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Functor sig_obj A := Build_Functor sig_obj A (@pr1_type _ _) (fun s d m => m) (fun _ _ _ _ _ => idpath) (fun _ => idpath). | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnObjects.v | pr1_obj | 2,100 |
PreCategory := @sig A Pobj (fun _ _ _ => Unit) _ (fun _ => tt) (fun _ _ _ _ _ _ _ => tt). | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnObjects.v | sig_obj_as_sig | 2,101 |
Functor sig_obj_as_sig sig_obj := Build_Functor sig_obj_as_sig sig_obj (fun x => x) (fun _ _ => @pr1_type _ _) (fun _ _ _ _ _ => idpath) (fun _ => idpath). | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnObjects.v | sig_functor_obj | 2,102 |
Functor sig_obj sig_obj_as_sig := Build_Functor sig_obj sig_obj_as_sig (fun x => x) (fun _ _ m => exist _ m tt) (fun _ _ _ _ _ => idpath) (fun _ => idpath). | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnObjects.v | sig_functor_obj_inv | 2,103 |
`{Funext} : sig_functor_obj o sig_functor_obj_inv = 1 /\ sig_functor_obj_inv o sig_functor_obj = 1. Proof. split; path_functor; trivial. apply path_forall; intros []. apply path_forall; intros []. apply path_forall; intros [? []]. reflexivity. Qed. | Lemma | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnObjects.v | sig_obj_eq | 2,104 |
pr1_obj o sig_functor_obj = pr1' := idpath. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core Category.Sigma.Core. Require Import Functor.Paths. Import Functor.Identity.FunctorIdentityNotations. Import Functor.Composition.Core.FunctorCompositionCoreNotations. | Categories\Category\Sigma\OnObjects.v | sig_obj_compat | 2,105 |
{s d} : @Isomorphic A' s d -> @Isomorphic A s d. Proof. refine ((issig_full_isomorphic A _ _) o _ o (issig_full_isomorphic A' _ _)^-1). exact (functor_sigma pr1_type (fun _ => functor_sigma pr1_type (fun _ => functor_sigma pr1_path (fun _ => pr1_path)))). Defined. | Definition | Require Import Category.Core Category.Morphisms. Require Import Category.Univalent. Require Import Category.Sigma.Core Category.Sigma.OnObjects Category.Sigma.OnMorphisms. Require Import HoTT.Types HoTT.Basics. | Categories\Category\Sigma\Univalent.v | iscategory_sig_mor_helper | 2,106 |
`{A'_cat : IsCategory A'} : IsCategory A. Proof. intros s d. refine (isequiv_homotopic (iscategory_sig_mor_helper o (@idtoiso A' _ _)) _). intro x; apply path_isomorphic; cbn. destruct x; reflexivity. Defined. | Definition | Require Import Category.Core Category.Morphisms. Require Import Category.Univalent. Require Import Category.Sigma.Core Category.Sigma.OnObjects Category.Sigma.OnMorphisms. Require Import HoTT.Types HoTT.Basics. | Categories\Category\Sigma\Univalent.v | iscategory_from_sig_mor | 2,107 |
Record | Require Import Functor.Core NaturalTransformation.Core. Require Import Category.Strict. Require Import Functor.Identity NaturalTransformation.Identity. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Functor.Paths. Require Import HoTT.Basics HoTT.Types. | Categories\CategoryOfSections\Core.v | SectionOfFunctor | 2,108 |
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section_of_functor_sig' : section_of_functor_sig <~> SectionOfFunctor. Proof. issig. Defined. | Lemma | Require Import Functor.Core NaturalTransformation.Core. Require Import Category.Strict. Require Import Functor.Identity NaturalTransformation.Identity. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Functor.Paths. Require Import HoTT.Basics HoTT.Types. | Categories\CategoryOfSections\Core.v | section_of_functor_sig' | 2,109 |
PreCategory. Proof. refine (@Build_PreCategory SectionOfFunctor (fun F G => NaturalTransformation F G) (fun F => 1) (fun _ _ _ T U => T o U) _ _ _ _); abstract (path_natural_transformation; auto with morphism). Defined. | Definition | Require Import Functor.Core NaturalTransformation.Core. Require Import Category.Strict. Require Import Functor.Identity NaturalTransformation.Identity. Require Import NaturalTransformation.Paths NaturalTransformation.Composition.Core. Require Import Functor.Paths. Require Import HoTT.Basics HoTT.Types. | Categories\CategoryOfSections\Core.v | category_of_sections | 2,110 |
Record | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | object | 2,111 |
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object_sig_T <~> object. Proof. issig. Defined. | Lemma | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | issig_object | 2,112 |
path_object' (x y : object) : forall (Ha : x.(a) = y.(a)) (Hb : x.(b) = y.(b)), transport (fun X => morphism C (S X) _) Ha (transport (fun Y => morphism C _ (T Y)) Hb x.(f)) = y.(f) -> x = y. Proof. destruct x, y; simpl. intros; path_induction; reflexivity. Defined. | Lemma | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | path_object' | 2,113 |
ap_a_path_object' x y Ha Hb Hf : ap (@a) (@path_object' x y Ha Hb Hf) = Ha. Proof. destruct x, y; simpl in *. destruct Ha, Hb, Hf; simpl in *. reflexivity. Qed. | Lemma | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | ap_a_path_object' | 2,114 |
