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posttensor 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_posttensor | 2,400 |
NaturalIsomorphism I_pretensor 1. | 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_unitor | 2,401 |
NaturalIsomorphism I_posttensor 1. Close Scope functor_scope. Variable alpha : associator. Variable lambda : left_unitor. Variable rho : . Notation alpha_nat_trans := ((@morphism_isomorphic (C * (C * C) -> C)%category right_assoc left_assoc) alpha). Notation lambda_nat_trans := ((@morphism_isomorphic _ _ _) lambda). Notation rho_nat_trans := ((@morphism_isomorphic _ _ _) rho). Section coherence_laws. Variable a b c d : C. Local P1 : (a ⊗ (b ⊗ (c ⊗ d))) --> (a ⊗ ((b ⊗ c) ⊗ d)). Proof. apply (morphism_of tensor); split; simpl. - exact (Core.identity a). - exact (alpha_nat_trans (b, (c, d))). Defined. | 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_unitor | 2,402 |
P3 o P2 o P1 = P5 o P4. Close Scope morphism_scope. Local Q1 : (a ⊗ (I ⊗ b)) --> a ⊗ b. Proof. apply (morphism_of tensor); split; simpl. - exact (Core.identity a). - exact (lambda_nat_trans _). Defined. | 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 | pentagon_eq | 2,403 |
Q1 = Q2. | 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 | triangle_eq | 2,404 |
Record | Require Import Category.Core Functor.Core. | Categories\NaturalTransformation\Core.v | NaturalTransformation | 2,405 |
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CO COM := ' CO COM (fun _ _ _ => symmetry _ _ (COM _ _ _)). | Definition | Require Import Category.Core Functor.Core. | Categories\NaturalTransformation\Core.v | Build_NaturalTransformation | 2,406 |
C D (F G : Functor C D) (T : NaturalTransformation F G) s d d' (m : morphism C s d) (m' : morphism D _ d') : (m' o T d) o F _1 m = (m' o G _1 m) o T s := ((Category. | Definition | Require Import Category.Core Functor.Core. | Categories\NaturalTransformation\Core.v | commutes_pT_F | 2,407 |
C D (F G : Functor C D) (T : NaturalTransformation F G) s d d' (m : morphism C s d) (m' : morphism D d' _) : T d o (F _1 m o m') = G _1 m o (T s o m') := ((Category. | Definition | Require Import Category.Core Functor.Core. | Categories\NaturalTransformation\Core.v | commutes_T_Fp | 2,408 |
C D (F G : Functor C D) (T : NaturalTransformation F G) : NaturalTransformation G^op F^op := Build_NaturalTransformation' (G^op) (F^op) (components_of T) (fun s d => commutes_sym T d s) (fun s d => commutes T d s). | Definition | Import Category.Dual.CategoryDualNotations Functor.Dual.FunctorDualNotations. Require Import Category.Core Functor.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Dual.v | opposite | 2,409 |
C D (F G : Functor C D) (T : NaturalTransformation F G) : (T^op)^op = T := idpath. | Definition | Import Category.Dual.CategoryDualNotations Functor.Dual.FunctorDualNotations. Require Import Category.Core Functor.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Dual.v | opposite_involutive | 2,410 |
s d (m : morphism C s d) : CO d o F _1 m = G _1 m o CO s. Proof. case HM. case HO. exact (left_identity _ _ _ _ @ (right_identity _ _ _ _)^). Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Identity.v | generalized_identity_commutes | 2,411 |
s d (m : morphism C s d) : G _1 m o CO s = CO d o F _1 m. Proof. case HM. case HO. exact (right_identity _ _ _ _ @ (left_identity _ _ _ _)^). Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Identity.v | generalized_identity_commutes_sym | 2,412 |
NaturalTransformation F G := Build_NaturalTransformation' F G (fun c => CO c) generalized_identity_commutes generalized_identity_commutes_sym. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Identity.v | generalized_identity | 2,413 |
generalized_identity' : NaturalTransformation F G. Proof. apply (generalized_identity F G (ap (@object_of C D) H)). case H. reflexivity. Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Identity.v | generalized_identity' | 2,414 |
(F : Functor C D) : NaturalTransformation F F := Eval simpl in @generalized_identity F F 1 1. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Identity.v | identity | 2,415 |
C D (F G : Functor C D) (T' : F = G) x : (Category.Morphisms.idtoiso (_ -> _) T' : morphism _ _ _) x = Category.Morphisms.idtoiso _ (ap10 (ap object_of T') x). Proof. destruct T'. reflexivity. Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import NaturalTransformation.Composition.Core. Require Import Functor.Composition.Core. Require Import Category.Morphisms. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import Basics.Tactics. | Categories\NaturalTransformation\Isomorphisms.v | idtoiso_components_of | 2,416 |
C D (F F' F'' : Functor C D) (T' : F' = F'') (T : F = F') : ((Category.Morphisms.idtoiso (_ -> _) T' : morphism _ _ _) o (Category.Morphisms.idtoiso (_ -> _) T : morphism _ _ _))%natural_transformation = (Category.Morphisms.idtoiso (_ -> _) (T @ T')%path : morphism _ _ _). Proof. path_natural_transformation; path_induction; simpl; auto with morphism. Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import NaturalTransformation.Composition.Core. Require Import Functor.Composition.Core. Require Import Category.Morphisms. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import Basics.Tactics. | Categories\NaturalTransformation\Isomorphisms.v | idtoiso_compose | 2,417 |
