Datasets:

phanerozoic

/

Coq-HoTT

like 2

(M : IsInitialMorphism Ap) : morphism C X (U (IsInitialMorphism_object M)) := CommaCategory.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics.

Categories\UniversalProperties.v

IsInitialMorphism_morphism

(M : IsInitialMorphism Ap) (Y : D) (f : morphism C X (U Y)) : morphism D (IsInitialMorphism_object M) Y := CommaCategory.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics.

Categories\UniversalProperties.v

IsInitialMorphism_property_morphism

(M : IsInitialMorphism Ap) (Y : D) (f : morphism C X (U Y)) : (U _1 (IsInitialMorphism_property_morphism M Y f)) o IsInitialMorphism_morphism M = f := concat (CommaCategory.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics.

Categories\UniversalProperties.v

IsInitialMorphism_property_morphism_property

(M : IsInitialMorphism Ap) (Y : D) (f : morphism C X (U Y)) m' (H : U _1 m' o IsInitialMorphism_morphism M = f) : IsInitialMorphism_property_morphism M Y f = m' := ap (@CommaCategory.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics.

Categories\UniversalProperties.v

IsInitialMorphism_property_morphism_unique

(M : IsInitialMorphism Ap) (Y : D) (f : morphism C X (U Y)) : Contr { m : morphism D (IsInitialMorphism_object M) Y | U _1 m o IsInitialMorphism_morphism M = f } := Build_Contr _ (IsInitialMorphism_property_morphism M Y f; IsInitialMorphism_property_morphism_property M Y f) (fun m' => path_sigma _ (IsInitialMorphism_property_morphism M Y f; IsInitialMorphism_property_morphism_property M Y f) m' (@IsInitialMorphism_property_morphism_unique M Y f m'.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics.

Categories\UniversalProperties.v

IsInitialMorphism_property

(Ap : object (U / X)) : Type := @IsInitialMorphism (C^op) _ X (U^op) (op_object Ap).

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics.

Categories\UniversalProperties.v

IsTerminalMorphism

forall (A : D)(* := CommaCategory.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics.

Categories\UniversalProperties.v

Build_IsTerminalMorphism

forall (A : D) (p : morphism C (U A) X) (Ap := CommaCategory.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics.

Categories\UniversalProperties.v

Build_IsTerminalMorphism_curried

forall (univ : { A : D | { p : morphism C (U A) X | let Ap := CommaCategory.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics.

Categories\UniversalProperties.v

Build_IsTerminalMorphism_uncurried

D := @IsInitialMorphism_object C^op D^op X U^op (op_object Ap) M.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics.

Categories\UniversalProperties.v

IsTerminalMorphism_object

morphism C (U IsTerminalMorphism_object) X := @IsInitialMorphism_morphism C^op D^op X U^op (op_object Ap) M.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics.

Categories\UniversalProperties.v

IsTerminalMorphism_morphism

forall (Y : D) (f : morphism C (U Y) X), Contr { m : morphism D Y IsTerminalMorphism_object | IsTerminalMorphism_morphism o U _1 m = f } := @IsInitialMorphism_property C^op D^op X U^op (op_object Ap) M.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics.

Categories\UniversalProperties.v

IsTerminalMorphism_property

forall (Y : D) (f : morphism C (U Y) X), morphism D Y IsTerminalMorphism_object := @IsInitialMorphism_property_morphism C^op D^op X U^op (op_object Ap) M.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics.

Categories\UniversalProperties.v

IsTerminalMorphism_property_morphism

forall (Y : D) (f : morphism C (U Y) X), IsTerminalMorphism_morphism o (U _1 (IsTerminalMorphism_property_morphism Y f)) = f := @IsInitialMorphism_property_morphism_property C^op D^op X U^op (op_object Ap) M.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics.

Categories\UniversalProperties.v

IsTerminalMorphism_property_morphism_property

forall (Y : D) (f : morphism C (U Y) X) m' (H : IsTerminalMorphism_morphism o U _1 m' = f), IsTerminalMorphism_property_morphism Y f = m' := @IsInitialMorphism_property_morphism_unique C^op D^op X U^op (op_object Ap) M.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics.

Categories\UniversalProperties.v

IsTerminalMorphism_property_morphism_unique

A : Functor A^op (A -> set_cat) := ExponentialLaws.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import SetCategory. Require Import Functor.Attributes. Require Import Functor.Composition.Functorial. Require Import Functor.Identity. Require Import HomFunctor. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import HoTT.Tactics.

Categories\Yoneda.v

coyoneda

A : Functor A (A^op -> set_cat) := coyoneda A^op.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import SetCategory. Require Import Functor.Attributes. Require Import Functor.Composition.Functorial. Require Import Functor.Identity. Require Import HomFunctor. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import HoTT.Tactics.

Categories\Yoneda.v

yoneda

Functor (A -> set_cat) (A -> set_cat) := (compose_functor _ _ set_cat (coyoneda A)^op) o (yoneda (A -> set_cat)).

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import SetCategory. Require Import Functor.Attributes. Require Import Functor.Composition.Functorial. Require Import Functor.Identity. Require Import HomFunctor. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import HoTT.Tactics.

Categories\Yoneda.v

coyoneda_functor

F : morphism (_ -> _) (coyoneda_functor A F) F. Proof. refine (Build_NaturalTransformation (coyoneda_functor A F) F (fun a phi => phi a 1%morphism) _). simpl. abstract ( repeat (intro || apply path_forall); simpl in *; match goal with | [ T : NaturalTransformation _ _ |- _ ] => simpl rewrite <- (fun s d m => apD10 (commutes T s d m)) end; rewrite ?left_identity, ?right_identity; reflexivity ). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import SetCategory. Require Import Functor.Attributes. Require Import Functor.Composition.Functorial. Require Import Functor.Identity. Require Import HomFunctor. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import HoTT.Tactics.

Categories\Yoneda.v

coyoneda_natural_transformation_helper

morphism (_ -> _) (coyoneda_functor A) 1. Proof. hnf. simpl. let F := match goal with |- NaturalTransformation ?F ?G => constr:(F) end in let G := match goal with |- NaturalTransformation ?F ?G => constr:(G) end in refine (Build_NaturalTransformation F G coyoneda_natural_transformation_helper _). simpl. abstract (repeat first [ intro | progress path_natural_transformation | reflexivity ]). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import SetCategory. Require Import Functor.Attributes. Require Import Functor.Composition.Functorial. Require Import Functor.Identity. Require Import HomFunctor. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import HoTT.Tactics.