ap_b_path_object' x y Ha Hb Hf : ap (@b) (@path_object' x y Ha Hb Hf) = Hb. Proof. destruct x, y; simpl in *. destruct Ha, Hb, Hf; simpl in *. reflexivity. Qed. | Lemma | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | ap_b_path_object' | 2,115 |
(abf a'b'f' : object) := | Record | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | morphism | 2,116 |
abf a'b'f' g h p : morphism abf a'b'f' := @' abf a'b'f' g h p p^. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | Build_morphism | 2,117 |
issig_morphism' abf a'b'f' : (morphism_sig_T' abf a'b'f') <~> morphism abf a'b'f'. Proof. issig. Defined. | Lemma | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | issig_morphism' | 2,118 |
{T0} `{IsHSet T0} (a b : T0) (pf : a = b) : Contr (b = a). Proof. destruct pf. apply contr_inhabited_hprop; try reflexivity. typeclasses eauto. Qed. | Lemma | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | issig_morphism_helper | 2,119 |
abf a'b'f' : (morphism_sig_T abf a'b'f') <~> morphism abf a'b'f'. Proof. etransitivity; [ | exact (' abf a'b'f') ]. repeat (apply equiv_functor_sigma_id; intro). symmetry; apply equiv_sigma_contr; intro. apply issig_morphism_helper; assumption. Defined. | Lemma | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | issig_morphism | 2,120 |
abf a'b'f' (gh g'h' : morphism abf a'b'f') : gh.(g) = g'h'.(g) -> gh.(h) = g'h'.(h) -> gh = g'h'. Proof. destruct gh, g'h'; simpl. intros; path_induction. f_ap. all:exact (center _). Qed. | Lemma | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | path_morphism | 2,121 |
s d d' (gh : morphism d d') (g'h' : morphism s d) : morphism s d' := Build_morphism' s d' (gh. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | compose | 2,122 |
x : morphism x x := Build_morphism' x x ( (x. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | identity | 2,123 |
A B C (S : Functor A C) (T : Functor B C) : PreCategory. Proof. refine (@Build_PreCategory (@object _ _ _ S T) (@morphism _ _ _ S T) (@identity _ _ _ S T) (@compose _ _ _ S T) _ _ _ _ ); abstract path_comma_t. Defined. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | comma_category | 2,124 |
(x y : comma_category) : x ≅ y -> x = y. Proof. intro i. destruct i as [i [i' ? ?]]. hnf in *. destruct i, i'. simpl in *. Global Instance comma_category_IsCategory `{IsCategory A, IsCategory B} : IsCategory comma_category. Proof. hnf. unfold IsStrictCategory in *. typeclasses eauto. Qed. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | comma_category_isotoid | 2,125 |
comma_category S (!a). | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | slice_category | 2,126 |
comma_category (!a) S. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | coslice_category | 2,127 |
slice_category a (Functor. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | slice_category_over | 2,128 |
coslice_category a (Functor. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | coslice_category_over | 2,129 |
comma_category (Functor. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | arrow_category | 2,130 |
CC_Functor' (C : PreCategory) (D : PreCategory) := Functor C D. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Functor.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Require Import Category.Strict. Import Functor.Identity.FunctorIdentityNotations. Module Import CommaCategory. | Categories\Comma\Core.v | CC_Functor' | 2,131 |
Functor (S / T) ((T^op / S^op)^op) := Build_Functor (S / T) ((T^op / S^op)^op) (fun x => obj_of x) (fun s d m => mor_of s d m) (fun _ _ _ _ _ => 1%path) (fun _ => 1%path). | Definition | Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core Functor.Identity. Import Comma.Core. | Categories\Comma\Dual.v | dual_functor | 2,132 |
A B C (S : Functor A C) (T : Functor B C) : dual_functor S T o (dual_functor T^op S^op)^op = 1 /\ (dual_functor T^op S^op)^op o dual_functor S T = 1 := (idpath, idpath)%core. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core Functor.Identity. Import Comma.Core. | Categories\Comma\Dual.v | dual_functor_involutive | 2,133 |
(S / T) -> (S' / T') := fun x => CommaCategory. | Definition | Require Import Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Functor.Paths. Require Import Category.Strict. Import Comma.Core. Import Functor.Identity.FunctorIdentityNotations NaturalTransformation.Identity.NaturalTransformationIdentityNotations. Require Import HoTT.Tactics PathGroupoids Types.Forall. | Categories\Comma\Functorial.v | functorial_morphism_of_object_of | 2,134 |
s d (m : morphism (S / T) s d) : morphism (S' / T') (functorial_morphism_of_object_of s) (functorial_morphism_of_object_of d). Proof. simpl in *. refine (CommaCategory.Build_morphism (functorial_morphism_of_object_of s) (functorial_morphism_of_object_of d) (AF _1 (CommaCategory.g m)) (BF _1 (CommaCategory.h m)) _). unfold functorial_morphism_of_object_of; simpl. clear. abstract helper (exact (CommaCategory.p m)) (commutes TA) (commutes TB). Defined. | Definition | Require Import Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Functor.Paths. Require Import Category.Strict. Import Comma.Core. Import Functor.Identity.FunctorIdentityNotations NaturalTransformation.Identity.NaturalTransformationIdentityNotations. Require Import HoTT.Tactics PathGroupoids Types.Forall. | Categories\Comma\Functorial.v | functorial_morphism_of_morphism_of | 2,135 |