C D E (F : Functor D E) (G G' : Functor C D) (T : G = G') : whisker_l F (Category.Morphisms.idtoiso (_ -> _) T : morphism _ _ _) = (Category.Morphisms.idtoiso (_ -> _) (ap _ T) : morphism _ _ _). Proof. path_natural_transformation; path_induction; simpl; auto with functor. Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import NaturalTransformation.Composition.Core. Require Import Functor.Composition.Core. Require Import Category.Morphisms. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import Basics.Tactics. | Categories\NaturalTransformation\Isomorphisms.v | idtoiso_whisker_l | 2,418 |
C D E (F F' : Functor D E) (T : F = F') (G : Functor C D) : whisker_r (Category.Morphisms.idtoiso (_ -> _) T : morphism _ _ _) G = (Category.Morphisms.idtoiso (_ -> _) (ap (fun _ => _ o _)%functor T) : morphism _ _ _). Proof. path_natural_transformation; path_induction; simpl; auto with functor. Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import NaturalTransformation.Composition.Core. Require Import Functor.Composition.Core. Require Import Category.Morphisms. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import Basics.Tactics. | Categories\NaturalTransformation\Isomorphisms.v | idtoiso_whisker_r | 2,419 |
`{IsIsomorphism C s d m} D (F G : Functor C D) (T : NaturalTransformation F G) : (G _1 m)^-1 o (T d o F _1 m) = T s. Proof. apply iso_moveR_Vp. apply commutes. Qed. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import NaturalTransformation.Composition.Core. Require Import Functor.Composition.Core. Require Import Category.Morphisms. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import Basics.Tactics. | Categories\NaturalTransformation\Isomorphisms.v | path_components_of_isisomorphism | 2,420 |
path_components_of_isisomorphism' `{IsIsomorphism C s d m} D (F G : Functor C D) (T : NaturalTransformation F G) : (G _1 m o T s) o (F _1 m)^-1 = T d. Proof. apply iso_moveR_pV. symmetry. apply commutes. Qed. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import NaturalTransformation.Composition.Core. Require Import Functor.Composition.Core. Require Import Category.Morphisms. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import Basics.Tactics. | Categories\NaturalTransformation\Isomorphisms.v | path_components_of_isisomorphism' | 2,421 |
`(m : @Isomorphic C s d) D (F G : Functor C D) (T : NaturalTransformation F G) : (G _1 m)^-1 o (T d o F _1 m) = T s := @path_components_of_isisomorphism _ _ _ m m D F G T. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import NaturalTransformation.Composition.Core. Require Import Functor.Composition.Core. Require Import Category.Morphisms. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import Basics.Tactics. | Categories\NaturalTransformation\Isomorphisms.v | path_components_of_isomorphic | 2,422 |
path_components_of_isomorphic' `(m : @Isomorphic C s d) D (F G : Functor C D) (T : NaturalTransformation F G) : (G _1 m o T s) o (F _1 m)^-1 = T d := @path_components_of_isisomorphism' _ _ _ m m D F G T. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import NaturalTransformation.Composition.Core. Require Import Functor.Composition.Core. Require Import Category.Morphisms. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import Basics.Tactics. | Categories\NaturalTransformation\Isomorphisms.v | path_components_of_isomorphic' | 2,423 |
{ CO : forall x, morphism D (F x) (G x) | forall s d (m : morphism C s d), CO d o F _1 m = G _1 m o CO s } <~> NaturalTransformation F G. Proof. let build := constr:(@Build_NaturalTransformation _ _ F G) in let pr1 := constr:(@components_of _ _ F G) in let pr2 := constr:(@commutes _ _ F G) in apply (equiv_adjointify (fun u => build u.1 u.2) (fun v => (pr1 v; pr2 v))); hnf; [ intros []; intros; simpl; expand; f_ap; exact (center _) | intros; apply eta_sigma ]. Defined. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Equivalences HoTT.Types Trunc Basics.Tactics. | Categories\NaturalTransformation\Paths.v | equiv_sig_natural_transformation | 2,424 |
components_of T = components_of U -> T = U. Proof. intros. destruct T, U; simpl in *. path_induction. f_ap; refine (center _). Qed. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Equivalences HoTT.Types Trunc Basics.Tactics. | Categories\NaturalTransformation\Paths.v | path'_natural_transformation | 2,425 |
components_of T == components_of U -> T = U. Proof. intros. apply path'_natural_transformation. apply path_forall; assumption. Qed. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Equivalences HoTT.Types Trunc Basics.Tactics. | Categories\NaturalTransformation\Paths.v | path_natural_transformation | 2,426 |