Categories\Yoneda.v

coyoneda_natural_transformation

`{Funext} A (F : object (A -> set_cat)) a : morphism set_cat (F a) (coyoneda_functor A F a). Proof. intro Fa. hnf. simpl in *. let F0 := match goal with |- NaturalTransformation ?F ?G => constr:(F) end in let G0 := match goal with |- NaturalTransformation ?F ?G => constr:(G) end in refine (Build_NaturalTransformation F0 G0 (fun a' : A => (fun f : morphism A a a' => F _1 f Fa)) _ ). simpl. abstract ( repeat first [ reflexivity | intro | apply path_forall | progress rewrite ?composition_of, ?identity_of ] ). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import SetCategory. Require Import Functor.Attributes. Require Import Functor.Composition.Functorial. Require Import Functor.Identity. Require Import HomFunctor. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import HoTT.Tactics.

Categories\Yoneda.v

coyoneda_lemma_morphism_inverse

Functor (A^op -> set_cat) (A^op -> set_cat) := coyoneda_functor A^op.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import SetCategory. Require Import Functor.Attributes. Require Import Functor.Composition.Functorial. Require Import Functor.Identity. Require Import HomFunctor. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import HoTT.Tactics.

Categories\Yoneda.v

yoneda_functor

morphism (_ -> _) 1 (yoneda_functor A) := @morphism_inverse _ _ _ _ (coyoneda_lemma A^op).

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import SetCategory. Require Import Functor.Attributes. Require Import Functor.Composition.Functorial. Require Import Functor.Identity. Require Import HomFunctor. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import HoTT.Tactics.

Categories\Yoneda.v

yoneda_natural_transformation

(A : PreCategory) : IsFullyFaithful (coyoneda A). Proof. intros a b. pose proof (@isisomorphism_inverse _ _ _ _ (@isisomorphism_components_of _ _ _ _ _ _ (@isisomorphism_components_of _ _ _ _ _ _ (@coyoneda_lemma _ A) (@coyoneda _ A b)) a)) as H'. simpl in *. unfold coyoneda_lemma_morphism_inverse in *; simpl in *. unfold Functors.inverse_object_of_morphism_of in *; simpl in *. let m := match type of H' with IsIsomorphism ?m => constr:(m) end in apply isisomorphism_set_cat_natural_transformation_paths with (T1 := m). - simpl. clear H'. intros; apply path_forall; intro; rewrite left_identity, right_identity; reflexivity. - destruct H' as [m' H0' H1']. (exists m'). + exact H0'. + exact H1'. Qed.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import SetCategory. Require Import Functor.Attributes. Require Import Functor.Composition.Functorial. Require Import Functor.Identity. Require Import HomFunctor. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import HoTT.Tactics.

Categories\Yoneda.v

coyoneda_embedding

(A : PreCategory) : IsFullyFaithful (yoneda A). Proof. intros a b. pose proof (@coyoneda_embedding A^op a b) as CYE. unfold yoneda. let T := type of CYE in let T' := (eval simpl in T) in pose proof ((fun x : T => (x : T')) CYE) as CYE'. let G := match goal with |- ?G => constr:(G) end in let G' := (eval simpl in G) in exact ((fun x : G' => (x : G)) CYE'). Qed.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import SetCategory. Require Import Functor.Attributes. Require Import Functor.Composition.Functorial. Require Import Functor.Identity. Require Import HomFunctor. Require Import FunctorCategory.Core. Require Import NaturalTransformation.Paths. Require Import HoTT.Tactics.

Categories\Yoneda.v

yoneda_embedding

C D (F : Functor C D) (G : Functor D C) (A : F -| G) : G^op -| F^op := @Build_AdjunctionUnitCounit _ _ (G^op) (F^op) ((counit A)^op) ((unit A)^op) (unit_counit_equation_2 A) (unit_counit_equation_1 A).

Definition

Require Import Category.Core Functor.Core. Require Import Functor.Dual NaturalTransformation.Dual. Require Import Adjoint.UnitCounit Adjoint.Core.

Categories\Adjoint\Dual.v

opposite

C D (F : Functor C D) (G : Functor D C) (A : F -| G) : (A^op)^op = A := idpath.

Definition

Require Import Category.Core Functor.Core. Require Import Functor.Dual NaturalTransformation.Dual. Require Import Adjoint.UnitCounit Adjoint.Core.

Categories\Adjoint\Dual.v

opposite_involutive

Record

Require Import Category.Core Functor.Core. Require Import Adjoint.UnitCounit. Require Import Functor.Dual. Require Import Functor.Prod.Core. Require Import HomFunctor. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity.

Categories\Adjoint\Hom.v

AdjunctionHom

adjunction_naturalityT. Proof. pose proof (ap10 (commutes A (c, d) (c, d') (1%morphism, g))^ f) as H'; simpl in *. rewrite ?identity_of, ?left_identity, ?right_identity in H'. exact H'. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Hom. Require Import Category.Morphisms. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity. Require Import SetCategory.Morphisms. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Equivalences.

Categories\Adjoint\HomCoercions.v

adjunction_naturality

adjunction_naturality' : adjunction_naturalityT'. Proof. pose proof (ap10 (commutes A (c', d) (c, d) (g, 1%morphism))^ f) as H'; simpl in *. rewrite ?identity_of, ?left_identity, ?right_identity in H'. exact H'. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Hom. Require Import Category.Morphisms. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity. Require Import SetCategory.Morphisms. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Equivalences.

Categories\Adjoint\HomCoercions.v

adjunction_naturality'

9.

Proposition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Hom. Require Import Category.Morphisms. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity. Require Import SetCategory.Morphisms. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Equivalences.

Categories\Adjoint\HomCoercions.v

9.4.

(A : AdjunctionHom F G) (c : C) (d : D) (f : morphism C c (G d)) : IsHProp {g : morphism D (F c) d & G _1 g o A (c, F c) 1 = f}. Proof. apply hprop_allpath. intros [g0 H0] [g1 H1]; apply path_sigma_hprop; simpl. destruct H1. rewrite !adjunction_naturality in H0. rewrite !right_identity in H0. change (idmap g0 = idmap g1). rewrite <- (ap10 (@left_inverse _ _ _ (A (c, d)) _)). simpl rewrite H0. let k := constr:(ap10 (@left_inverse _ _ _ (A (c, d)) _)) in simpl rewrite k. reflexivity. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Hom. Require Import Category.Morphisms. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity. Require Import SetCategory.Morphisms. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Equivalences.

Categories\Adjoint\HomCoercions.v

adjunction_unit__of__adjunction_hom_helper

(A : AdjunctionHom F G) (s d : C) (m : morphism C s d) : A (d, F d) 1 o m = G _1 (F _1 m) o A (s, F s) 1. Proof. simpl; rewrite adjunction_naturality', adjunction_naturality. rewrite ?left_identity, ?right_identity. reflexivity. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Hom. Require Import Category.Morphisms. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity. Require Import SetCategory.Morphisms. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Equivalences.

