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Require Import TestSuite.admit.
Generalizable All Variables.
Set Implicit Arguments.
Set Universe Polymorphism.
Axiom admit : forall {T}, T.
Reserved Infix "o" (at level 40, left associativity).
Class IsEquiv {A B : Type} (f : A -> B) := { equiv_inv : B -> A }.
Record Equiv A B := { equiv_fun :> A -> B ; equiv_isequiv :> IsEquiv equiv_fun }.
#[export] Existing Instance equiv_isequiv.
Delimit Scope equiv_scope with equiv.
Local Open Scope equiv_scope.
Notation "A <~> B" := (Equiv A B) (at level 85) : equiv_scope.
Notation "f ^-1" := (@equiv_inv _ _ f _) (at level 3) : equiv_scope.
Axiom IsHSet : Type -> Type.
Existing Class IsHSet.
Definition trunc_equiv' `(f : A <~> B) `{IsHSet A} : IsHSet B := admit.
Delimit Scope morphism_scope with morphism.
Delimit Scope category_scope with category.
Delimit Scope object_scope with object.
Record PreCategory :=
{ object :> Type;
morphism : object -> object -> Type;
compose : forall s d d',
morphism d d'
-> morphism s d
-> morphism s d'
where "f 'o' g" := (compose f g);
trunc_morphism : forall s d, IsHSet (morphism s d) }.
Bind Scope category_scope with PreCategory.
Infix "o" := (@compose _ _ _ _) : morphism_scope.
Delimit Scope functor_scope with functor.
Record Functor (C D : PreCategory) :=
{
object_of :> C -> D;
morphism_of : forall s d, morphism C s d
-> morphism D (object_of s) (object_of d)
}.
Bind Scope functor_scope with Functor.
Arguments morphism_of [C%_category] [D%_category] F%_functor [s%_object d%_object] m%_morphism : rename, simpl nomatch.
Notation "F '_1' m" := (morphism_of F m) (at level 10, no associativity) : morphism_scope.
Local Open Scope morphism_scope.
Class IsIsomorphism {C : PreCategory} {s d} (m : morphism C s d) := { morphism_inverse : morphism C d s }.
Local Notation "m ^-1" := (morphism_inverse (m := m)) : morphism_scope.
Class Isomorphic {C : PreCategory} s d :=
{
morphism_isomorphic :: morphism C s d;
isisomorphism_isomorphic :: IsIsomorphism morphism_isomorphic
}.
Coercion morphism_isomorphic : Isomorphic >-> morphism.
Local Infix "<~=~>" := Isomorphic (at level 70, no associativity) : category_scope.
Definition isisomorphism_inverse `(@IsIsomorphism C x y m) : IsIsomorphism m^-1 := {| morphism_inverse := m |}.
Global Instance isisomorphism_compose `(@IsIsomorphism C y z m0) `(@IsIsomorphism C x y m1)
: IsIsomorphism (m0 o m1).
admit.
Defined.
Section composition.
Variable C : PreCategory.
Variable D : PreCategory.
Variable E : PreCategory.
Variable G : Functor D E.
Variable F : Functor C D.
Definition composeF : Functor C E
:= Build_Functor
C E
(fun c => G (F c))
(fun _ _ m => morphism_of G (morphism_of F m)).
End composition.
Infix "o" := composeF : functor_scope.
Delimit Scope natural_transformation_scope with natural_transformation.
Record NaturalTransformation C D (F G : Functor C D) := { components_of :> forall c, morphism D (F c) (G c) }.
Section compose.
Variable C : PreCategory.
Variable D : PreCategory.
Variables F F' F'' : Functor C D.
Variable T' : NaturalTransformation F' F''.
Variable T : NaturalTransformation F F'.
Local Notation CO c := (T' c o T c).
Definition composeT
: NaturalTransformation F F'' := Build_NaturalTransformation F F'' (fun c => CO c).
End compose.
Section whisker.
Variable C : PreCategory.
Variable D : PreCategory.