path_inv o path_natural_transformation == idmap. Proof. repeat intro. refine (center _). Qed. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Equivalences HoTT.Types Trunc Basics.Tactics. | Categories\NaturalTransformation\Paths.v | eisretr_path_natural_transformation | 2,427 |
path_natural_transformation o path_inv == idmap. Proof. repeat intro. refine (center _). Qed. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Equivalences HoTT.Types Trunc Basics.Tactics. | Categories\NaturalTransformation\Paths.v | eissect_path_natural_transformation | 2,428 |
forall x, @eisretr_path_natural_transformation (path_inv x) = ap path_inv (eissect_path_natural_transformation x). Proof. repeat intro. refine (center _). Qed. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Equivalences HoTT.Types Trunc Basics.Tactics. | Categories\NaturalTransformation\Paths.v | eisadj_path_natural_transformation | 2,429 |
T = U <~> (components_of T == components_of U). Proof. econstructor. econstructor. exact eisadj_path_natural_transformation. Defined. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Equivalences HoTT.Types Trunc Basics.Tactics. | Categories\NaturalTransformation\Paths.v | equiv_path_natural_transformation | 2,430 |
(F G : Functor C D) (T : NaturalTransformation F G) (F' : Functor C' D') : NaturalTransformation (pointwise F F') (pointwise G F'). Proof. refine (Build_NaturalTransformation (pointwise F F') (pointwise G F') (fun f : object (D -> C') => (F' o f) oL T)%natural_transformation _). abstract t. Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import FunctorCategory.Core NaturalTransformation.Paths Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import Functor.Pointwise.Core. | Categories\NaturalTransformation\Pointwise.v | pointwise_l | 2,431 |
(F : Functor C D) (F' G' : Functor C' D') (T' : NaturalTransformation F' G') : NaturalTransformation (pointwise F F') (pointwise F G'). Proof. refine (Build_NaturalTransformation (pointwise F F') (pointwise F G') (fun f : object (D -> C') => T' oR f oR F)%natural_transformation _). abstract t. Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import FunctorCategory.Core NaturalTransformation.Paths Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import Functor.Pointwise.Core. | Categories\NaturalTransformation\Pointwise.v | pointwise_r | 2,432 |
NaturalTransformation (F * G) (F' * G') := Build_NaturalTransformation (F * G) (F' * G') (fun x : A => (T x, U x)) (fun _ _ _ => path_prod' (commutes T _ _ _) (commutes U _ _ _)). | Definition | Require Import Category.Core Functor.Core Category.Prod Functor.Prod.Core NaturalTransformation.Core. Require Import Types.Prod. | Categories\NaturalTransformation\Prod.v | prod | 2,433 |
s d (m : morphism C s d) : NaturalTransformation (Functor.Prod.Core.induced_snd F s) (Functor.Prod.Core.induced_snd F d). Proof. let F0 := match goal with |- NaturalTransformation ?F0 ?G0 => constr:(F0) end in let G0 := match goal with |- NaturalTransformation ?F0 ?G0 => constr:(G0) end in refine (Build_NaturalTransformation F0 G0 (fun d => @morphism_of _ _ F (_, _) (_, _) (m, @identity D d)) _). abstract t. Defined. | Definition | Require Import Category.Core Functor.Core Category.Prod Functor.Prod.Core NaturalTransformation.Core. Require Import Types.Prod. | Categories\NaturalTransformation\Prod.v | induced_fst | 2,434 |
s d (m : morphism D s d) : NaturalTransformation (Functor.Prod.Core.induced_fst F s) (Functor.Prod.Core.induced_fst F d). Proof. let F0 := match goal with |- NaturalTransformation ?F0 ?G0 => constr:(F0) end in let G0 := match goal with |- NaturalTransformation ?F0 ?G0 => constr:(G0) end in refine (Build_NaturalTransformation F0 G0 (fun c => @morphism_of _ _ F (_, _) (_, _) (@identity C c, m)) _). abstract t. Defined. | Definition | Require Import Category.Core Functor.Core Category.Prod Functor.Prod.Core NaturalTransformation.Core. Require Import Types.Prod. | Categories\NaturalTransformation\Prod.v | induced_snd | 2,435 |
C C' D F G F' G' (T : @NaturalTransformation C D F G) (T' : @NaturalTransformation C' D F' G') : NaturalTransformation (F + F') (G + G'). Proof. refine (Build_NaturalTransformation (F + F') (G + G') (fun x => match x with | Basics.Overture.inl c => T c | Basics.Overture.inr c' => T' c' end) _). abstract ( repeat (intros [] || intro); simpl; auto with natural_transformation ). Defined. | Definition | Require Import Category.Sum Functor.Sum NaturalTransformation.Core. | Categories\NaturalTransformation\Sum.v | sum | 2,436 |
s d (m : morphism C s d) : CO d o F _1 m = F'' _1 m o CO s := (associativity _ _ _ _ _ _ _ _) @ ap (fun x => _ o x) (commutes T _ _ m) @ (associativity_sym _ _ _ _ _ _ _ _) @ ap (fun x => x o _) (commutes T' _ _ m) @ (associativity _ _ _ _ _ _ _ _). | Definition | Require Import Category.Core Functor.Core Functor.Composition.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Composition\Core.v | compose_commutes | 2,437 |