Categories\Adjoint\HomCoercions.v

adjunction_unit__of__adjunction_hom__mate_of__commutes

(A : AdjunctionHom F G) : AdjunctionUnit F G. Proof. exists (Build_NaturalTransformation 1 (G o F) (fun c => A (c, F c) 1) (adjunction_unit__of__adjunction_hom__mate_of__commutes A)). simpl in *. intros c d f. apply contr_inhabited_hprop. - apply adjunction_unit__of__adjunction_hom_helper. - exact ((A (c, d))^-1%morphism f; ((adjunction_naturality A _ _ _ _ _) @ (ap (A (c, d)) (right_identity _ _ _ _)) @ (ap10 (@right_inverse _ _ _ (A (c, d)) _) f))%path). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Hom. Require Import Category.Morphisms. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity. Require Import SetCategory.Morphisms. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Equivalences.

Categories\Adjoint\HomCoercions.v

adjunction_unit__of__adjunction_hom

(c : C) (d : D) : morphism C c (G d) <~> morphism D (F c) d. Proof. refine (equiv_adjointify (fun f => (@center _ (T.2 _ _ f)).1) (fun g => G _1 g o T.1 c) _ _); intro. - match goal with | [ |- @pr1 ?A ?P ?x = ?y ] => change (x.1 = (exist P y idpath).1) end. apply (ap pr1). apply contr. - match goal with | [ |- context[?x.1] ] => apply x.2 end. Defined.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Hom. Require Import Category.Morphisms. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity. Require Import SetCategory.Morphisms. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Equivalences.

Categories\Adjoint\HomCoercions.v

equiv_hom_set_adjunction

(T : AdjunctionUnit F G) sc sd dc dd (mc : morphism C dc sc) (md : morphism D sd dd) : (fun x : morphism D (F sc) sd => G _1 (md o x o F _1 mc) o T .1 dc) = (fun x : morphism D (F sc) sd => G _1 md o (G _1 x o T .1 sc) o mc). Proof. apply path_forall; intro. rewrite !composition_of, !associativity. simpl rewrite (commutes T.1). reflexivity. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Hom. Require Import Category.Morphisms. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity. Require Import SetCategory.Morphisms. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Equivalences.

Categories\Adjoint\HomCoercions.v

adjunction_hom__of__adjunction_unit__commutes

(T : AdjunctionUnit F G) : AdjunctionHom F G. Proof. constructor. (eexists (Build_NaturalTransformation _ _ _ _)). apply (@isisomorphism_natural_transformation _); simpl. exact (fun cd => @isiso_isequiv _ _ _ _ (equiv_isequiv (equiv_hom_set_adjunction T (fst cd) (snd cd))^-1)). Unshelve. simpl. intros. exact (adjunction_hom__of__adjunction_unit__commutes T _ _ _ _ _ _). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Hom. Require Import Category.Morphisms. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity. Require Import SetCategory.Morphisms. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Equivalences.

Categories\Adjoint\HomCoercions.v

adjunction_hom__of__adjunction_unit

`{Funext} C D F G := @AdjunctionUnit C D F G.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Hom. Require Import Category.Morphisms. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity. Require Import SetCategory.Morphisms. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Equivalences.

Categories\Adjoint\HomCoercions.v

AdjunctionUnitWithFunext

`{Funext} C D F G := @AdjunctionCounit C D F G.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Hom. Require Import Category.Morphisms. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity. Require Import SetCategory.Morphisms. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Equivalences.

Categories\Adjoint\HomCoercions.v

AdjunctionCounitWithFunext

`{Funext} C D F G := @AdjunctionUnitCounit C D F G.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Hom. Require Import Category.Morphisms. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity. Require Import SetCategory.Morphisms. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Equivalences.

Categories\Adjoint\HomCoercions.v

AdjunctionUnitCounitWithFunext

`{Funext} C D F G (A : AdjunctionUnitWithFunext _ _) : AdjunctionHom _ _ := @adjunction_hom__of__adjunction_unit _ C D F G A.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Hom. Require Import Category.Morphisms. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity. Require Import SetCategory.Morphisms. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Equivalences.

Categories\Adjoint\HomCoercions.v

adjunction_hom__of__adjunction_unit_Funext

`{Funext} C D F G (A : AdjunctionCounitWithFunext _ _) : AdjunctionHom _ _ := @adjunction_hom__of__adjunction_unit _ C D F G (adjunction_unit_counit__of__adjunction_counit A).

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Hom. Require Import Category.Morphisms. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity. Require Import SetCategory.Morphisms. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Equivalences.

Categories\Adjoint\HomCoercions.v

AdjunctionHomOfAdjunctionCounit_Funext

`{Funext} C D F G (A : AdjunctionUnitCounitWithFunext _ _) : AdjunctionHom _ _ := @adjunction_hom__of__adjunction_unit _ C D F G A.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Hom. Require Import Category.Morphisms. Require Import Functor.Composition.Core. Require Import FunctorCategory.Morphisms. Require Import Functor.Identity. Require Import SetCategory.Morphisms. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Equivalences.

Categories\Adjoint\HomCoercions.v

adjunction_hom__of__adjunction_unitCounit_Funext

C : @Adjunction C C 1 1 := @Build_AdjunctionUnitCounit C C 1 1 1 1 (fun _ => identity_identity _ _) (fun _ => identity_identity _ _).

Definition

Require Import Category.Core. Require Import Functor.Identity NaturalTransformation.Identity. Require Import Adjoint.UnitCounit Adjoint.Core.

Categories\Adjoint\Identity.v

identity

adjunction_sig <~> (F -| G). Proof. issig. Defined.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core Functor.Identity. Require Import Adjoint.UnitCounit Adjoint.Core NaturalTransformation.Paths. Require Import Types Trunc. Require Import Basics.Tactics.

Categories\Adjoint\Paths.v

equiv_sig_adjunction

path_adjunction' (A A' : F -| G) : unit A = unit A' -> counit A = counit A' -> A = A'. Proof. intros. destruct A, A'; simpl in *. path_induction. f_ap; exact (center _). Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core Functor.Identity. Require Import Adjoint.UnitCounit Adjoint.Core NaturalTransformation.Paths. Require Import Types Trunc. Require Import Basics.Tactics.

Categories\Adjoint\Paths.v

path_adjunction'

(A A' : F -| G) : components_of (unit A) == components_of (unit A') -> components_of (counit A) == components_of (counit A') -> A = A'. Proof. intros. apply '; apply path_natural_transformation; assumption. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core Functor.Identity. Require Import Adjoint.UnitCounit Adjoint.Core NaturalTransformation.Paths. Require Import Types Trunc. Require Import Basics.Tactics.