Functor (S / T) (S' / T'). Proof. refine (Build_Functor (S / T) (S' / T') functorial_morphism_of_object_of functorial_morphism_of_morphism_of _ _); abstract ( intros; apply CommaCategory.path_morphism; simpl; auto with functor ). Defined. | Definition | Require Import Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Functor.Paths. Require Import Category.Strict. Import Comma.Core. Import Functor.Identity.FunctorIdentityNotations NaturalTransformation.Identity.NaturalTransformationIdentityNotations. Require Import HoTT.Tactics PathGroupoids Types.Forall. | Categories\Comma\Functorial.v | functorial_morphism_of | 2,136 |
x : @functorial_morphism_of_object_of _ _ _ S T 1 1 1 1 1 x = x. Proof. let A := match goal with |- ?A = ?B => constr:(A) end in let B := match goal with |- ?A = ?B => constr:(B) end in refine (@CommaCategory.path_object' _ _ _ _ _ A B 1%path 1%path _). exact (Category.Core.right_identity _ _ _ _ @ Category.Core.left_identity _ _ _ _)%path. Defined. | Definition | Require Import Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Functor.Paths. Require Import Category.Strict. Import Comma.Core. Import Functor.Identity.FunctorIdentityNotations NaturalTransformation.Identity.NaturalTransformationIdentityNotations. Require Import HoTT.Tactics PathGroupoids Types.Forall. | Categories\Comma\Functorial.v | functorial_identity_of_helper | 2,137 |
`{Funext} : @functorial_morphism_of _ _ _ S T 1 1 1 1 1 = 1%functor. Proof. path_functor; simpl. exists (path_forall _ _ functorial_identity_of_helper). simpl. functorial_helper_t functorial_identity_of_helper. Qed. | Definition | Require Import Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Functor.Paths. Require Import Category.Strict. Import Comma.Core. Import Functor.Identity.FunctorIdentityNotations NaturalTransformation.Identity.NaturalTransformationIdentityNotations. Require Import HoTT.Tactics PathGroupoids Types.Forall. | Categories\Comma\Functorial.v | functorial_identity_of | 2,138 |
x : (functorial_morphism_of TA' TB' o functorial_morphism_of TA TB)%functor x = functorial_morphism_of TA'' TB'' x. Proof. let A := match goal with |- ?A = ?B => constr:(A) end in let B := match goal with |- ?A = ?B => constr:(B) end in refine (@CommaCategory.path_object' _ _ _ _ _ A B 1%path 1%path _). subst AF'' BF'' CF'' TA'' TB''. simpl in *. abstract ( autorewrite with morphism; simpl; helper (exact idpath) (commutes TA') (commutes TB') ). Defined. | Definition | Require Import Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Functor.Paths. Require Import Category.Strict. Import Comma.Core. Import Functor.Identity.FunctorIdentityNotations NaturalTransformation.Identity.NaturalTransformationIdentityNotations. Require Import HoTT.Tactics PathGroupoids Types.Forall. | Categories\Comma\Functorial.v | functorial_composition_of_helper | 2,139 |
`{Funext} : (functorial_morphism_of TA' TB' o functorial_morphism_of TA TB)%functor = functorial_morphism_of TA'' TB''. Proof. path_functor; simpl. exists (path_forall _ _ functorial_composition_of_helper). functorial_helper_t functorial_composition_of_helper. Qed. | Definition | Require Import Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import NaturalTransformation.Composition.Laws. Require Import Functor.Paths. Require Import Category.Strict. Import Comma.Core. Import Functor.Identity.FunctorIdentityNotations NaturalTransformation.Identity.NaturalTransformationIdentityNotations. Require Import HoTT.Tactics PathGroupoids Types.Forall. | Categories\Comma\Functorial.v | functorial_composition_of | 2,140 |
s d (m : morphism ((A -> C)^op * (B -> C)) s d) (x : fst s / snd s) : (fst d / snd d) := CommaCategory. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | comma_category_induced_functor_object_of | 2,141 |
s x : comma_category_induced_functor_object_of (Category.Core.identity s) x = x. Proof. let x1 := match goal with |- ?x1 = ?x2 => constr:(x1) end in let x2 := match goal with |- ?x1 = ?x2 => constr:(x2) end in apply (CommaCategory.path_object' x1 x2 idpath idpath). simpl. abstract (rewrite ?left_identity, ?right_identity; reflexivity). Defined. | Lemma | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | comma_category_induced_functor_object_of_identity | 2,142 |
s d d' (m : morphism ((A -> C)^op * (B -> C)) d d') (m' : morphism ((A -> C)^op * (B -> C)) s d) x : comma_category_induced_functor_object_of (m o m') x = comma_category_induced_functor_object_of m (comma_category_induced_functor_object_of m' x). Proof. let x1 := match goal with |- ?x1 = ?x2 => constr:(x1) end in let x2 := match goal with |- ?x1 = ?x2 => constr:(x2) end in apply (CommaCategory.path_object' x1 x2 idpath idpath). abstract ( destruct m', m, x; simpl in *; rewrite !associativity; reflexivity ). Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | comma_category_induced_functor_object_of_compose | 2,143 |
