.

Categories\Functor\Pointwise\Core.v

pointwise_whiskerR_morphism_of

Functor (C -> D) (C' -> D'). Proof. refine (Build_Functor (C -> D) (C' -> D') (fun x => pointwise_object_of x) pointwise_morphism_of _ _); abstract (intros; simpl; path_natural_transformation; auto with functor). Defined.

Definition

Require Import Category.Core Functor.Core FunctorCategory.Core NaturalTransformation.Paths Functor.Composition.Core NaturalTransformation.Composition.Core.

Categories\Functor\Pointwise\Core.v

pointwise

Functor (C -> D) (C -> D'). Proof. refine (Build_Functor (C -> D) (C -> D') (fun x => pointwise_whiskerL_object_of x) pointwise_whiskerL_morphism_of _ _); abstract (intros; simpl; path_natural_transformation; auto with functor). Defined.

Definition

Require Import Category.Core Functor.Core FunctorCategory.Core NaturalTransformation.Paths Functor.Composition.Core NaturalTransformation.Composition.Core.

Categories\Functor\Pointwise\Core.v

pointwise_whiskerL

Functor (C -> D) (C' -> D). Proof. refine (Build_Functor (C -> D) (C' -> D) (fun x => pointwise_whiskerR_object_of x) pointwise_whiskerR_morphism_of _ _); abstract (intros; simpl; path_natural_transformation; auto with functor). Defined.

Definition

Require Import Category.Core Functor.Core FunctorCategory.Core NaturalTransformation.Paths Functor.Composition.Core NaturalTransformation.Composition.Core.

Categories\Functor\Pointwise\Core.v

pointwise_whiskerR

(x : Functor C D) : 1 o x o 1 = x. Proof. path_functor. Defined.

Lemma

Require Import Category.Core Functor.Core Functor.Pointwise.Core NaturalTransformation.Core NaturalTransformation.Paths Functor.Composition.Core Functor.Identity Functor.Paths. Require Import PathGroupoids Types.Forall HoTT.Tactics. Require Import Basics.Tactics.

Categories\Functor\Pointwise\Properties.v

identity_of_helper_helper

x : ap object_of (identity_of_helper_helper x) = idpath := path_functor_uncurried_fst _ _ _.

Definition

Require Import Category.Core Functor.Core Functor.Pointwise.Core NaturalTransformation.Core NaturalTransformation.Paths Functor.Composition.Core Functor.Identity Functor.Paths. Require Import PathGroupoids Types.Forall HoTT.Tactics. Require Import Basics.Tactics.

Categories\Functor\Pointwise\Properties.v

identity_of_helper_helper_object_of

(fun x : Functor C D => 1 o x o 1) = idmap. Proof. apply path_forall; intro x. apply identity_of_helper_helper. Defined.

Lemma

Require Import Category.Core Functor.Core Functor.Pointwise.Core NaturalTransformation.Core NaturalTransformation.Paths Functor.Composition.Core Functor.Identity Functor.Paths. Require Import PathGroupoids Types.Forall HoTT.Tactics. Require Import Basics.Tactics.

Categories\Functor\Pointwise\Properties.v

identity_of_helper

pointwise (identity C) (identity D) = identity _. Proof. path_functor. exists identity_of_helper. unfold identity_of_helper. abstract functor_pointwise_t identity_of_helper_helper identity_of_helper_helper_object_of. Defined.

Lemma

Require Import Category.Core Functor.Core Functor.Pointwise.Core NaturalTransformation.Core NaturalTransformation.Paths Functor.Composition.Core Functor.Identity Functor.Paths. Require Import PathGroupoids Types.Forall HoTT.Tactics. Require Import Basics.Tactics.

Categories\Functor\Pointwise\Properties.v

identity_of

(x : Functor C'' D) : G' o G o x o (F' o F) = G' o (G o x o F') o F. Proof. path_functor. Defined.

Lemma

Require Import Category.Core Functor.Core Functor.Pointwise.Core NaturalTransformation.Core NaturalTransformation.Paths Functor.Composition.Core Functor.Identity Functor.Paths. Require Import PathGroupoids Types.Forall HoTT.Tactics. Require Import Basics.Tactics.

Categories\Functor\Pointwise\Properties.v

composition_of_helper_helper

x : ap object_of (composition_of_helper_helper x) = idpath := path_functor_uncurried_fst _ _ _.

Definition

Require Import Category.Core Functor.Core Functor.Pointwise.Core NaturalTransformation.Core NaturalTransformation.Paths Functor.Composition.Core Functor.Identity Functor.Paths. Require Import PathGroupoids Types.Forall HoTT.Tactics. Require Import Basics.Tactics.