Categories\Adjoint\Paths.v

path_adjunction

NaturalTransformation (identity (E -> C)) ((pointwise (identity E) G) o (pointwise (identity E) F)). Proof. pose proof (A : AdjunctionUnit _ _) as A''. refine (_ o (((idtoiso (C := (_ -> _)) (Functor.Pointwise.Properties.identity_of _ _))^-1)%morphism : morphism _ _ _)). refine (_ o NaturalTransformation.Pointwise.pointwise_r (Functor.Identity.identity E) (proj1 A'')). refine (((idtoiso (C := (_ -> _)) (Functor.Pointwise.Properties.composition_of (Functor.Identity.identity E) F (Functor.Identity.identity E) G)) : morphism _ _ _) o _). refine (NaturalTransformation.Pointwise.pointwise_l _ _). exact (NaturalTransformation.Composition.Laws.left_identity_natural_transformation_2 _). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions. Require Import Functor.Pointwise.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import FunctorCategory.Core. Import NaturalTransformation.Identity.NaturalTransformationIdentityNotations. Require Import NaturalTransformation.Paths Functor.Paths. Require Import Basics.PathGroupoids HoTT.Tactics Types.Arrow.

Categories\Adjoint\Pointwise.v

unit_l

NaturalTransformation (pointwise (identity E) F o pointwise (identity E) G) (identity (E -> D)). Proof. pose proof (A : AdjunctionCounit _ _) as A''. refine ((((idtoiso (C := (_ -> _)) (Functor.Pointwise.Properties.identity_of _ _)))%morphism : morphism _ _ _) o _). refine (NaturalTransformation.Pointwise.pointwise_r (Functor.Identity.identity E) (proj1 A'') o _). refine (_ o (((idtoiso (C := (_ -> _)) (Functor.Pointwise.Properties.composition_of (Functor.Identity.identity E) G (Functor.Identity.identity E) F))^-1)%morphism : morphism _ _ _)). refine (NaturalTransformation.Pointwise.pointwise_l _ _). exact (NaturalTransformation.Composition.Laws.left_identity_natural_transformation_1 _). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions. Require Import Functor.Pointwise.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import FunctorCategory.Core. Import NaturalTransformation.Identity.NaturalTransformationIdentityNotations. Require Import NaturalTransformation.Paths Functor.Paths. Require Import Basics.PathGroupoids HoTT.Tactics Types.Arrow.

Categories\Adjoint\Pointwise.v

counit_l

pointwise (identity E) F -| pointwise (identity E) G. Proof. exists unit_l counit_l; abstract ( path_natural_transformation; intros; destruct A; simpl in *; repeat match goal with | _ => progress simpl | _ => progress autorewrite with adjoint_pointwise | [ |- context[ap object_of (path_functor_uncurried ?F ?G (?HO; ?HM))] ] => rewrite (@path_functor_uncurried_fst _ _ _ F G HO HM) | _ => progress unfold Functor.Pointwise.Properties.identity_of | _ => progress unfold Functor.Pointwise.Properties.composition_of | _ => progress unfold Functor.Pointwise.Properties.identity_of_helper | _ => progress unfold Functor.Pointwise.Properties.composition_of_helper | _ => progress unfold Functor.Pointwise.Properties.identity_of_helper_helper | _ => progress unfold Functor.Pointwise.Properties.composition_of_helper_helper | [ H : _ |- _ ] => apply H end ). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions. Require Import Functor.Pointwise.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import FunctorCategory.Core. Import NaturalTransformation.Identity.NaturalTransformationIdentityNotations. Require Import NaturalTransformation.Paths Functor.Paths. Require Import Basics.PathGroupoids HoTT.Tactics Types.Arrow.

Categories\Adjoint\Pointwise.v

pointwise_l

NaturalTransformation (identity (C -> E)) ((pointwise F (identity E)) o (pointwise G (identity E))). Proof. pose proof (A : AdjunctionUnit _ _) as A''. refine (_ o (((idtoiso (C := (_ -> _)) (Functor.Pointwise.Properties.identity_of _ _))^-1)%morphism : morphism _ _ _)). refine (_ o NaturalTransformation.Pointwise.pointwise_l (proj1 A'') (Functor.Identity.identity E)). refine (((idtoiso (C := (_ -> _)) (Functor.Pointwise.Properties.composition_of G (Functor.Identity.identity E) F (Functor.Identity.identity E))) : morphism _ _ _) o _). refine (NaturalTransformation.Pointwise.pointwise_r _ _). exact (NaturalTransformation.Composition.Laws.left_identity_natural_transformation_2 _). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions. Require Import Functor.Pointwise.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import FunctorCategory.Core. Import NaturalTransformation.Identity.NaturalTransformationIdentityNotations. Require Import NaturalTransformation.Paths Functor.Paths. Require Import Basics.PathGroupoids HoTT.Tactics Types.Arrow.

Categories\Adjoint\Pointwise.v

unit_r

NaturalTransformation (pointwise G (identity E) o pointwise F (identity E)) (identity (D -> E)). Proof. pose proof (A : AdjunctionCounit _ _) as A''. refine ((((idtoiso (C := (_ -> _)) (Functor.Pointwise.Properties.identity_of _ _)))%morphism : morphism _ _ _) o _). refine (NaturalTransformation.Pointwise.pointwise_l (proj1 A'') (Functor.Identity.identity E) o _). refine (_ o (((idtoiso (C := (_ -> _)) (Functor.Pointwise.Properties.composition_of F (Functor.Identity.identity E) G (Functor.Identity.identity E)))^-1)%morphism : morphism _ _ _)). refine (NaturalTransformation.Pointwise.pointwise_r _ _). exact (NaturalTransformation.Composition.Laws.left_identity_natural_transformation_1 _). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions. Require Import Functor.Pointwise.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import FunctorCategory.Core. Import NaturalTransformation.Identity.NaturalTransformationIdentityNotations. Require Import NaturalTransformation.Paths Functor.Paths. Require Import Basics.PathGroupoids HoTT.Tactics Types.Arrow.

Categories\Adjoint\Pointwise.v

counit_r

pointwise G (identity E) -| pointwise F (identity E). Proof. exists unit_r counit_r; abstract ( path_natural_transformation; intros; destruct A; simpl in *; repeat match goal with | _ => reflexivity | _ => progress simpl | _ => progress autorewrite with adjoint_pointwise | [ |- context[ap object_of (path_functor_uncurried ?F ?G (?HO; ?HM))] ] => rewrite (@path_functor_uncurried_fst _ _ _ F G HO HM) | _ => progress unfold Functor.Pointwise.Properties.identity_of | _ => progress unfold Functor.Pointwise.Properties.composition_of | _ => progress unfold Functor.Pointwise.Properties.identity_of_helper | _ => progress unfold Functor.Pointwise.Properties.composition_of_helper | _ => progress unfold Functor.Pointwise.Properties.identity_of_helper_helper | _ => progress unfold Functor.Pointwise.Properties.composition_of_helper_helper | _ => rewrite <- composition_of; progress rewrite_hyp end ). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions. Require Import Functor.Pointwise.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import FunctorCategory.Core. Import NaturalTransformation.Identity.NaturalTransformationIdentityNotations. Require Import NaturalTransformation.Paths Functor.Paths. Require Import Basics.PathGroupoids HoTT.Tactics Types.Arrow.