s d m s0 d0 (m0 : morphism (fst s / snd s) s0 d0) : morphism (fst d / snd d) (@comma_category_induced_functor_object_of s d m s0) (@comma_category_induced_functor_object_of s d m d0). Proof. simpl. let s := match goal with |- CommaCategory.morphism ?s ?d => constr:(s) end in let d := match goal with |- CommaCategory.morphism ?s ?d => constr:(d) end in refine (CommaCategory.Build_morphism s d (CommaCategory.g m0) (CommaCategory.h m0) _); simpl in *; clear. abstract ( destruct_head prod; destruct_head CommaCategory.morphism; destruct_head CommaCategory.object; simpl in *; repeat (try_associativity_quick (rewrite <- !commutes || (progress f_ap))); repeat (try_associativity_quick (rewrite !commutes || (progress f_ap))); assumption ). Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | comma_category_induced_functor_morphism_of | 2,144 |
s d (m : morphism ((A -> C)^op * (B -> C)) s d) : Functor (fst s / snd s) (fst d / snd d). Proof. refine (Build_Functor (fst s / snd s) (fst d / snd d) (@comma_category_induced_functor_object_of s d m) (@comma_category_induced_functor_morphism_of s d m) _ _ ); abstract ( intros; apply CommaCategory.path_morphism; reflexivity ). Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | comma_category_induced_functor | 2,145 |
s d (m : morphism D s d) : NaturalTransformation !s !d. Proof. exists (fun _ : Unit => m); simpl; intros; clear; abstract (autorewrite with category; reflexivity). Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | slice_category_induced_functor_nt | 2,146 |
F' a' (m : morphism D a a') (T : NaturalTransformation F' F) : Functor (F / a) (F' / a') := comma_category_induced_functor (s := (F, !a)) (d := (F', !a')) (T, @slice_category_induced_functor_nt a a' m). | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | slice_category_induced_functor | 2,147 |
F' T := @slice_category_induced_functor F' a 1 T. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | slice_category_nt_induced_functor | 2,148 |
a' m := @slice_category_induced_functor F a' m 1. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | slice_category_morphism_induced_functor | 2,149 |
F' a' (m : morphism D a' a) (T : NaturalTransformation F F') : Functor (a / F) (a' / F') := comma_category_induced_functor (s := (!a, F)) (d := (!a', F')) (@slice_category_induced_functor_nt a' a m, T). | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | coslice_category_induced_functor | 2,150 |
F' T := @coslice_category_induced_functor F' a 1 T. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | coslice_category_nt_induced_functor | 2,151 |
a' m := @coslice_category_induced_functor F a' m 1. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | coslice_category_morphism_induced_functor | 2,152 |
a a' (m : morphism C a a') : Functor (C / a) (C / a') := Eval hnf in slice_category_morphism_induced_functor _ _ _ m. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | slice_category_over_induced_functor | 2,153 |
a a' (m : morphism C a' a) : Functor (a \ C) (a' \ C) := Eval hnf in coslice_category_morphism_induced_functor _ _ _ m. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | coslice_category_over_induced_functor | 2,154 |
a a' (m : morphism cat a a') : Functor (cat / a) (cat / a') := slice_category_over_induced_functor cat a a' m. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | cat_over_induced_functor | 2,155 |
a a' (m : morphism cat a' a) : Functor (a \ cat) (a' \ cat) := coslice_category_over_induced_functor cat a a' m. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual. Require Import Category.Prod. Require Import NaturalTransformation.Identity. Require Import FunctorCategory.Core Cat.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics. | Categories\Comma\InducedFunctors.v | over_cat_induced_functor | 2,156 |
Functor (S / T) (A * B) := Build_Functor (S / T) (A * B) (fun abf => (CommaCategory. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Prod Functor.Prod.Core. Require Import Functor.Composition.Core Functor.Identity. Require Import InitialTerminalCategory.Functors. Require Import Types.Prod. Import Comma.Core. | Categories\Comma\Projection.v | comma_category_projection | 2,157 |
Functor (arrow_category A) A := Eval simpl in fst o comma_category_projection _ 1. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Prod Functor.Prod.Core. Require Import Functor.Composition.Core Functor.Identity. Require Import InitialTerminalCategory.Functors. Require Import Types.Prod. Import Comma.Core. | Categories\Comma\Projection.v | arrow_category_projection | 2,158 |
(a : A) : Functor (A / a) A := Eval simpl in fst o comma_category_projection 1 _. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Prod Functor.Prod.Core. Require Import Functor.Composition.Core Functor.Identity. Require Import InitialTerminalCategory.Functors. Require Import Types.Prod. Import Comma.Core. | Categories\Comma\Projection.v | slice_category_over_projection | 2,159 |
(a : A) : Functor (a \ A) A := Eval simpl in snd o comma_category_projection _ 1. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Prod Functor.Prod.Core. Require Import Functor.Composition.Core Functor.Identity. Require Import InitialTerminalCategory.Functors. Require Import Types.Prod. Import Comma.Core. | Categories\Comma\Projection.v | coslice_category_over_projection | 2,160 |
