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Unset Strict Universe Declaration.
Require Import TestSuite.admit.
(* File reduced by coq-bug-finder from original input, then from 12106 lines to 1070 lines *)
Set Universe Polymorphism.
Definition setleq (A : Type@{i}) (B : Type@{j}) := A : Type@{j}.
Inductive foo : Type@{l} := bar : foo .
Section MakeEq.
Variables (a : foo@{i}) (b : foo@{j}).
Let t := ltac:(let ty := type of b in exact ty).
Definition make_eq (x:=b) := a : t.
End MakeEq.
Definition same (x : foo@{i}) (y : foo@{i}) := x.
Section foo.
Variables x : foo@{i}.
Variables y : foo@{j}.
Let AleqB := let foo := make_eq x y in (Type * Type)%type.
Definition baz := same x y.
End foo.
Definition baz' := Eval unfold baz in baz@{i j k l}.
Module Export HoTT_DOT_Overture.
Module Export HoTT.
Module Export Overture.
Definition relation (A : Type) := A -> A -> Type.
Class Symmetric {A} (R : relation A) :=
symmetry : forall x y, R x y -> R y x.
Definition compose {A B C : Type} (g : B -> C) (f : A -> B) :=
fun x => g (f x).
Notation "g 'o' f" := (compose g f) (at level 40, left associativity) : function_scope.
Open Scope function_scope.
Set Printing Universes. Set Printing All.
Inductive paths {A : Type} (a : A) : A -> Type :=
idpath : paths a a.
Arguments idpath {A a} , [A] a.
Notation "x = y :> A" := (@paths A x y) : type_scope.
Notation "x = y" := (x = y :>_) : type_scope.
Delimit Scope path_scope with path.
Local Open Scope path_scope.
Definition concat {A : Type} {x y z : A} (p : x = y) (q : y = z) : x = z :=
match p, q with idpath, idpath => idpath end.
Definition inverse {A : Type} {x y : A} (p : x = y) : y = x
:= match p with idpath => idpath end.
Instance symmetric_paths {A} : Symmetric (@paths A) | 0 := @inverse A.
Notation "1" := idpath : path_scope.
Notation "p @ q" := (concat p q) (at level 20) : path_scope.
Notation "p ^" := (inverse p) (at level 3) : path_scope.
Definition transport {A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) : P y :=
match p with idpath => u end.
Definition ap {A B:Type} (f:A -> B) {x y:A} (p:x = y) : f x = f y
:= match p with idpath => idpath end.
Definition pointwise_paths {A} {P:A->Type} (f g:forall x:A, P x) : Type
:= forall x:A, f x = g x.
Hint Unfold pointwise_paths : typeclass_instances.
Notation "f == g" := (pointwise_paths f g) (at level 70, no associativity) : type_scope.
Definition apD10 {A} {B:A->Type} {f g : forall x, B x} (h:f=g)
: f == g
:= fun x => match h with idpath => 1 end.
Definition Sect {A B : Type} (s : A -> B) (r : B -> A) :=
forall x : A, r (s x) = x.
Class IsEquiv {A B : Type} (f : A -> B) := BuildIsEquiv {
equiv_inv : B -> A ;
eisretr : Sect equiv_inv f;
eissect : Sect f equiv_inv;
eisadj : forall x : A, eisretr (f x) = ap f (eissect x)
}.
Delimit Scope equiv_scope with equiv.
Local Open Scope equiv_scope.
Notation "f ^-1" := (@equiv_inv _ _ f _) (at level 3) : equiv_scope.
Class Contr_internal (A : Type) := BuildContr {
center : A ;
contr : (forall y : A, center = y)
}.
Inductive trunc_index : Type :=
| minus_two : trunc_index
| trunc_S : trunc_index -> trunc_index.
Fixpoint nat_to_trunc_index (n : nat) : trunc_index
:= match n with
| 0 => trunc_S (trunc_S minus_two)
| S n' => trunc_S (nat_to_trunc_index n')
end.
Coercion nat_to_trunc_index : nat >-> trunc_index.
Fixpoint IsTrunc_internal (n : trunc_index) (A : Type) : Type :=
match n with
| minus_two => Contr_internal A
| trunc_S n' => forall (x y : A), IsTrunc_internal n' (x = y)
end.
Class IsTrunc (n : trunc_index) (A : Type) : Type :=
Trunc_is_trunc : IsTrunc_internal n A.
Notation IsHSet := (IsTrunc 0).
Class Funext :=
{ isequiv_apD10 :> forall (A : Type) (P : A -> Type) f g, IsEquiv (@apD10 A P f g) }.
Definition path_forall `{Funext} {A : Type} {P : A -> Type} (f g : forall x : A, P x) :
f == g -> f = g
:=
(@apD10 A P f g)^-1.
End Overture.
End HoTT.
End HoTT_DOT_Overture.
Module Export HoTT_DOT_categories_DOT_Category_DOT_Core.
Module Export HoTT.
Module Export categories.
Module Export Category.
Module Export Core.
Set Universe Polymorphism.
Set Implicit Arguments.
Delimit Scope morphism_scope with morphism.
Delimit Scope category_scope with category.
Delimit Scope object_scope with object.
Set Printing Universes.
Set Printing All.