Categories\Adjoint\Pointwise.v

pointwise_r

{ T : NaturalTransformation 1 (G o F) | forall (c : C) (d : D) (f : morphism C c (G d)), Contr { g : morphism D (F c) d | G _1 g o T c = f } }.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual NaturalTransformation.Dual. Require Import Functor.Composition.Core Functor.Identity.

Categories\Adjoint\UnitCounit.v

AdjunctionUnit

{ U : NaturalTransformation (F o G) 1 | forall (c : C) (d : D) (g : morphism D (F c) d), Contr { f : morphism C c (G d) | U d o F _1 f = g } }.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual NaturalTransformation.Dual. Require Import Functor.Composition.Core Functor.Identity.

Categories\Adjoint\UnitCounit.v

AdjunctionCounit

(A : AdjunctionUnit G^op F^op) : AdjunctionCounit F G := exist (fun U : NaturalTransformation (F o G) 1 => forall (c : C) (d : D) (g : morphism D (F c) d), Contr {f : morphism C c (G d) | U d o F _1 f = g }) (A.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual NaturalTransformation.Dual. Require Import Functor.Composition.Core Functor.Identity.

Categories\Adjoint\UnitCounit.v

adjunction_counit__op__adjunction_unit

(A : AdjunctionUnit G F) : AdjunctionCounit F^op G^op := exist (fun U : NaturalTransformation (F^op o G^op) 1 => forall (c : C^op) (d : D^op) (g : morphism D^op ((F^op)%functor c) d), Contr {f : morphism C^op c ((G^op)%functor d) | U d o F^op _1 f = g }) (A.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual NaturalTransformation.Dual. Require Import Functor.Composition.Core Functor.Identity.

Categories\Adjoint\UnitCounit.v

adjunction_counit__op__adjunction_unit__inv

(A : AdjunctionCounit G^op F^op) : AdjunctionUnit F G := exist (fun T : NaturalTransformation 1 (G o F) => forall (c : C) (d : D) (f : morphism C c (G d)), Contr { g : morphism D (F c) d | G _1 g o T c = f }) (A.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual NaturalTransformation.Dual. Require Import Functor.Composition.Core Functor.Identity.

Categories\Adjoint\UnitCounit.v

adjunction_unit__op__adjunction_counit

(A : AdjunctionCounit G F) : AdjunctionUnit F^op G^op := exist (fun T : NaturalTransformation 1 (G^op o F^op) => forall (c : C^op) (d : D^op) (f : morphism C^op c ((G^op)%functor d)), Contr {g : morphism D^op ((F^op)%functor c) d | G^op _1 g o T c = f }) (A.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual NaturalTransformation.Dual. Require Import Functor.Composition.Core Functor.Identity.

Categories\Adjoint\UnitCounit.v

adjunction_unit__op__adjunction_counit__inv

Record

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Category.Dual Functor.Dual NaturalTransformation.Dual. Require Import Functor.Composition.Core Functor.Identity.

Categories\Adjoint\UnitCounit.v

AdjunctionUnitCounit

C D F G (A : @AdjunctionUnitCounit C D F G) : AdjunctionUnit F G. Proof. exists (unit A). intros c d f. apply contr_inhabited_hprop; [ apply hprop_allpath | (exists (counit A d o F _1 f)); abstract unit_counit_of_t ]. intros [? ?] [? ?]. apply path_sigma_uncurried. let A := match goal with |- @sig ?A ?P => constr:(A) end in let H := fresh in assert (H : A); [ | exists H; exact (center _) ]. simpl. let x := match goal with |- ?x = ?y => constr:(x) end in let y := match goal with |- ?x = ?y => constr:(y) end in rewrite <- (right_identity _ _ _ x), <- (right_identity _ _ _ y), <- !(unit_counit_equation_1 A), <- ?associativity; repeat simpl rewrite <- (commutes (counit A)); (try_associativity_quick rewrite <- !composition_of); repeat apply ap; etransitivity; [ | symmetry ]; eassumption. Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.Dual. Require Import Functor.Composition.Core Functor.Identity. Require Import HoTT.Tactics Basics.Trunc Types.Sigma. Require Import Basics.Tactics.

Categories\Adjoint\UnitCounitCoercions.v

adjunction_unit__of__adjunction_unit_counit

C D F G (A : @AdjunctionUnitCounit C D F G) : AdjunctionCounit F G := adjunction_counit__op__adjunction_unit (adjunction_unit__of__adjunction_unit_counit A^op).

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.Dual. Require Import Functor.Composition.Core Functor.Identity. Require Import HoTT.Tactics Basics.Trunc Types.Sigma. Require Import Basics.Tactics.

Categories\Adjoint\UnitCounitCoercions.v

adjunction_counit__of__adjunction_unit_counit

(A : AdjunctionUnit F G) s d (m : morphism D s d) (eta := A.1) (eps := fun X => (@center _ (A.2 (G X) X 1)).1) : G _1 (eps d o F _1 (G _1 m)) o eta (G s) = G _1 m -> G _1 (m o eps s) o eta (G s) = G _1 m -> eps d o F _1 (G _1 m) = m o eps s. Proof. intros. transitivity (@center _ (A.2 _ _ (G _1 m))).1; [ symmetry | ]; let x := match goal with |- _ = ?x => constr:(x) end in refine ((fun H => ap pr1 (@contr _ (A.2 _ _ (G _1 m)) (x; H))) _); assumption. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.Dual. Require Import Functor.Composition.Core Functor.Identity. Require Import HoTT.Tactics Basics.Trunc Types.Sigma. Require Import Basics.Tactics.

Categories\Adjoint\UnitCounitCoercions.v

counit_natural_transformation__of__adjunction_unit_helper

(A : AdjunctionUnit F G) : NaturalTransformation (F o G) 1. Proof. refine (Build_NaturalTransformation (F o G) 1 (fun d => (@center _ (A.2 (G d) d 1)).1) _). abstract ( to_unit_counit_nt counit_natural_transformation__of__adjunction_unit_helper ltac:(fun H => try_associativity_quick rewrite <- H) ). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.Dual. Require Import Functor.Composition.Core Functor.Identity. Require Import HoTT.Tactics Basics.Trunc Types.Sigma. Require Import Basics.Tactics.