Variable E : PreCategory.
Section L.
Variable F : Functor D E.
Variables G G' : Functor C D.
Variable T : NaturalTransformation G G'.
Local Notation CO c := (morphism_of F (T c)).
Definition whisker_l
:= Build_NaturalTransformation
(F o G) (F o G')
(fun c => CO c).
End L.
Section R.
Variables F F' : Functor D E.
Variable T : NaturalTransformation F F'.
Variable G : Functor C D.
Local Notation CO c := (T (G c)).
Definition whisker_r
:= Build_NaturalTransformation
(F o G) (F' o G)
(fun c => CO c).
End R.
End whisker.
Infix "o" := composeT : natural_transformation_scope.
Infix "oL" := whisker_l (at level 40, left associativity) : natural_transformation_scope.
Infix "oR" := whisker_r (at level 40, left associativity) : natural_transformation_scope.
Section path_natural_transformation.
Variable C : PreCategory.
Variable D : PreCategory.
Variables F G : Functor C D.
Lemma equiv_sig_natural_transformation
: { CO : forall x, morphism D (F x) (G x)
| forall s d (m : morphism C s d),
CO d o F _1 m = G _1 m o CO s }
<~> NaturalTransformation F G.
admit.
Defined.
Global Instance trunc_natural_transformation
: IsHSet (NaturalTransformation F G).
Proof.
eapply trunc_equiv'; [ exact equiv_sig_natural_transformation | ].
admit.
Qed.
End path_natural_transformation.
Definition functor_category (C D : PreCategory) : PreCategory
:= @Build_PreCategory (Functor C D) (@NaturalTransformation C D) (@composeT C D) _.
Notation "C -> D" := (functor_category C D) : category_scope.
Definition NaturalIsomorphism (C D : PreCategory) F G := @Isomorphic (C -> D) F G.
Coercion natural_transformation_of_natural_isomorphism C D F G (T : @NaturalIsomorphism C D F G) : NaturalTransformation F G
:= T : morphism _ _ _.
Local Infix "<~=~>" := NaturalIsomorphism : natural_transformation_scope.
Global Instance isisomorphism_compose'
`(T' : @NaturalTransformation C D F' F'')
`(T : @NaturalTransformation C D F F')
`{@IsIsomorphism (C -> D) F' F'' T'}
`{@IsIsomorphism (C -> D) F F' T}
: @IsIsomorphism (C -> D) F F'' (T' o T)%natural_transformation
:= @isisomorphism_compose (C -> D) _ _ T' _ _ T _.
Arguments isisomorphism_compose' {C D F' F''} T' {F} T {H H0}.
Section lemmas.
Local Open Scope natural_transformation_scope.
Variable C : PreCategory.
Variable F : C -> PreCategory.
Context
{w x y z}
{f : Functor (F w) (F z)} {f0 : Functor (F w) (F y)}
{f1 : Functor (F x) (F y)} {f2 : Functor (F y) (F z)}
{f3 : Functor (F w) (F x)} {f4 : Functor (F x) (F z)}
{f5 : Functor (F w) (F z)} {n : f5 <~=~> (f4 o f3)%functor}
{n0 : f4 <~=~> (f2 o f1)%functor} {n1 : f0 <~=~> (f1 o f3)%functor}
{n2 : f <~=~> (f2 o f0)%functor}.
Lemma p_composition_of_coherent_iso_for_rewrite__isisomorphism_helper'
: @IsIsomorphism
(_ -> _) _ _
(n2 ^-1 o (f2 oL n1 ^-1 o (admit o (n0 oR f3 o n))))%natural_transformation.
Proof.
eapply isisomorphism_compose';
[ eapply isisomorphism_inverse
| eapply isisomorphism_compose';
[ admit
| eapply isisomorphism_compose';
[ admit |
eapply isisomorphism_compose'; [ admit | ]]]].
Set Printing All. Set Printing Universes.
apply @isisomorphism_isomorphic.
Qed.
End lemmas.
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