Functor (S / a) A := Eval simpl in fst o comma_category_projection S !a. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Prod Functor.Prod.Core. Require Import Functor.Composition.Core Functor.Identity. Require Import InitialTerminalCategory.Functors. Require Import Types.Prod. Import Comma.Core. | Categories\Comma\Projection.v | slice_category_projection | 2,161 |
Functor (a / S) A := Eval simpl in snd o comma_category_projection !a S. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Prod Functor.Prod.Core. Require Import Functor.Composition.Core Functor.Identity. Require Import InitialTerminalCategory.Functors. Require Import Types.Prod. Import Comma.Core. | Categories\Comma\Projection.v | coslice_category_projection | 2,162 |
(ST : object ((A -> C)^op * (B -> C))) : Cat / !((A * B; PAB) : Cat). Proof. exists (Basics.Overture.fst ST / Basics.Overture.snd ST; P_comma _ _) (center _). exact (comma_category_projection (Basics.Overture.fst ST) (Basics.Overture.snd ST)). Defined. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Prod Functor.Prod.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors NatCategory. Require Import FunctorCategory.Core. Require Import Cat.Core. Require Import Functor.Paths. Import Comma.Core. Require Import Comma.InducedFunctors Comma.Projection. Require Import Types.Forall PathGroupoids HoTT.Tactics. | Categories\Comma\ProjectionFunctors.v | comma_category_projection_functor_object_of | 2,163 |
s d (m : morphism ((A -> C)^op * (B -> C)) s d) : morphism (Cat / !((A * B; PAB) : Cat)) (comma_category_projection_functor_object_of s) (comma_category_projection_functor_object_of d). Proof. hnf. refine (CommaCategory.Build_morphism (comma_category_projection_functor_object_of s) (comma_category_projection_functor_object_of d) (comma_category_induced_functor m) (center _) _). simpl. destruct_head_hnf Basics.Overture.prod. path_functor. Defined. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Prod Functor.Prod.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors NatCategory. Require Import FunctorCategory.Core. Require Import Cat.Core. Require Import Functor.Paths. Import Comma.Core. Require Import Comma.InducedFunctors Comma.Projection. Require Import Types.Forall PathGroupoids HoTT.Tactics. | Categories\Comma\ProjectionFunctors.v | comma_category_projection_functor_morphism_of | 2,164 |
x : comma_category_projection_functor_morphism_of (Category.Core.identity x) = 1. Proof. apply CommaCategory.path_morphism; simpl; [ | reflexivity ]. path_functor. exists (path_forall _ _ (comma_category_induced_functor_object_of_identity _)). comma_laws_t. Qed. | Lemma | Require Import Category.Core Functor.Core. Require Import Category.Prod Functor.Prod.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors NatCategory. Require Import FunctorCategory.Core. Require Import Cat.Core. Require Import Functor.Paths. Import Comma.Core. Require Import Comma.InducedFunctors Comma.Projection. Require Import Types.Forall PathGroupoids HoTT.Tactics. | Categories\Comma\ProjectionFunctors.v | comma_category_projection_functor_identity_of | 2,165 |
s d d' m m' : comma_category_projection_functor_morphism_of (@Category.Core.compose _ s d d' m' m) = (comma_category_projection_functor_morphism_of m') o (comma_category_projection_functor_morphism_of m). Proof. apply CommaCategory.path_morphism; simpl; [ | reflexivity ]. path_functor. simpl. exists (path_forall _ _ (comma_category_induced_functor_object_of_compose m' m)). comma_laws_t. Qed. | Lemma | Require Import Category.Core Functor.Core. Require Import Category.Prod Functor.Prod.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors NatCategory. Require Import FunctorCategory.Core. Require Import Cat.Core. Require Import Functor.Paths. Import Comma.Core. Require Import Comma.InducedFunctors Comma.Projection. Require Import Types.Forall PathGroupoids HoTT.Tactics. | Categories\Comma\ProjectionFunctors.v | comma_category_projection_functor_composition_of | 2,166 |
Functor ((A -> C)^op * (B -> C)) (Cat / !((A * B; PAB) : Cat)) := Build_Functor ((A -> C)^op * (B -> C)) (Cat / !((A * B; PAB) : Cat)) comma_category_projection_functor_object_of comma_category_projection_functor_morphism_of comma_category_projection_functor_composition_of comma_category_projection_functor_identity_of. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Prod Functor.Prod.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors NatCategory. Require Import FunctorCategory.Core. Require Import Cat.Core. Require Import Functor.Paths. Import Comma.Core. Require Import Comma.InducedFunctors Comma.Projection. Require Import Types.Forall PathGroupoids HoTT.Tactics. | Categories\Comma\ProjectionFunctors.v | comma_category_projection_functor | 2,167 |