Categories\Adjoint\UnitCounitCoercions.v

counit_natural_transformation__of__adjunction_unit

(A : AdjunctionUnit F G) (Y : C) (eta := A.1) (eps := fun X => (@center _ (A.2 (G X) X 1)).1) : G _1 (eps (F Y) o F _1 (eta Y)) o eta Y = eta Y -> eps (F Y) o F _1 (eta Y) = 1. Proof. intros. etransitivity; [ symmetry | ]; simpl_do_clear ltac:(fun H => apply H) (fun y H => (@contr _ (A.2 _ _ (A.1 Y)) (y; H))..1); try assumption. simpl. rewrite ?identity_of, ?left_identity, ?right_identity; reflexivity. Qed.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.Dual. Require Import Functor.Composition.Core Functor.Identity. Require Import HoTT.Tactics Basics.Trunc Types.Sigma. Require Import Basics.Tactics.

Categories\Adjoint\UnitCounitCoercions.v

zig__of__adjunction_unit

(A : AdjunctionUnit F G) : AdjunctionUnitCounit F G. Proof. exists A.1 (counit_natural_transformation__of__adjunction_unit A); simpl; intros; try match goal with | [ |- context[?x.1] ] => exact x.2 end; []. abstract (to_unit_counit_nt zig__of__adjunction_unit ltac:(fun H => try_associativity_quick rewrite <- H)). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.Dual. Require Import Functor.Composition.Core Functor.Identity. Require Import HoTT.Tactics Basics.Trunc Types.Sigma. Require Import Basics.Tactics.

Categories\Adjoint\UnitCounitCoercions.v

adjunction_unit_counit__of__adjunction_unit

C D F G (A : @AdjunctionCounit C D F G) : AdjunctionUnitCounit F G := ((adjunction_unit_counit__of__adjunction_unit (adjunction_unit__op__adjunction_counit__inv A))^op)%adjunction.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Adjoint.UnitCounit Adjoint.Dual. Require Import Functor.Composition.Core Functor.Identity. Require Import HoTT.Tactics Basics.Trunc Types.Sigma. Require Import Basics.Tactics.

Categories\Adjoint\UnitCounitCoercions.v

adjunction_unit_counit__of__adjunction_counit

((_, _); (AH o AG) o AF) = ((_, _); AH o (AG o AF)) :> AdjunctionWithFunctors B E. Proof. apply path_sigma_uncurried; simpl. (exists (path_prod' (Functor.Composition.Laws. _ _ _) (symmetry _ _ (Functor.Composition.Laws. _ _ _)))); Adjoint.Composition.LawsTactic.law_t. Qed.

Lemma

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

Categories\Adjoint\Composition\AssociativityLaw.v

associativity

NaturalTransformation 1 ((G o G') o (F' o F)). Proof. pose (unit A) as eta. pose (unit A') as eta'. refine ((fun (T : NaturalTransformation _ _) (U : NaturalTransformation _ _) => T o (G oL eta' oR F) o U o eta) _ _); NaturalTransformation.Composition.Laws.nt_solve_associator. Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import Functor.Identity. Require Import Adjoint.UnitCounit Adjoint.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics.

Categories\Adjoint\Composition\Core.v

compose_unit

NaturalTransformation ((F' o F) o (G o G')) 1. Proof. pose (counit A) as eps. pose (counit A') as eps'. refine ((fun (T : NaturalTransformation _ _) (U : NaturalTransformation _ _) => eps' o U o (F' oL eps oR G') o T) _ _); NaturalTransformation.Composition.Laws.nt_solve_associator. Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import Functor.Identity. Require Import Adjoint.UnitCounit Adjoint.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics.

Categories\Adjoint\Composition\Core.v

compose_counit

F' o F -| G o G'. Proof. exists compose_unit compose_counit; simpl; abstract ( repeat match goal with | _ => intro | _ => reflexivity | _ => progress rewrite ?identity_of, ?left_identity, ?right_identity | _ => rewrite <- ?composition_of, unit_counit_equation_1 | _ => rewrite <- ?composition_of, unit_counit_equation_2 | [ A : _ -| _ |- _ = 1%morphism ] => (etransitivity; [ | apply (unit_counit_equation_1 A) ]; try_associativity_quick f_ap) | [ A : _ -| _ |- _ = 1%morphism ] => (etransitivity; [ | apply (unit_counit_equation_2 A) ]; try_associativity_quick f_ap) | _ => repeat (try_associativity_quick rewrite <- !composition_of); progress repeat apply ap; rewrite ?composition_of | [ |- context[components_of ?T] ] => (try_associativity_quick simpl rewrite <- (commutes T)); try_associativity_quick (apply concat_right_identity || apply concat_left_identity) | [ |- context[components_of ?T] ] => (try_associativity_quick simpl rewrite (commutes T)); try_associativity_quick (apply concat_right_identity || apply concat_left_identity) end ). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Composition.Core NaturalTransformation.Composition.Core. Require Import Functor.Identity. Require Import Adjoint.UnitCounit Adjoint.Core. Require Import HoTT.Tactics. Require Import Basics.Tactics.

Categories\Adjoint\Composition\Core.v

compose

((_, _); 1 o A) = ((_, _); A) :> AdjunctionWithFunctors C D. Proof. apply path_sigma_uncurried; simpl. (exists (path_prod' (Functor.Composition.Laws. _) (Functor.Composition.Laws.right_identity _))). Adjoint.Composition.LawsTactic.law_t. Qed.

Lemma

Require Import Category.Core Functor.Core. Require Import Adjoint.Composition.Core Adjoint.Core Adjoint.Identity. Require Import Types.Sigma Types.Prod.

Categories\Adjoint\Composition\IdentityLaws.v

left_identity

((_, _); A o 1) = ((_, _); A) :> AdjunctionWithFunctors C D. Proof. apply path_sigma_uncurried; simpl. (exists (path_prod' (Functor.Composition.Laws. _) (Functor.Composition.Laws.left_identity _))). Adjoint.Composition.LawsTactic.law_t. Qed.

Lemma

Require Import Category.Core Functor.Core. Require Import Adjoint.Composition.Core Adjoint.Core Adjoint.Identity. Require Import Types.Sigma Types.Prod.

Categories\Adjoint\Composition\IdentityLaws.v

right_identity

Functor (sig_obj (D -> C) (fun G => { F : Functor C D & F -| G })) (sig_obj ((C -> D)^op * (D -> C)) (fun FG => fst FG -| snd FG)) := Build_Functor (sig_obj (D -> C) (fun G => { F : Functor C D & F -| G })) (sig_obj ((C -> D)^op * (D -> C)) (fun FG => fst FG -| snd FG)) (fun GFA => ((GFA.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual. Require Import FunctorCategory.Core. Require Import Category.Sigma.OnObjects Category.Prod. Require Import Adjoint.Core. Require Import Adjoint.Functorial.Parts Adjoint.Functorial.Laws. Require Import HoTT.Types.Prod.