Categories\Functor\Pointwise\Properties.v

composition_of_helper_helper_object_of

(fun x => G' o G o x o (F' o F)) = (fun x => G' o (G o x o F') o F). Proof. apply path_forall; intro x. apply composition_of_helper_helper. Defined.

Lemma

Require Import Category.Core Functor.Core Functor.Pointwise.Core NaturalTransformation.Core NaturalTransformation.Paths Functor.Composition.Core Functor.Identity Functor.Paths. Require Import PathGroupoids Types.Forall HoTT.Tactics. Require Import Basics.Tactics.

Categories\Functor\Pointwise\Properties.v

composition_of_helper

pointwise (F' o F) (G' o G) = pointwise F G' o pointwise F' G. Proof. path_functor. exists composition_of_helper. unfold composition_of_helper. abstract functor_pointwise_t composition_of_helper_helper composition_of_helper_helper_object_of. Defined.

Lemma

Require Import Category.Core Functor.Core Functor.Pointwise.Core NaturalTransformation.Core NaturalTransformation.Paths Functor.Composition.Core Functor.Identity Functor.Paths. Require Import PathGroupoids Types.Forall HoTT.Tactics. Require Import Basics.Tactics.

Categories\Functor\Pointwise\Properties.v

composition_of

Functor (C * D) C := Build_Functor (C * D) C (@ _ _) (fun _ _ => @ _ _) (fun _ _ _ _ _ => idpath) (fun _ => idpath).

Definition

Require Import Category.Core Functor.Core Category.Prod Functor.Composition.Core. Require Import Types.Prod.

Categories\Functor\Prod\Core.v

fst

Functor (C * D) D := Build_Functor (C * D) D (@ _ _) (fun _ _ => @ _ _) (fun _ _ _ _ _ => idpath) (fun _ => idpath).

Definition

Require Import Category.Core Functor.Core Category.Prod Functor.Composition.Core. Require Import Types.Prod.

Categories\Functor\Prod\Core.v

snd

(F : Functor C D) (F' : Functor C D') : Functor C (D * D') := Build_Functor C (D * D') (fun c => (F c, F' c)) (fun s d m => (F _1 m, F' _1 m)) (fun _ _ _ _ _ => path_prod' (composition_of F _ _ _ _ _) (composition_of F' _ _ _ _ _)) (fun _ => path_prod' (identity_of F _) (identity_of F' _)).

Definition

Require Import Category.Core Functor.Core Category.Prod Functor.Composition.Core. Require Import Types.Prod.

Categories\Functor\Prod\Core.v

prod

Functor (C * C') (D * D') := (F o fst) * (F' o snd).

Definition

Require Import Category.Core Functor.Core Category.Prod Functor.Composition.Core. Require Import Types.Prod.

Categories\Functor\Prod\Core.v

pair

(d : D) : Functor C E. Proof. refine (Build_Functor C E (fun c => F (c, d)) (fun _ _ m => @morphism_of _ _ F (_, _) (_, _) (m, identity d)) _ _); abstract t. Defined.

Definition

Require Import Category.Core Functor.Core Category.Prod Functor.Composition.Core. Require Import Types.Prod.

Categories\Functor\Prod\Core.v

induced_fst

(c : C) : Functor D E. Proof. refine (Build_Functor D E (fun d => F (c, d)) (fun _ _ m => @morphism_of _ _ F (_, _) (_, _) (identity c, m)) _ _); abstract t. Defined.

Definition

Require Import Category.Core Functor.Core Category.Prod Functor.Composition.Core. Require Import Types.Prod.

Categories\Functor\Prod\Core.v

induced_snd

s d (m : morphism ((C -> D) * (C -> D')) s d) : morphism (_ -> _) (fst s * snd s)%functor (fst d * snd d)%functor := fst_type m * snd_type m.

Definition

Require Import Category.Core Functor.Core Functor.Prod.Core FunctorCategory.Core Category.Prod NaturalTransformation.Prod NaturalTransformation.Composition.Core. Require Import NaturalTransformation.Paths.

Categories\Functor\Prod\Functorial.v

functor_morphism_of

s d d' (m1 : morphism ((C -> D) * (C -> D')) s d) (m2 : morphism ((C -> D) * (C -> D')) d d') : functor_morphism_of (m2 o m1) = functor_morphism_of m2 o functor_morphism_of m1. Proof. path_natural_transformation; reflexivity. Qed.