s d (m : morphism C s d) : F'' _1 m o CO s = CO d o F _1 m := (associativity_sym _ _ _ _ _ _ _ _) @ ap (fun x => x o _) (commutes_sym T' _ _ m) @ (associativity _ _ _ _ _ _ _ _) @ ap (fun x => _ o x) (commutes_sym T _ _ m) @ (associativity_sym _ _ _ _ _ _ _ _). | Definition | Require Import Category.Core Functor.Core Functor.Composition.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Composition\Core.v | compose_commutes_sym | 2,438 |
NaturalTransformation F F'' := Build_NaturalTransformation' F F'' (fun c => CO c) compose_commutes compose_commutes_sym. | Definition | Require Import Category.Core Functor.Core Functor.Composition.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Composition\Core.v | compose | 2,439 |
s d (m : morphism C s d) : F _1 (T d) o (F o G) _1 m = (F o G') _1 m o F _1 (T s) := ((composition_of F _ _ _ _ _)^) @ (ap (fun m => F _1 m) (commutes T _ _ _)) @ (composition_of F _ _ _ _ _). | Definition | Require Import Category.Core Functor.Core Functor.Composition.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Composition\Core.v | whisker_l_commutes | 2,440 |
s d (m : morphism C s d) : (F o G') _1 m o F _1 (T s) = F _1 (T d) o (F o G) _1 m := ((composition_of F _ _ _ _ _)^) @ (ap (fun m => F _1 m) (commutes_sym T _ _ _)) @ (composition_of F _ _ _ _ _). | Definition | Require Import Category.Core Functor.Core Functor.Composition.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Composition\Core.v | whisker_l_commutes_sym | 2,441 |
Build_NaturalTransformation' (F o G) (F o G') (fun c => CO c) whisker_l_commutes whisker_l_commutes_sym. | Definition | Require Import Category.Core Functor.Core Functor.Composition.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Composition\Core.v | whisker_l | 2,442 |
s d (m : morphism C s d) : T (G d) o (F o G) _1 m = (F' o G) _1 m o T (G s) := commutes T _ _ _. | Definition | Require Import Category.Core Functor.Core Functor.Composition.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Composition\Core.v | whisker_r_commutes | 2,443 |
s d (m : morphism C s d) : (F' o G) _1 m o T (G s) = T (G d) o (F o G) _1 m := commutes_sym T _ _ _. | Definition | Require Import Category.Core Functor.Core Functor.Composition.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Composition\Core.v | whisker_r_commutes_sym | 2,444 |
Build_NaturalTransformation' (F o G) (F' o G) (fun c => CO c) whisker_r_commutes whisker_r_commutes_sym. | Definition | Require Import Category.Core Functor.Core Functor.Composition.Core NaturalTransformation.Core. | Categories\NaturalTransformation\Composition\Core.v | whisker_r | 2,445 |
(F : (D -> E)%category) : ((C -> D) -> (C -> E))%category := Build_Functor (C -> D) (C -> E) (fun G => F o G)%functor (fun _ _ T => F oL T) (fun _ _ _ _ _ => composition_of_whisker_l _ _ _) (fun _ => whisker_l_right_identity _ _). | Definition | Require Import Category.Core Functor.Core. Require Import FunctorCategory.Core Functor.Composition.Core NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. | Categories\NaturalTransformation\Composition\Functorial.v | whiskerL_functor | 2,446 |
(G : (C -> D)%category) : ((D -> E) -> (C -> E))%category := Build_Functor (D -> E) (C -> E) (fun F => F o G)%functor (fun _ _ T => T oR G) (fun _ _ _ _ _ => composition_of_whisker_r _ _ _) (fun _ => whisker_r_left_identity _ _). | Definition | Require Import Category.Core Functor.Core. Require Import FunctorCategory.Core Functor.Composition.Core NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. | Categories\NaturalTransformation\Composition\Functorial.v | whiskerR_functor | 2,447 |
(F F' : Functor C D) (T : NaturalTransformation F F') : 1 o T = T. Proof. path_natural_transformation; auto with morphism. Qed. | Lemma | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | left_identity | 2,448 |
(F F' : Functor C D) (T : NaturalTransformation F F') : T o 1 = T. Proof. path_natural_transformation; auto with morphism. Qed. | Lemma | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | right_identity | 2,449 |
E (G : Functor D E) (F : Functor C D) : identity G oR F = 1. Proof. path_natural_transformation; auto with morphism. Qed. | Definition | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | whisker_r_left_identity | 2,450 |
E (G : Functor D E) (F : Functor C D) : G oL identity F = 1. Proof. path_natural_transformation; auto with functor. Qed. | Definition | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | whisker_l_right_identity | 2,451 |
(G' oL T) o (T' oR F) = (T' oR F') o (G oL T). Proof. path_natural_transformation; simpl. symmetry. apply NaturalTransformation.Core.commutes. Qed. | Lemma | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | exchange_whisker | 2,452 |
E (I : Functor D E) : I oL (T o T') = (I oL T) o (I oL T'). Proof. path_natural_transformation; apply composition_of. Qed. | Lemma | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | composition_of_whisker_l | 2,453 |
B (I : Functor B C) : (T o T') oR I = (T oR I) o (T' oR I). Proof. path_natural_transformation; apply idpath. Qed. | Lemma | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | composition_of_whisker_r | 2,454 |