Categories\Adjoint\Functorial\Core.v

left_functor_nondep

Functor (sig_obj (C -> D) (fun F => { G : Functor D C & F -| G })) (sig_obj ((C -> D) * (D -> C)^op) (fun FG => fst FG -| snd FG)) := Build_Functor (sig_obj (C -> D) (fun F => { G : Functor D C & F -| G })) (sig_obj ((C -> D) * (D -> C)^op) (fun FG => fst FG -| snd FG)) (fun GFA => ((GFA.

Definition

Require Import Category.Core Functor.Core. Require Import Category.Dual. Require Import FunctorCategory.Core. Require Import Category.Sigma.OnObjects Category.Prod. Require Import Adjoint.Core. Require Import Adjoint.Functorial.Parts Adjoint.Functorial.Laws. Require Import HoTT.Types.Prod.

Categories\Adjoint\Functorial\Core.v

right_functor_nondep

@left_morphism_of C C 1 D D 1 G F A G F A 1 = ((left_identity_natural_transformation_2 _) o (right_identity_natural_transformation_1 _))%natural_transformation. Proof. t. Qed.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. Require Import NaturalTransformation.Paths. Require Import Functor.Dual NaturalTransformation.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.Dual. Require Import Adjoint.Functorial.Parts. Require Import HoTT.Tactics. Require Import Basics.Tactics.

Categories\Adjoint\Functorial\Laws.v

left_identity_of

@left_morphism_of_nondep C D G F A G F A 1 = 1%natural_transformation. Proof. t. Qed.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. Require Import NaturalTransformation.Paths. Require Import Functor.Dual NaturalTransformation.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.Dual. Require Import Adjoint.Functorial.Parts. Require Import HoTT.Tactics. Require Import Basics.Tactics.

Categories\Adjoint\Functorial\Laws.v

left_identity_of_nondep

(@left_morphism_of _ _ _ _ _ _ _ _ A _ _ A'' ((associator_1 _ _ _) o (T' oR DF) o (associator_2 _ _ _) o (CF' oL T) o (associator_1 _ _ _))) = (associator_2 _ _ _) o (DF' oL left_morphism_of A A' T) o (associator_1 _ _ _) o (left_morphism_of A' A'' T' oR CF) o (associator_2 _ _ _). Proof. t. Qed.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. Require Import NaturalTransformation.Paths. Require Import Functor.Dual NaturalTransformation.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.Dual. Require Import Adjoint.Functorial.Parts. Require Import HoTT.Tactics. Require Import Basics.Tactics.

Categories\Adjoint\Functorial\Laws.v

left_composition_of

(@left_morphism_of_nondep _ _ _ _ A _ _ A'' (T' o T)) = ((left_morphism_of_nondep A A' T) o (left_morphism_of_nondep A' A'' T')). Proof. t. Qed.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. Require Import NaturalTransformation.Paths. Require Import Functor.Dual NaturalTransformation.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.Dual. Require Import Adjoint.Functorial.Parts. Require Import HoTT.Tactics. Require Import Basics.Tactics.

Categories\Adjoint\Functorial\Laws.v

left_composition_of_nondep

@right_morphism_of C C 1 D D 1 F G A F G A 1 = ((right_identity_natural_transformation_2 _) o (left_identity_natural_transformation_1 _))%natural_transformation := ap (@NaturalTransformation.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. Require Import NaturalTransformation.Paths. Require Import Functor.Dual NaturalTransformation.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.Dual. Require Import Adjoint.Functorial.Parts. Require Import HoTT.Tactics. Require Import Basics.Tactics.

Categories\Adjoint\Functorial\Laws.v

right_identity_of

@right_morphism_of_nondep C D F G A F G A 1 = 1%natural_transformation := ap (@NaturalTransformation.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. Require Import NaturalTransformation.Paths. Require Import Functor.Dual NaturalTransformation.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.Dual. Require Import Adjoint.Functorial.Parts. Require Import HoTT.Tactics. Require Import Basics.Tactics.

Categories\Adjoint\Functorial\Laws.v

right_identity_of_nondep

right_morphism_of A A'' ((associator_2 DF' DF F) o ((DF' oL T) o ((associator_1 DF' F' CF) o ((T' oR CF) o (associator_2 F'' CF' CF))))) = (associator_1 G'' DF' DF) o ((right_morphism_of A' A'' T' oR DF) o ((associator_2 CF' G' DF) o ((CF' oL right_morphism_of A A' T) o (associator_1 CF' CF G)))) := ap (@NaturalTransformation.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. Require Import NaturalTransformation.Paths. Require Import Functor.Dual NaturalTransformation.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.Dual. Require Import Adjoint.Functorial.Parts. Require Import HoTT.Tactics. Require Import Basics.Tactics.

Categories\Adjoint\Functorial\Laws.v

right_composition_of

(@right_morphism_of_nondep _ _ _ _ A'' _ _ A (T' o T)) = ((right_morphism_of_nondep A' A T) o (right_morphism_of_nondep A'' A' T')) := ap (@NaturalTransformation.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. Require Import NaturalTransformation.Paths. Require Import Functor.Dual NaturalTransformation.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.Dual. Require Import Adjoint.Functorial.Parts. Require Import HoTT.Tactics. Require Import Basics.Tactics.

Categories\Adjoint\Functorial\Laws.v

right_composition_of_nondep

NaturalTransformation (F' o CF) (DF o F). Proof. refine ((_) o (counit A' oR (DF o F)) o _ o (F' oL ((T oR F) o _ o (CF oL unit A) o _)))%natural_transformation; nt_solve_associator. Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import Functor.Dual NaturalTransformation.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.Dual.

Categories\Adjoint\Functorial\Parts.v

left_morphism_of

NaturalTransformation F' F. Proof. refine (_ o (@left_morphism_of C C 1 D D 1 G F A G' F' A' (_ o T o _)) o _)%natural_transformation; nt_solve_associator. Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import Functor.Dual NaturalTransformation.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.Dual.

Categories\Adjoint\Functorial\Parts.v

left_morphism_of_nondep

C C' CF D D' DF (F : Functor C D) (G : Functor D C) (A : F -| G) (F' : Functor C' D') (G' : Functor D' C') (A' : F' -| G') (T : NaturalTransformation (F' o CF) (DF o F)) : NaturalTransformation (CF o G) (G' o DF) := (@left_morphism_of _ _ DF^op _ _ CF^op F^op G^op A^op F'^op G'^op A'^op T^op)^op.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import Functor.Dual NaturalTransformation.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.Dual.