object (((C -> D)^op) -> (D -> (Cat / ((C; PC) : Cat)))). Proof. refine ((ExponentialLaws.Law4.Functors.inverse _ _ _) _). refine (_ o (Functor.Identity.identity (C -> D)^op, inv D)). refine (_ o @comma_category_projection_functor _ P HF C 1 D PC1 P_comma). refine (cat_over_induced_functor _). hnf. exact (ProductLaws.Law1.functor _). Defined. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Prod Functor.Prod.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors NatCategory. Require Import FunctorCategory.Core. Require Import Cat.Core. Require Import Functor.Paths. Import Comma.Core. Require Import Comma.InducedFunctors Comma.Projection. Require Import Types.Forall PathGroupoids HoTT.Tactics. | Categories\Comma\ProjectionFunctors.v | slice_category_projection_functor | 2,168 |
object ((C -> D)^op -> (D -> (Cat / ((C; PC) : Cat)))). Proof. refine ((ExponentialLaws.Law4.Functors.inverse _ _ _) _). refine (_ o (Functor.Identity.identity (C -> D)^op, inv D)). refine (_ o @comma_category_projection_functor _ P HF C 1 D PC1 P_comma). refine (cat_over_induced_functor _). hnf. exact (ProductLaws.Law1.functor _). Defined. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Prod Functor.Prod.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors NatCategory. Require Import FunctorCategory.Core. Require Import Cat.Core. Require Import Functor.Paths. Import Comma.Core. Require Import Comma.InducedFunctors Comma.Projection. Require Import Types.Forall PathGroupoids HoTT.Tactics. | Categories\Comma\ProjectionFunctors.v | coslice_category_projection_functor | 2,169 |
slice_category_projection_functor' : object ((C -> D) -> (D^op -> (Cat / ((C; PC) : Cat)))). Proof. refine ((ExponentialLaws.Law4.Functors.inverse _ _ _) _). refine (_ o (Functor.Identity.identity (C -> D), (inv D)^op)). refine (_ o ProductLaws.Swap.functor _ _). refine (_ o @comma_category_projection_functor _ P HF 1 C D P1C P_comma'). refine (cat_over_induced_functor _). hnf. exact (ProductLaws.Law1.functor' _). Defined. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Prod Functor.Prod.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors NatCategory. Require Import FunctorCategory.Core. Require Import Cat.Core. Require Import Functor.Paths. Import Comma.Core. Require Import Comma.InducedFunctors Comma.Projection. Require Import Types.Forall PathGroupoids HoTT.Tactics. | Categories\Comma\ProjectionFunctors.v | slice_category_projection_functor' | 2,170 |
coslice_category_projection_functor' : object ((C -> D) -> (D^op -> (Cat / ((C; PC) : Cat)))). Proof. refine ((ExponentialLaws.Law4.Functors.inverse _ _ _) _). refine (_ o (Functor.Identity.identity (C -> D), (inv D)^op)). refine (_ o ProductLaws.Swap.functor _ _). refine (_ o @comma_category_projection_functor _ P HF 1 C D P1C P_comma'). refine (cat_over_induced_functor _). hnf. exact (ProductLaws.Law1.functor' _). Defined. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Prod Functor.Prod.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors NatCategory. Require Import FunctorCategory.Core. Require Import Cat.Core. Require Import Functor.Paths. Import Comma.Core. Require Import Comma.InducedFunctors Comma.Projection. Require Import Types.Forall PathGroupoids HoTT.Tactics. | Categories\Comma\ProjectionFunctors.v | coslice_category_projection_functor' | 2,171 |
Functor (0 -> C) 1 := center _. | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Identity. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors InitialTerminalCategory.NaturalTransformations. Require Import HoTT.Basics HoTT.Types. | Categories\ExponentialLaws\Law0.v | functor | 2,172 |
Functor 1 (0 -> C) := center _. | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Identity. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors InitialTerminalCategory.NaturalTransformations. Require Import HoTT.Basics HoTT.Types. | Categories\ExponentialLaws\Law0.v | inverse | 2,173 |
functor o inverse = 1 /\ inverse o functor = 1 := center _. | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Identity. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors InitialTerminalCategory.NaturalTransformations. Require Import HoTT.Basics HoTT.Types. | Categories\ExponentialLaws\Law0.v | law | 2,174 |
functor' : Functor (C -> 0) 0 := center _. | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Identity. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors InitialTerminalCategory.NaturalTransformations. Require Import HoTT.Basics HoTT.Types. | Categories\ExponentialLaws\Law0.v | functor' | 2,175 |
inverse' : Functor 0 (C -> 0) := center _. | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Identity. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors InitialTerminalCategory.NaturalTransformations. Require Import HoTT.Basics HoTT.Types. | Categories\ExponentialLaws\Law0.v | inverse' | 2,176 |
law' : functor' o inverse' = 1 /\ inverse' o functor' = 1 := center _. | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Identity. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors InitialTerminalCategory.NaturalTransformations. Require Import HoTT.Basics HoTT.Types. | Categories\ExponentialLaws\Law0.v | law' | 2,177 |
Functor (0 -> 0) 1 := functor _. | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Identity. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors InitialTerminalCategory.NaturalTransformations. Require Import HoTT.Basics HoTT.Types. | Categories\ExponentialLaws\Law0.v | functor00 | 2,178 |