NaturalTransformation F0 F1 := Eval simpl in generalized_identity F0 F1 idpath idpath. | Definition | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | associator_1 | 2,455 |
NaturalTransformation F1 F0 := Eval simpl in generalized_identity F1 F0 idpath idpath. | Definition | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | associator_2 | 2,456 |
C D F G H I (V : @NaturalTransformation C D F G) (U : @NaturalTransformation C D G H) (T : @NaturalTransformation C D H I) : (T o U) o V = T o (U o V). Proof. path_natural_transformation. apply . Qed. | Definition | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | associativity | 2,457 |
NaturalTransformation (1 o F) F := Eval simpl in generalized_identity (1 o F) F idpath idpath. | Definition | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | left_identity_natural_transformation_1 | 2,458 |
NaturalTransformation F (1 o F) := Eval simpl in generalized_identity F (1 o F) idpath idpath. | Definition | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | left_identity_natural_transformation_2 | 2,459 |
left_identity_natural_transformation_1 o left_identity_natural_transformation_2 = 1 /\ left_identity_natural_transformation_2 o left_identity_natural_transformation_1 = 1. Proof. nt_id_t. Qed. | Theorem | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | left_identity_isomorphism | 2,460 |
NaturalTransformation (F o 1) F := Eval simpl in generalized_identity (F o 1) F idpath idpath. | Definition | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | right_identity_natural_transformation_1 | 2,461 |
NaturalTransformation F (F o 1) := Eval simpl in generalized_identity F (F o 1) idpath idpath. | Definition | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | right_identity_natural_transformation_2 | 2,462 |
right_identity_natural_transformation_1 o right_identity_natural_transformation_2 = 1 /\ right_identity_natural_transformation_2 o right_identity_natural_transformation_1 = 1. Proof. nt_id_t. Qed. | Theorem | Require Import Category.Core Functor.Core Functor.Identity Functor.Composition.Core NaturalTransformation.Core NaturalTransformation.Identity NaturalTransformation.Composition.Core NaturalTransformation.Paths. | Categories\NaturalTransformation\Composition\Laws.v | right_identity_isomorphism | 2,463 |
Functor (D^op * C) set_cat. | Definition | Require Import Category.Core Functor.Core Category.Prod Category.Dual SetCategory.Core. | Categories\Profunctor\Core.v | Profunctor | 2,464 |
`{Funext} (C : PreCategory) : C -|-> C := hom_functor C. | Definition | Require Import Category.Core Profunctor.Core HomFunctor. | Categories\Profunctor\Identity.v | identity | 2,465 |
C D (F : Functor C D) : C -|-> D := 1%profunctor o (1, F). | Definition | Require Import Category.Core Functor.Core Functor.Prod.Core Profunctor.Core Functor.Dual Profunctor.Identity Functor.Composition.Core Functor.Identity. | Categories\Profunctor\Representable.v | representable | 2,466 |
C D (F : Functor C D) : D -|-> C := 1%profunctor o (F^op, 1). | Definition | Require Import Category.Core Functor.Core Functor.Prod.Core Profunctor.Core Functor.Dual Profunctor.Identity Functor.Composition.Core Functor.Identity. | Categories\Profunctor\Representable.v | corepresentable | 2,467 |
Record | Require Import Category.Core Functor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core Functor.Identity. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import FunctorCategory.Core. | Categories\Pseudofunctor\Core.v | Pseudofunctor | 2,468 |
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A B (F1 F2 : Functor A B) (pf1 pf2 : F1 = F2) : P A -> P B -> pf1 = pf2 := fun PA PB => @path_ishprop _ (@HP A B PA PB F1 F2) _ _. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import Functor.Identity. Require Import Pseudofunctor.Core. Require Import Cat.Core. Require Import FunctorCategory.Core. Require Import FunctorCategory.Morphisms NaturalTransformation.Isomorphisms. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Basics.PathGroupoids Basics.Trunc. | Categories\Pseudofunctor\FromFunctor.v | path_functor_helper | 2,469 |
{x0 x1 x2 x : PreCategory} {x7 x11 : Functor x0 x1} {x12 : x7 = x11} {x6 : Functor x0 x2} {x9 : Functor x2 x1} {x14 : x11 = (x9 o x6)%functor} {x4 : Functor x0 x} {x5 : Functor x x1} {x8 : x7 = (x5 o x4)%functor} {x10 : Functor x x2} {x13 : x6 = (x10 o x4)%functor} {x15 : x5 = (x9 o x10)%functor} (H0' : P x0) (H1' : P x1) (H2' : P x2) (H' : P x) : ((associator_1 x9 x10 x4) o ((idtoiso (x -> x1) x15 : morphism _ _ _) oR x4 o (idtoiso (x0 -> x1) x8 : morphism _ _ _)))%natural_transformation = (x9 oL (idtoiso (x0 -> x2) x13 : morphism _ _ _) o ((idtoiso (x0 -> x1) x14 : morphism _ _ _) o (idtoiso (x0 -> x1) x12 : morphism _ _ _)))%natural_transformation. Proof. clear F. symmetry; simpl; pseudofunctor_t. Qed. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import Functor.Identity. Require Import Pseudofunctor.Core. Require Import Cat.Core. Require Import FunctorCategory.Core. Require Import FunctorCategory.Morphisms NaturalTransformation.Isomorphisms. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Basics.PathGroupoids Basics.Trunc. | Categories\Pseudofunctor\FromFunctor.v | pseudofunctor_of_functor__composition_of | 2,470 |