Categories\Adjoint\Functorial\Parts.v

right_morphism_of

C D (F : Functor C D) (G : Functor D C) (A : F -| G) (F' : Functor C D) (G' : Functor D C) (A' : F' -| G') (T : NaturalTransformation F' F) : NaturalTransformation G G' := (@left_morphism_of_nondep _ _ F^op G^op A^op F'^op G'^op A'^op T^op)^op.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import Functor.Dual NaturalTransformation.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.Dual.

Categories\Adjoint\Functorial\Parts.v

right_morphism_of_nondep

object (Y / G) := @CommaCategory.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import Functor.Dual Category.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Dual. Import Comma.Core. Require Import UniversalProperties Comma.Dual InitialTerminalCategory.Core InitialTerminalCategory.Functors.

Categories\Adjoint\UniversalMorphisms\Core.v

initial_morphism__of__adjunction

IsInitialMorphism initial_morphism__of__adjunction := Build_IsInitialMorphism _ _ _ _ ((A : AdjunctionUnit _ _).

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import Functor.Dual Category.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Dual. Import Comma.Core. Require Import UniversalProperties Comma.Dual InitialTerminalCategory.Core InitialTerminalCategory.Functors.

Categories\Adjoint\UniversalMorphisms\Core.v

is_initial_morphism__of__adjunction

object (F / X) := Eval simpl in dual_functor (! X)^op F^op (initial_morphism__of__adjunction A^op X).

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import Functor.Dual Category.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Dual. Import Comma.Core. Require Import UniversalProperties Comma.Dual InitialTerminalCategory.Core InitialTerminalCategory.Functors.

Categories\Adjoint\UniversalMorphisms\Core.v

terminal_morphism__of__adjunction

IsTerminalMorphism terminal_morphism__of__adjunction := is_initial_morphism__of__adjunction A^op X.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import Functor.Dual Category.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Dual. Import Comma.Core. Require Import UniversalProperties Comma.Dual InitialTerminalCategory.Core InitialTerminalCategory.Functors.

Categories\Adjoint\UniversalMorphisms\Core.v

is_terminal_morphism__of__adjunction

Y : mor_of Y Y 1 = 1. Proof. simpl. erewrite IsInitialMorphism_property_morphism_unique; [ reflexivity | ]. rewrite ?, ?left_identity, ?right_identity. reflexivity. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import Functor.Dual Category.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Dual. Import Comma.Core. Require Import UniversalProperties Comma.Dual InitialTerminalCategory.Core InitialTerminalCategory.Functors.

Categories\Adjoint\UniversalMorphisms\Core.v

identity_of

x y z g f : mor_of _ _ (f o g) = mor_of y z f o mor_of x y g. Proof. simpl. erewrite IsInitialMorphism_property_morphism_unique; [ reflexivity | ]. rewrite ?. repeat try_associativity_quick rewrite IsInitialMorphism_property_morphism_property. reflexivity. Qed.

Lemma

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import Functor.Dual Category.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Dual. Import Comma.Core. Require Import UniversalProperties Comma.Dual InitialTerminalCategory.Core InitialTerminalCategory.Functors.

Categories\Adjoint\UniversalMorphisms\Core.v

composition_of

Functor C D := Build_Functor C D (fun x => obj_of x) (fun s d m => mor_of s d m) composition_of identity_of.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import Functor.Dual Category.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Dual. Import Comma.Core. Require Import UniversalProperties Comma.Dual InitialTerminalCategory.Core InitialTerminalCategory.Functors.

Categories\Adjoint\UniversalMorphisms\Core.v

functor__of__initial_morphism

functor__of__initial_morphism -| G. Proof. refine (adjunction_unit_counit__of__adjunction_unit _). eexists (Build_NaturalTransformation 1 (G o functor__of__initial_morphism) (fun x => IsInitialMorphism_morphism (@HM x)) (fun s d m => symmetry _ _ (IsInitialMorphism_property_morphism_property (@HM s) _ _))). simpl. exact (fun c => @IsInitialMorphism_property _ _ c G (M c) (@HM c)). Defined.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import Functor.Dual Category.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Dual. Import Comma.Core. Require Import UniversalProperties Comma.Dual InitialTerminalCategory.Core InitialTerminalCategory.Functors.

Categories\Adjoint\UniversalMorphisms\Core.v

adjunction__of__initial_morphism

Functor D C := (@functor__of__initial_morphism (D^op) (C^op) (F^op) (fun x : D => dual_functor F !x (M x)) HM)^op.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import Functor.Dual Category.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Dual. Import Comma.Core. Require Import UniversalProperties Comma.Dual InitialTerminalCategory.Core InitialTerminalCategory.Functors.

Categories\Adjoint\UniversalMorphisms\Core.v

functor__of__terminal_morphism

F -| functor__of__terminal_morphism := ((@adjunction__of__initial_morphism (D^op) (C^op) (F^op) (fun x : D => dual_functor F !x (M x)) HM)^op)%adjunction.

Definition

Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Functor.Identity Functor.Composition.Core. Require Import Functor.Dual Category.Dual. Require Import Adjoint.Core Adjoint.UnitCounit Adjoint.UnitCounitCoercions Adjoint.Dual. Import Comma.Core. Require Import UniversalProperties Comma.Dual InitialTerminalCategory.Core InitialTerminalCategory.Functors.

Categories\Adjoint\UniversalMorphisms\Core.v

adjunction__of__terminal_morphism

PreCategory := @Build_PreCategory { C : PreCategory | P C } (fun C D => Functor C.

Definition

Require Import Category.Objects InitialTerminalCategory.Core InitialTerminalCategory.Functors Functor.Core Category.Strict Functor.Paths. Require Import Functor.Identity Functor.Composition.Core Functor.Composition.Laws.

Categories\Cat\Core.v

sub_pre_cat

`{Funext} : PreCategory := sub_pre_cat (fun C => IsStrictCategory C).

Definition

Require Import Category.Objects InitialTerminalCategory.Core InitialTerminalCategory.Functors Functor.Core Category.Strict Functor.Paths. Require Import Functor.Identity Functor.Composition.Core Functor.Composition.Laws.

Categories\Cat\Core.v

strict_cat

`(IsTerminalCategory one) (HT : P one) : IsTerminalObject (sub_pre_cat P HF) (one; HT). Proof. typeclasses eauto. Defined.

Lemma

Require Import Category.Objects InitialTerminalCategory.Core InitialTerminalCategory.Functors Functor.Core Category.Strict Functor.Paths. Require Import Functor.Identity Functor.Composition.Core Functor.Composition.Laws.

Categories\Cat\Core.v

is_terminal_object__is_terminal_category