Definition

Require Import Category.Core Functor.Core Functor.Prod.Core FunctorCategory.Core Category.Prod NaturalTransformation.Prod NaturalTransformation.Composition.Core. Require Import NaturalTransformation.Paths.

Categories\Functor\Prod\Functorial.v

functor_composition_of

(x : object ((C -> D) * (C -> D'))) : functor_morphism_of (identity x) = identity _. Proof. path_natural_transformation; reflexivity. Qed.

Definition

Require Import Category.Core Functor.Core Functor.Prod.Core FunctorCategory.Core Category.Prod NaturalTransformation.Prod NaturalTransformation.Composition.Core. Require Import NaturalTransformation.Paths.

Categories\Functor\Prod\Functorial.v

functor_identity_of

object ((C -> D) * (C -> D') -> (C -> D * D')) := Build_Functor ((C -> D) * (C -> D')) (C -> D * D') _ functor_morphism_of functor_composition_of functor_identity_of.

Definition

Require Import Category.Core Functor.Core Functor.Prod.Core FunctorCategory.Core Category.Prod NaturalTransformation.Prod NaturalTransformation.Composition.Core. Require Import NaturalTransformation.Paths.

Categories\Functor\Prod\Functorial.v

functor

fst o (a * b) = a. Proof. path_functor; trivial. Defined.

Lemma

Require Import Category.Core Functor.Core Category.Prod Functor.Composition.Core Functor.Prod.Core. Require Import Functor.Paths. Require Import Types.Prod HoTT.Tactics Types.Forall Types.Sigma. Require Import Basics.Tactics.

Categories\Functor\Prod\Universal.v

compose_fst_prod

snd o (a * b) = b. Proof. path_functor; trivial. Defined.

Lemma

Require Import Category.Core Functor.Core Category.Prod Functor.Composition.Core Functor.Prod.Core. Require Import Functor.Paths. Require Import Types.Prod HoTT.Tactics Types.Forall Types.Sigma. Require Import Basics.Tactics.

Categories\Functor\Prod\Universal.v

compose_snd_prod

c : (a * b) c = F c. Proof. pose proof (ap (fun F => object_of F c) H1). pose proof (ap (fun F => object_of F c) H2). simpl in *. path_induction. apply eta_prod. Defined.

Lemma

Require Import Category.Core Functor.Core Category.Prod Functor.Composition.Core Functor.Prod.Core. Require Import Functor.Paths. Require Import Types.Prod HoTT.Tactics Types.Forall Types.Sigma. Require Import Basics.Tactics.

Categories\Functor\Prod\Universal.v

unique_helper

transport (fun GO : C -> prod_type A B => forall s d : C, morphism C s d -> prod_type (morphism A (fst_type (GO s)) (fst_type (GO d))) (morphism B (snd_type (GO s)) (snd_type (GO d)))) (path_forall (a * b) F unique_helper) (fun (s d : C) (m : morphism C s d) => pair_type (a _1 m) (b _1 m)) = morphism_of F. Proof. repeat (apply path_forall; intro). repeat match goal with | _ => reflexivity | _ => progress simpl | _ => rewrite !transport_forall_constant end. transport_path_forall_hammer. unfold unique_helper. repeat match goal with | [ H : _ = _ |- _ ] => case H; simpl; clear H end. repeat match goal with | [ |- context[@morphism_of ?C ?D ?F ?s ?d ?m] ] => destruct (@morphism_of C D F s d m); clear m | [ |- context[@object_of ?C ?D ?F ?x] ] => destruct (@object_of C D F x); clear x end. reflexivity. Qed.

Lemma

Require Import Category.Core Functor.Core Category.Prod Functor.Composition.Core Functor.Prod.Core. Require Import Functor.Paths. Require Import Types.Prod HoTT.Tactics Types.Forall Types.Sigma. Require Import Basics.Tactics.

Categories\Functor\Prod\Universal.v

unique_helper2

a * b = F. Proof. path_functor. exists (path_forall _ _ unique_helper). apply unique_helper2. Defined.

Lemma

Require Import Category.Core Functor.Core Category.Prod Functor.Composition.Core Functor.Prod.Core. Require Import Functor.Paths. Require Import Types.Prod HoTT.Tactics Types.Forall Types.Sigma. Require Import Basics.Tactics.

Categories\Functor\Prod\Universal.v

unique

(F G : Functor C (A * B)) (H1 : fst o F = fst o G) (H2 : snd o F = snd o G) : F = G. Proof. etransitivity; [ symmetry | ]; apply unique; try eassumption; reflexivity. Defined.