{x0 x : PreCategory} {x2 : Functor x x} {x3 : x2 = 1%functor} {x4 x5 : Functor x0 x} {x6 : x4 = x5} {x7 : x4 = (x2 o x5)%functor} (H0' : P x0) (H' : P x) : ((Category.Morphisms.idtoiso (x -> x) x3 : morphism _ _ _) oR x5 o (Category.Morphisms.idtoiso (x0 -> x) x7 : morphism _ _ _))%natural_transformation = ((NaturalTransformation.Composition.Laws.left_identity_natural_transformation_2 x5) o (Category.Morphisms.idtoiso (x0 -> x) x6 : morphism _ _ _))%natural_transformation. Proof. clear F. simpl; pseudofunctor_t. Qed. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import Functor.Identity. Require Import Pseudofunctor.Core. Require Import Cat.Core. Require Import FunctorCategory.Core. Require Import FunctorCategory.Morphisms NaturalTransformation.Isomorphisms. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Basics.PathGroupoids Basics.Trunc. | Categories\Pseudofunctor\FromFunctor.v | pseudofunctor_of_functor__left_identity_of | 2,471 |
{x0 x : PreCategory} {x4 : Functor x0 x0} {x5 : x4 = 1%functor} {x2 x3 : Functor x0 x} {x6 : x2 = x3} {x7 : x2 = (x3 o x4)%functor} (H0' : P x0) (H' : P x) : (x3 oL (Category.Morphisms.idtoiso (x0 -> x0) x5 : morphism _ _ _) o (Category.Morphisms.idtoiso (x0 -> x) x7 : morphism _ _ _))%natural_transformation = ((NaturalTransformation.Composition.Laws.right_identity_natural_transformation_2 x3) o (Category.Morphisms.idtoiso (x0 -> x) x6 : morphism _ _ _))%natural_transformation. Proof. clear F. simpl; pseudofunctor_t. Qed. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import Functor.Identity. Require Import Pseudofunctor.Core. Require Import Cat.Core. Require Import FunctorCategory.Core. Require Import FunctorCategory.Morphisms NaturalTransformation.Isomorphisms. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Basics.PathGroupoids Basics.Trunc. | Categories\Pseudofunctor\FromFunctor.v | pseudofunctor_of_functor__right_identity_of | 2,472 |
Pseudofunctor C := Build_Pseudofunctor C (fun x => pr1 (F x)) (fun s d m => F _1 m) (fun s d d' m0 m1 => Category. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import Functor.Identity. Require Import Pseudofunctor.Core. Require Import Cat.Core. Require Import FunctorCategory.Core. Require Import FunctorCategory.Morphisms NaturalTransformation.Isomorphisms. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Basics.PathGroupoids Basics.Trunc. | Categories\Pseudofunctor\FromFunctor.v | pseudofunctor_of_functor | 2,473 |
`{Funext} {C} `{HP : forall C D, P C -> P D -> IsHSet (Functor C D)} := Functor C (@sub_pre_cat _ P HP). | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import Functor.Identity. Require Import Pseudofunctor.Core. Require Import Cat.Core. Require Import FunctorCategory.Core. Require Import FunctorCategory.Morphisms NaturalTransformation.Isomorphisms. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Basics.PathGroupoids Basics.Trunc. | Categories\Pseudofunctor\FromFunctor.v | FunctorToCat | 2,474 |
`(F : @FunctorToCat H C P HP) := @pseudofunctor_of_functor _ C P HP F. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import Functor.Identity. Require Import Pseudofunctor.Core. Require Import Cat.Core. Require Import FunctorCategory.Core. Require Import FunctorCategory.Morphisms NaturalTransformation.Isomorphisms. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Basics.PathGroupoids Basics.Trunc. | Categories\Pseudofunctor\FromFunctor.v | pseudofunctor_of_functor_to_cat | 2,475 |
(w x y z : PreCategory) (f : Functor w x) (g : Functor x y) (h : Functor y z) : associator_1 h g f o (1 oR f o 1) = h oL 1 o (1 o @morphism_isomorphic _ _ _ (idtoiso (w -> z) (ap idmap (Functor.Composition.Laws.associativity f g h)))). Proof. t. Defined. | Lemma | Require Import FunctorCategory.Morphisms. Require Import Category.Core Functor.Core. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import Cat.Core. Require Import Pseudofunctor.Core. Import NaturalTransformation.Identity. Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Import Category.Morphisms. Import FunctorCategory.Core. Require Import PathGroupoids. | Categories\Pseudofunctor\Identity.v | identity_associativity | 2,476 |
(x y : PreCategory) (f : Functor x y) : 1 oR f o 1 = (left_identity_natural_transformation_2 f) o @morphism_isomorphic _ _ _ (idtoiso (x -> y) (ap idmap (Functor.Composition.Laws.left_identity f))). Proof. t. Defined. | Lemma | Require Import FunctorCategory.Morphisms. Require Import Category.Core Functor.Core. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import Cat.Core. Require Import Pseudofunctor.Core. Import NaturalTransformation.Identity. Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Import Category.Morphisms. Import FunctorCategory.Core. Require Import PathGroupoids. | Categories\Pseudofunctor\Identity.v | identity_left_identity | 2,477 |