Functor 1 (0 -> 0) := inverse _. | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Identity. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors InitialTerminalCategory.NaturalTransformations. Require Import HoTT.Basics HoTT.Types. | Categories\ExponentialLaws\Law0.v | inverse00 | 2,179 |
functor00 o inverse00 = 1 /\ inverse00 o functor00 = 1 := law _. | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Identity. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors InitialTerminalCategory.NaturalTransformations. Require Import HoTT.Basics HoTT.Types. | Categories\ExponentialLaws\Law0.v | law00 | 2,180 |
Functor (1 -> C) C := Build_Functor (1 -> C) C (fun F => F (center _)) (fun s d m => m (center _)) (fun _ _ _ _ _ => idpath) (fun _ => idpath). | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Identity NaturalTransformation.Core NaturalTransformation.Paths. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors InitialTerminalCategory.NaturalTransformations. Require Import HoTT.Basics HoTT.Types. | Categories\ExponentialLaws\Law1\Functors.v | functor | 2,181 |
s d (m : morphism C s d) : morphism (1 -> C) (Functors.from_terminal _ s) (Functors.from_terminal _ d). Proof. refine (Build_NaturalTransformation (Functors.from_terminal _ s) (Functors.from_terminal _ d) (fun _ => m) _). simpl; intros. etransitivity; [ apply right_identity | symmetry; apply left_identity ]. Defined. | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Identity NaturalTransformation.Core NaturalTransformation.Paths. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors InitialTerminalCategory.NaturalTransformations. Require Import HoTT.Basics HoTT.Types. | Categories\ExponentialLaws\Law1\Functors.v | inverse_morphism_of | 2,182 |
Functor C (1 -> C). Proof. refine (Build_Functor C (1 -> C) (@Functors.from_terminal _ _ _ _ _) inverse_morphism_of _ _ ); abstract (path_natural_transformation; trivial). Defined. | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Identity NaturalTransformation.Core NaturalTransformation.Paths. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors InitialTerminalCategory.NaturalTransformations. Require Import HoTT.Basics HoTT.Types. | Categories\ExponentialLaws\Law1\Functors.v | inverse | 2,183 |
functor' : Functor (C -> 1) 1 := Functors. | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Identity NaturalTransformation.Core NaturalTransformation.Paths. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors InitialTerminalCategory.NaturalTransformations. Require Import HoTT.Basics HoTT.Types. | Categories\ExponentialLaws\Law1\Functors.v | functor' | 2,184 |
inverse' : Functor 1 (C -> 1) := Functors. | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Identity NaturalTransformation.Core NaturalTransformation.Paths. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors InitialTerminalCategory.NaturalTransformations. Require Import HoTT.Basics HoTT.Types. | Categories\ExponentialLaws\Law1\Functors.v | inverse' | 2,185 |
law' : functor' o inverse' = 1 /\ inverse' o functor' = 1 := center _. | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Functor.Identity NaturalTransformation.Core NaturalTransformation.Paths. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors InitialTerminalCategory.NaturalTransformations. Require Import HoTT.Basics HoTT.Types. | Categories\ExponentialLaws\Law1\Functors.v | law' | 2,186 |
(c : Functor 1 C) : Functors.from_terminal C (c (center _)) = c. Proof. path_functor. exists (path_forall _ _ (fun x => ap (object_of c) (contr _))). abstract ( exp_laws_t; simpl; rewrite <- identity_of; f_ap; symmetry; apply contr ). Defined. | Definition | Require Import Category.Core Functor.Core Functor.Identity Functor.Paths ExponentialLaws.Law1.Functors Functor.Composition.Core. Require Import InitialTerminalCategory.Core. Require Import Basics.Trunc ExponentialLaws.Tactics. Require Import Basics.Tactics. | Categories\ExponentialLaws\Law1\Law.v | helper | 2,187 |
@functor _ one _ C o inverse C = 1 /\ inverse C o @functor _ one _ C = 1. Proof. split; path_functor. exists (path_forall _ _ helper). unfold helper. exp_laws_t. Qed. | Lemma | Require Import Category.Core Functor.Core Functor.Identity Functor.Paths ExponentialLaws.Law1.Functors Functor.Composition.Core. Require Import InitialTerminalCategory.Core. Require Import Basics.Trunc ExponentialLaws.Tactics. Require Import Basics.Tactics. | Categories\ExponentialLaws\Law1\Law.v | law | 2,188 |
Functor (C1 + C2 -> D) ((C1 -> D) * (C2 -> D)) := pointwise (inl C1 C2) 1 * pointwise (inr C1 C2) 1. | Definition | Require Import Functor.Core FunctorCategory.Core Functor.Identity NaturalTransformation.Core Category.Sum Category.Prod Functor.Sum Functor.Prod.Core NaturalTransformation.Sum Functor.Pointwise.Core NaturalTransformation.Paths. | Categories\ExponentialLaws\Law2\Functors.v | functor | 2,189 |