Definition

Require Import Category.Core Functor.Core Category.Prod Functor.Composition.Core Functor.Prod.Core. Require Import Functor.Paths. Require Import Types.Prod HoTT.Tactics Types.Forall Types.Sigma. Require Import Basics.Tactics.

Categories\Functor\Prod\Universal.v

path_prod

PreCategory := @Build_PreCategory (Functor C D) (@NaturalTransformation C D) (@identity C D) (@compose C D) (@associativity _ C D) (@left_identity _ C D) (@right_identity _ C D) _.

Definition

Require Import Category.Strict Functor.Core NaturalTransformation.Core Functor.Paths. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Identity NaturalTransformation.Composition.Laws NaturalTransformation.Paths.

Categories\FunctorCategory\Core.v

functor_category

`{Funext} C `{IsStrictCategory D} : IsStrictCategory (C -> D). Proof. typeclasses eauto. Defined.

Lemma

Require Import Category.Strict Functor.Core NaturalTransformation.Core Functor.Paths. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Identity NaturalTransformation.Composition.Laws NaturalTransformation.Paths.

Categories\FunctorCategory\Core.v

isstrict_functor_category

(C D : PreCategory) : Functor (C -> D) (C^op -> D^op)^op := Build_Functor (C -> D) ((C^op -> D^op)^op) (fun F => F^op)%functor (fun _ _ T => T^op)%natural_transformation (fun _ _ _ _ _ => idpath) (fun _ => idpath).

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual NaturalTransformation.Dual. Require Import Functor.Identity. Require Import FunctorCategory.Core. Require Import Functor.Paths. Require Import HoTT.Tactics.

Categories\FunctorCategory\Dual.v

opposite_functor

C D : opposite_functor C D o (opposite_functor C^op D^op)^op = 1 /\ (opposite_functor C^op D^op)^op o opposite_functor C D = 1. Proof. op_t C D. Qed.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual NaturalTransformation.Dual. Require Import Functor.Identity. Require Import FunctorCategory.Core. Require Import Functor.Paths. Require Import HoTT.Tactics.

Categories\FunctorCategory\Dual.v

opposite_functor_law

object ((cat^op * cat) -> cat) := Eval cbv zeta in let object_of := (fun CD => (((fst CD).

Definition

Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Pointwise.Core Functor.Pointwise.Properties Category.Dual Category.Prod Cat.Core ExponentialLaws.Law4.Functors.

Categories\FunctorCategory\Functorial.v

functor_uncurried

object (cat^op -> (cat -> cat)) := ExponentialLaws.

Definition

Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Pointwise.Core Functor.Pointwise.Properties Category.Dual Category.Prod Cat.Core ExponentialLaws.Law4.Functors.

Categories\FunctorCategory\Functorial.v

functor

`{Funext} (C D : PreCategory) F G := @Isomorphic (C -> D) F G.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Paths FunctorCategory.Core Category.Morphisms NaturalTransformation.Core. Require Import Basics.Tactics.

Categories\FunctorCategory\Morphisms.v

NaturalIsomorphism

`{Funext} `{@IsIsomorphism (C -> D) F G T} x : IsIsomorphism (T x). Proof. exists (T^-1 x). - exact (apD10 (ap components_of left_inverse) x). - exact (apD10 (ap components_of right_inverse) x). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Paths FunctorCategory.Core Category.Morphisms NaturalTransformation.Core. Require Import Basics.Tactics.

Categories\FunctorCategory\Morphisms.v

isisomorphism_components_of

C D (F G : Functor C D) (T : NaturalTransformation F G) `{forall x, IsIsomorphism (T x)} : NaturalTransformation G F. Proof. exists (fun x => (T x)^-1); abstract ( intros; iso_move_inverse; first [ apply commutes | symmetry; apply commutes ] ). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Paths FunctorCategory.Core Category.Morphisms NaturalTransformation.Core. Require Import Basics.Tactics.

Categories\FunctorCategory\Morphisms.v

inverse

`{Funext} C D F G (T : NaturalTransformation F G) `{forall x, IsIsomorphism (T x)} : @IsIsomorphism (C -> D) F G T. Proof. exists (inverse _); abstract ( path_natural_transformation; first [ apply left_inverse | apply right_inverse ] ). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Paths FunctorCategory.Core Category.Morphisms NaturalTransformation.Core. Require Import Basics.Tactics.