(x y : PreCategory) (f : Functor x y) : f oL 1 o 1 = (right_identity_natural_transformation_2 f) o @morphism_isomorphic _ _ _ (idtoiso (x -> y) (ap idmap (Functor.Composition.Laws.right_identity f))). Proof. t. Defined. | Lemma | Require Import FunctorCategory.Morphisms. Require Import Category.Core Functor.Core. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import Cat.Core. Require Import Pseudofunctor.Core. Import NaturalTransformation.Identity. Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Import Category.Morphisms. Import FunctorCategory.Core. Require Import PathGroupoids. | Categories\Pseudofunctor\Identity.v | identity_right_identity | 2,478 |
Pseudofunctor cat := Build_Pseudofunctor cat (fun C => C. | Definition | Require Import FunctorCategory.Morphisms. Require Import Category.Core Functor.Core. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import Cat.Core. Require Import Pseudofunctor.Core. Import NaturalTransformation.Identity. Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Import Category.Morphisms. Import FunctorCategory.Core. Require Import PathGroupoids. | Categories\Pseudofunctor\Identity.v | identity | 2,479 |
w x y z (f : morphism C w x) (g : morphism C x y) (h : morphism C y z) (p p0 p1 p2 : PreCategory) (f0 : morphism C w z -> Functor p2 p1) (f1 : Functor p0 p1) (f2 : Functor p2 p) (f3 : Functor p p0) (f4 : Functor p2 p0) `(@IsIsomorphism (_ -> _) f4 (f3 o f2)%functor n) `(@IsIsomorphism (_ -> _) (f0 (h o (g o f))%morphism) (f1 o f4)%functor n0) : @paths (NaturalTransformation _ _) (@morphism_isomorphic _ _ _ (Category.Morphisms.idtoiso (p2 -> p1) (ap f0 (Category.Core.associativity C w x y z f g h)))) (n0^-1 o ((f1 oL n^-1) o ((f1 oL n) o (n0 o (@morphism_isomorphic _ _ _ (Category.Morphisms.idtoiso (p2 -> p1) (ap f0 (Category.Core.associativity C w x y z f g h))))))))%natural_transformation. Proof. simpl in *. let C := match goal with |- @paths (@NaturalTransformation ?C ?D ?F ?G) _ _ => constr:((C -> D)%category) end in apply (@iso_moveL_Vp C); apply (@iso_moveL_Mp C _ _ _ _ _ _ (iso_whisker_l _ _ _ _ _ _ _)). path_natural_transformation. reflexivity. Qed. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import HoTT.Tactics. | Categories\Pseudofunctor\RewriteLaws.v | p_composition_of_coherent_for_rewrite_helper | 2,480 |
p_composition_of_coherent_iso_for_rewrite__isisomorphism_helper' : @IsIsomorphism (_ -> _) _ _ (n2 ^-1 o (f2 oL n1 ^-1 o (associator_1 f2 f1 f3 o (n0 oR f3 o n))))%natural_transformation. Proof. eapply isisomorphism_compose; [ eapply isisomorphism_inverse | eapply isisomorphism_compose; [ eapply iso_whisker_l; eapply isisomorphism_inverse | eapply isisomorphism_compose; [ typeclasses eauto | eapply isisomorphism_compose; [ eapply iso_whisker_r; typeclasses eauto | typeclasses eauto ] ] ] ]. Defined. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import HoTT.Tactics. | Categories\Pseudofunctor\RewriteLaws.v | p_composition_of_coherent_iso_for_rewrite__isisomorphism_helper' | 2,481 |
Eval hnf in '. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import HoTT.Tactics. | Categories\Pseudofunctor\RewriteLaws.v | p_composition_of_coherent_iso_for_rewrite__isisomorphism_helper | 2,482 |
X (H' : X = @Build_Isomorphic (_ -> _) _ _ _ p_composition_of_coherent_iso_for_rewrite__isisomorphism_helper) : @morphism_inverse _ _ _ _ X = inv := ap (fun i => @morphism_inverse _ _ _ _ (@isisomorphism_isomorphic _ _ _ i)) H'. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import HoTT.Tactics. | Categories\Pseudofunctor\RewriteLaws.v | p_composition_of_coherent_iso_for_rewrite__isisomorphism_helper__to_inverse | 2,483 |
w x y z (f : morphism C w x) (g : morphism C x y) (h : morphism C y z) : (Category.Morphisms.idtoiso (_ -> _) (ap (@p_morphism_of _ _ F w z) (Category.Core.associativity C w x y z f g h))) = @Build_Isomorphic (_ -> _) _ _ ((((p_composition_of F w y z h (g o f))^-1) o ((p_morphism_of F h oL (p_composition_of F w x y g f)^-1) o ((associator_1 (p_morphism_of F h) (p_morphism_of F g) (p_morphism_of F f)) o ((p_composition_of F x y z h g oR p_morphism_of F f) o p_composition_of F w x z (h o g) f)))))%natural_transformation p_composition_of_coherent_iso_for_rewrite__isisomorphism_helper. Proof. apply path_isomorphic; simpl. simpl rewrite (@p_composition_of_coherent _ C F w x y z f g h). exact p_composition_of_coherent_for_rewrite_helper. Qed. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import HoTT.Tactics. | Categories\Pseudofunctor\RewriteLaws.v | p_composition_of_coherent_iso_for_rewrite | 2,484 |
x y (f : morphism C x y) : (Category.Morphisms.idtoiso (_ -> _) (ap (@p_morphism_of _ _ F x y) (Category.Core.left_identity C x y f))) = @Build_Isomorphic (_ -> _) _ _ ((left_identity_natural_transformation_1 (p_morphism_of F f)) o ((p_identity_of F y oR p_morphism_of F f) o p_composition_of F x y y 1 f))%natural_transformation _. Proof. apply path_isomorphic; simpl. simpl rewrite (@p_left_identity_of_coherent _ C F x y f). path_natural_transformation. symmetry. etransitivity; apply Category.Core.left_identity. Qed. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import HoTT.Tactics. | Categories\Pseudofunctor\RewriteLaws.v | p_left_identity_of_coherent_iso_for_rewrite | 2,485 |