Functor ((C1 -> D) * (C2 -> D)) (C1 + C2 -> D). Proof. refine (Build_Functor ((C1 -> D) * (C2 -> D)) (C1 + C2 -> D) (fun FG => fst FG + snd FG)%functor (fun _ _ m => fst_type m + snd_type m)%natural_transformation _ _); simpl in *; abstract ( repeat (intros [?|?] || intros [? ?]); simpl in *; apply path_natural_transformation; intros [?|?]; reflexivity ). Defined. | Definition | Require Import Functor.Core FunctorCategory.Core Functor.Identity NaturalTransformation.Core Category.Sum Category.Prod Functor.Sum Functor.Prod.Core NaturalTransformation.Sum Functor.Pointwise.Core NaturalTransformation.Paths. | Categories\ExponentialLaws\Law2\Functors.v | inverse | 2,190 |
(c : Functor C1 D * Functor C2 D) : ((1 o (Basics.Overture.fst c + Basics.Overture.snd c) o inl C1 C2)%functor, (1 o (Basics.Overture.fst c + Basics.Overture.snd c) o inr C1 C2)%functor)%core = c. Proof. apply path_prod; simpl; path_functor. Defined. | Lemma | Require Import Functor.Core. Require Import Category.Sum Functor.Sum. Require Import Functor.Paths. Require Import Functor.Identity Functor.Composition.Core. Require Import Types.Prod ExponentialLaws.Tactics. Require Import ExponentialLaws.Law2.Functors. | Categories\ExponentialLaws\Law2\Law.v | helper1 | 2,191 |
(c : Functor (C1 + C2) D) x : (1 o c o inl C1 C2 + 1 o c o inr C1 C2) x = c x. Proof. destruct x; reflexivity. Defined. | Lemma | Require Import Functor.Core. Require Import Category.Sum Functor.Sum. Require Import Functor.Paths. Require Import Functor.Identity Functor.Composition.Core. Require Import Types.Prod ExponentialLaws.Tactics. Require Import ExponentialLaws.Law2.Functors. | Categories\ExponentialLaws\Law2\Law.v | helper2_helper | 2,192 |
(c : Functor (C1 + C2) D) : 1 o c o inl C1 C2 + 1 o c o inr C1 C2 = c. Proof. path_functor. (exists (path_forall _ _ (@helper2_helper c))). abstract exp_laws_t. Defined. | Lemma | Require Import Functor.Core. Require Import Category.Sum Functor.Sum. Require Import Functor.Paths. Require Import Functor.Identity Functor.Composition.Core. Require Import Types.Prod ExponentialLaws.Tactics. Require Import ExponentialLaws.Law2.Functors. | Categories\ExponentialLaws\Law2\Law.v | helper2 | 2,193 |
functor D C1 C2 o inverse D C1 C2 = 1 /\ inverse D C1 C2 o functor D C1 C2 = 1. Proof. split; path_functor; [ (exists (path_forall _ _ helper1)) | (exists (path_forall _ _ helper2)) ]; exp_laws_t; unfold helper1, helper2; exp_laws_t. Qed. | Lemma | Require Import Functor.Core. Require Import Category.Sum Functor.Sum. Require Import Functor.Paths. Require Import Functor.Identity Functor.Composition.Core. Require Import Types.Prod ExponentialLaws.Tactics. Require Import ExponentialLaws.Law2.Functors. | Categories\ExponentialLaws\Law2\Law.v | law | 2,194 |
Functor (D -> C1 * C2) ((D -> C1) * (D -> C2)) := Build_Functor (D -> C1 * C2) ((D -> C1) * (D -> C2)) (fun H => (fst o H, snd o H)%core) (fun s d m => (fst oL m, snd oL m)%core) (fun _ _ _ _ _ => path_prod' (composition_of_whisker_l _ _ _) (composition_of_whisker_l _ _ _)) (fun _ => path_prod' (whisker_l_right_identity _ _) (whisker_l_right_identity _ _)). | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Category.Prod. Require Import Functor.Prod Functor.Composition.Core NaturalTransformation.Composition.Laws NaturalTransformation.Composition.Core. Require Import Types.Prod. | Categories\ExponentialLaws\Law3\Functors.v | functor | 2,195 |
Functor ((D -> C1) * (D -> C2)) (D -> C1 * C2) := Functor. | Definition | Require Import Category.Core Functor.Core FunctorCategory.Core Category.Prod. Require Import Functor.Prod Functor.Composition.Core NaturalTransformation.Composition.Laws NaturalTransformation.Composition.Core. Require Import Types.Prod. | Categories\ExponentialLaws\Law3\Functors.v | inverse | 2,196 |
(c : Functor D C1 * Functor D C2) : ((fst o (Basics.Overture.fst c * Basics.Overture.snd c))%functor, (snd o (Basics.Overture.fst c * Basics.Overture.snd c))%functor)%core = c. Proof. apply path_prod; [ apply compose_fst_prod | apply compose_snd_prod ]. Defined. | Lemma | Require Import Category.Core Functor.Core. Require Import Functor.Prod. Require Import Functor.Paths. Require Import Functor.Identity Functor.Composition.Core. Require Import ExponentialLaws.Law3.Functors. Require Import Types.Prod ExponentialLaws.Tactics. | Categories\ExponentialLaws\Law3\Law.v | helper | 2,197 |
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 _ _ helper)) | (exists (path_forall _ _ (fun _ => Functor.Prod.Universal.unique idpath idpath))) ]; exp_laws_t; unfold helper, compose_fst_prod, compose_snd_prod, Functor.Prod.Universal.unique, Functor.Prod.Universal.unique_helper; exp_laws_t. Qed. | Lemma | Require Import Category.Core Functor.Core. Require Import Functor.Prod. Require Import Functor.Paths. Require Import Functor.Identity Functor.Composition.Core. Require Import ExponentialLaws.Law3.Functors. Require Import Types.Prod ExponentialLaws.Tactics. | Categories\ExponentialLaws\Law3\Law.v | Law | 2,198 |
(C1 -> (C2 -> D))%category -> (C1 * C2 -> D)%category. Proof. intro F; hnf in F |- *. refine (Build_Functor (C1 * C2) D (fun c1c2 => F (fst c1c2) (snd c1c2)) (fun s d m => F (fst d) _1 (snd m) o (F _1 (fst m)) (snd s)) _ _); abstract do_exponential4. 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_object_of | 2,199 |