Categories\FunctorCategory\Morphisms.v

isisomorphism_natural_transformation

(F G : object (C -> D)) (T : F = G) : NaturalTransformation F G. Proof. refine (Build_NaturalTransformation' F G (fun x => idtoiso _ (ap10 (ap object_of T) x)) _ _); intros; case T; simpl; [ exact (left_identity _ _ _ _ @ (right_identity _ _ _ _)^) | exact (right_identity _ _ _ _ @ (left_identity _ _ _ _)^) ]. Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Paths FunctorCategory.Core Category.Morphisms NaturalTransformation.Core. Require Import Basics.Tactics.

Categories\FunctorCategory\Morphisms.v

idtoiso_natural_transformation

(F G : object (C -> D)) (T : F = G) : F <~=~> G. Proof. exists (idtoiso_natural_transformation T). exists (idtoiso_natural_transformation (T^)%path); abstract (path_natural_transformation; case T; simpl; auto with morphism). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Paths FunctorCategory.Core Category.Morphisms NaturalTransformation.Core. Require Import Basics.Tactics.

Categories\FunctorCategory\Morphisms.v

idtoiso

(F G : object (C -> D)) (T : F = G) : Category.Morphisms.idtoiso _ T = idtoiso T. Proof. case T. expand; f_ap. exact (center _). Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Paths FunctorCategory.Core Category.Morphisms NaturalTransformation.Core. Require Import Basics.Tactics.

Categories\FunctorCategory\Morphisms.v

eta_idtoiso

Record

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

Pair

s d d' (m1 : morphism d d') (m2 : morphism s d) : morphism s d'. Proof. exists (m1.1 o m2.1). refine (m1.2 o ((p_morphism_of F m1.1) _1 m2.2 o _)). apply (p_composition_of 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

compose

s : morphism s s. Proof. exists 1. apply (p_identity_of 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

identity

forall {x x0 : C} {x2 : Category.Core.morphism C x x0} {x1 : C} {x5 : Category.Core.morphism C x0 x1} {x4 : C} {x7 : Category.Core.morphism C x1 x4} {p p0 : PreCategory} {f : Category.Core.morphism C x x4 -> Functor p0 p} {p1 p2 : PreCategory} {f0 : Functor p2 p} {f1 : Functor p1 p2} {f2 : Functor p0 p2} {f3 : Functor p0 p1} {f4 : Functor p1 p} {x16 : Category.Core.morphism (_ -> _) (f (x7 o x5 o x2)) (f4 o f3)%functor} {x15 : Category.Core.morphism (_ -> _) f2 (f1 o f3)%functor} {H2 : IsIsomorphism x15} {x11 : Category.Core.morphism (_ -> _) (f (x7 o (x5 o x2))) (f0 o f2)%functor} {H1 : IsIsomorphism x11} {x9 : Category.Core.morphism (_ -> _) f4 (f0 o f1)%functor} {fst_hyp : x7 o x5 o x2 = x7 o (x5 o x2)} (rew_hyp : forall x3 : p0, (idtoiso (p0 -> p) (ap f fst_hyp) : Category.Core.morphism _ _ _) x3 = x11^-1 x3 o (f0 _1 (x15^-1 x3) o (1 o (x9 (f3 x3) o x16 x3)))) {H0' : IsIsomorphism x16} {H1' : IsIsomorphism x9} {x13 : p} {x3 : p0} {x6 : p1} {x10 : p2} {x14 : Category.Core.morphism p (f0 x10) x13} {x12 : Category.Core.morphism p2 (f1 x6) x10} {x8 : Category.Core.morphism p1 (f3 x3) x6}, exist (fun f5 : Category.Core.morphism C x x4 => Category.Core.morphism p ((f f5) x3) x13) (x7 o x5 o x2) (x14 o (f0 _1 x12 o x9 x6) o (f4 _1 x8 o x16 x3)) = (x7 o (x5 o x2); x14 o (f0 _1 (x12 o (f1 _1 x8 o x15 x3)) o x11 x3)). Proof. helper_t assoc_before_commutes_tac. assoc_fin_tac. 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_assoc_helper

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 p p} {x0 : Category.Core.morphism (_ -> _) (f0 (1 o f)) (f1 o f0 f)%functor} {x : Category.Core.morphism (_ -> _) f1 1%functor} {fst_hyp : 1 o f = f} (rewrite_hyp : forall x3 : p0, (idtoiso (p0 -> p) (ap f0 fst_hyp) : Category.Core.morphism _ _ _) x3 = 1 o (x ((f0 f) x3) o x0 x3)) {H0' : IsIsomorphism x0} {H1' : IsIsomorphism x} {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) (1 o f) (x x3 o (f1 _1 f' 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_left_identity_helper