x y (f : morphism C x y) : (Category.Morphisms.idtoiso (_ -> _) (ap (@p_morphism_of _ _ F x y) (Category.Core.right_identity C x y f))) = @Build_Isomorphic (_ -> _) _ _ ((right_identity_natural_transformation_1 (p_morphism_of F f)) o ((p_morphism_of F f oL p_identity_of F x) o p_composition_of F x x y f 1))%natural_transformation _. Proof. apply path_isomorphic; simpl. simpl rewrite (@p_right_identity_of_coherent _ C F x y f). path_natural_transformation. symmetry. etransitivity; apply Category.Core.left_identity. Qed. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import HoTT.Tactics. | Categories\Pseudofunctor\RewriteLaws.v | p_right_identity_of_coherent_iso_for_rewrite | 2,486 |
w x y z f g h : p_composition_of_coherent_for_rewrite_type w x y z f g h := ap (@morphism_isomorphic _ _ _) (@p_composition_of_coherent_iso_for_rewrite w x y z f g h). | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import HoTT.Tactics. | Categories\Pseudofunctor\RewriteLaws.v | p_composition_of_coherent_for_rewrite | 2,487 |
w x y z f g h : p_composition_of_coherent_inverse_for_rewrite_type w x y z f g h := p_composition_of_coherent_iso_for_rewrite__isisomorphism_helper__to_inverse (p_composition_of_coherent_iso_for_rewrite w x y z f g h). | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import HoTT.Tactics. | Categories\Pseudofunctor\RewriteLaws.v | p_composition_of_coherent_inverse_for_rewrite | 2,488 |
x y f : p_left_identity_of_coherent_for_rewrite_type x y f := ap (@morphism_isomorphic _ _ _) (@p_left_identity_of_coherent_iso_for_rewrite x y f). | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import HoTT.Tactics. | Categories\Pseudofunctor\RewriteLaws.v | p_left_identity_of_coherent_for_rewrite | 2,489 |
x y f : p_left_identity_of_coherent_inverse_for_rewrite_type x y f := ap (fun i => @morphism_inverse _ _ _ _ (@isisomorphism_isomorphic _ _ _ i)) (@p_left_identity_of_coherent_iso_for_rewrite x y f). | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import HoTT.Tactics. | Categories\Pseudofunctor\RewriteLaws.v | p_left_identity_of_coherent_inverse_for_rewrite | 2,490 |
x y f : p_right_identity_of_coherent_for_rewrite_type x y f := Eval simpl in ap (@morphism_isomorphic _ _ _) (@p_right_identity_of_coherent_iso_for_rewrite x y f). | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import HoTT.Tactics. | Categories\Pseudofunctor\RewriteLaws.v | p_right_identity_of_coherent_for_rewrite | 2,491 |
x y f : p_right_identity_of_coherent_inverse_for_rewrite_type x y f := ap (fun i => @morphism_inverse _ _ _ _ (@isisomorphism_isomorphic _ _ _ i)) (@p_right_identity_of_coherent_iso_for_rewrite x y f). | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Isomorphisms. Require Import NaturalTransformation.Paths. Require Import FunctorCategory.Core. Require Import Pseudofunctor.Core. Require Import HoTT.Tactics. | Categories\Pseudofunctor\RewriteLaws.v | p_right_identity_of_coherent_inverse_for_rewrite | 2,492 |
PreCategory := (forall x : X, F x -> G x)%category. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. | Categories\PseudonaturalTransformation\Core.v | A | 2,493 |
PreCategory := (forall x y (m : morphism X x y), F x -> G y)%category. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. | Categories\PseudonaturalTransformation\Core.v | B | 2,494 |
PreCategory := (forall x y z (m1 : morphism X y z) (m2 : morphism X x y), F x -> G z)%category. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. | Categories\PseudonaturalTransformation\Core.v | C | 2,495 |
Eval simpl in object A. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. | Categories\PseudonaturalTransformation\Core.v | a_part | 2,496 |
Functor A B. Proof. refine (Build_Functor A B (fun x__Fx_to_Gx => fun x y m => x__Fx_to_Gx y o p_morphism_of F m)%functor (fun x__s x__d x__m => fun x y m => x__m y oR p_morphism_of F m) _ _); simpl; repeat (intro || apply path_forall); [ apply composition_of_whisker_r | apply whisker_r_left_identity ]. Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. | Categories\PseudonaturalTransformation\Core.v | A_to_B_1 | 2,497 |
Functor A B. Proof. refine (Build_Functor A B (fun x__Fx_to_Gx => fun x y m => p_morphism_of G m o x__Fx_to_Gx x)%functor (fun x__s x__d x__m => fun x y m => p_morphism_of G m oL x__m x) _ _); simpl; repeat (intro || apply path_forall); [ apply composition_of_whisker_l | apply whisker_l_right_identity ]. Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. | Categories\PseudonaturalTransformation\Core.v | A_to_B_2 | 2,498 |
(a : a_part) := Eval simpl in forall x y m, (A_to_B_1 a x y m <~=~> A_to_B_2 a x y m). | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. | Categories\PseudonaturalTransformation\Core.v | b_part | 2,499 |