Record PreCategory :=
Build_PreCategory' {
object :> Type;
morphism : object -> object -> Type;
identity : forall x, morphism x x;
compose : forall s d d',
morphism d d'
-> morphism s d
-> morphism s d'
where "f 'o' g" := (compose f g);
associativity : forall x1 x2 x3 x4
(m1 : morphism x1 x2)
(m2 : morphism x2 x3)
(m3 : morphism x3 x4),
(m3 o m2) o m1 = m3 o (m2 o m1);
associativity_sym : forall x1 x2 x3 x4
(m1 : morphism x1 x2)
(m2 : morphism x2 x3)
(m3 : morphism x3 x4),
m3 o (m2 o m1) = (m3 o m2) o m1;
left_identity : forall a b (f : morphism a b), identity b o f = f;
right_identity : forall a b (f : morphism a b), f o identity a = f;
identity_identity : forall x, identity x o identity x = identity x;
trunc_morphism : forall s d, IsHSet (morphism s d)
}.
Bind Scope category_scope with PreCategory.
Arguments identity [!C%category] x%object : rename.
Arguments compose [!C%category s%object d%object d'%object] m1%morphism m2%morphism : rename.
Definition Build_PreCategory
object morphism compose identity
associativity left_identity right_identity
:= @Build_PreCategory'
object
morphism
compose
identity
associativity
(fun _ _ _ _ _ _ _ => symmetry _ _ (associativity _ _ _ _ _ _ _))
left_identity
right_identity
(fun _ => left_identity _ _ _).
Existing Instance trunc_morphism.
Hint Resolve @left_identity @right_identity @associativity : category morphism.
Module Export CategoryCoreNotations.
Infix "o" := compose : morphism_scope.
End CategoryCoreNotations.
End Core.
End Category.
End categories.
End HoTT.
End HoTT_DOT_categories_DOT_Category_DOT_Core.
Module Export HoTT_DOT_types_DOT_Forall.
Module Export HoTT.
Module Export types.
Module Export Forall.
Generalizable Variables A B f g e n.
Section AssumeFunext.
Global Instance trunc_forall `{P : A -> Type} `{forall a, IsTrunc n (P a)}
: IsTrunc n (forall a, P a) | 100.
admit.
Defined.
End AssumeFunext.
End Forall.
End types.
End HoTT.
End HoTT_DOT_types_DOT_Forall.
Module Export HoTT_DOT_types_DOT_Prod.
Module Export HoTT.
Module Export types.
Module Export Prod.
Local Open Scope path_scope.
Definition path_prod_uncurried {A B : Type} (z z' : A * B)
(pq : (fst z = fst z') * (snd z = snd z'))
: (z = z')
:= match pq with (p,q) =>
match z, z' return
(fst z = fst z') -> (snd z = snd z') -> (z = z') with
| (a,b), (a',b') => fun p q =>
match p, q with
idpath, idpath => 1
end
end p q
end.
Definition path_prod {A B : Type} (z z' : A * B) :
(fst z = fst z') -> (snd z = snd z') -> (z = z')
:= fun p q => path_prod_uncurried z z' (p,q).
Definition path_prod' {A B : Type} {x x' : A} {y y' : B}
: (x = x') -> (y = y') -> ((x,y) = (x',y'))
:= fun p q => path_prod (x,y) (x',y') p q.
End Prod.
End types.
End HoTT.
End HoTT_DOT_types_DOT_Prod.
Module Export HoTT_DOT_categories_DOT_Functor_DOT_Core.
Module Export HoTT.
Module Export categories.
Module Export Functor.
Module Export Core.
Set Universe Polymorphism.
Set Implicit Arguments.
Delimit Scope functor_scope with functor.
Local Open Scope morphism_scope.
Section Functor.
Variable C : PreCategory.
Variable D : PreCategory.
Record Functor :=
{
object_of :> C -> D;
morphism_of : forall s d, morphism C s d
-> morphism D (object_of s) (object_of d);
composition_of : forall s d d'
(m1 : morphism C s d) (m2: morphism C d d'),
morphism_of _ _ (m2 o m1)
= (morphism_of _ _ m2) o (morphism_of _ _ m1);
identity_of : forall x, morphism_of _ _ (identity x)
= identity (object_of x)
}.
End Functor.
Bind Scope functor_scope with Functor.
Arguments morphism_of [C%category] [D%category] F%functor [s%object d%object] m%morphism : rename, simpl nomatch.
Module Export FunctorCoreNotations.
Notation "F '_1' m" := (morphism_of F m) (at level 10, no associativity) : morphism_scope.
End FunctorCoreNotations.
End Core.
End Functor.
End categories.
End HoTT.
End HoTT_DOT_categories_DOT_Functor_DOT_Core.
Module Export HoTT_DOT_categories_DOT_Category_DOT_Morphisms.
Module Export HoTT.
Module Export categories.
Module Export Category.
Module Export Morphisms.
Set Universe Polymorphism.
Local Open Scope morphism_scope.
Class IsIsomorphism {C : PreCategory} {s d} (m : morphism C s d) :=
{
morphism_inverse : morphism C d s;
left_inverse : morphism_inverse o m = identity _;
right_inverse : m o morphism_inverse = identity _
}.
Class Isomorphic {C : PreCategory} s d :=
{
morphism_isomorphic :> morphism C s d;
isisomorphism_isomorphic :> IsIsomorphism morphism_isomorphic
}.
Module Export CategoryMorphismsNotations.
Infix "<~=~>" := Isomorphic (at level 70, no associativity) : category_scope.
End CategoryMorphismsNotations.
End Morphisms.
End Category.
End categories.
End HoTT.
End HoTT_DOT_categories_DOT_Category_DOT_Morphisms.
Module Export HoTT_DOT_categories_DOT_Category_DOT_Dual.
Module Export HoTT.
Module Export categories.
Module Export Category.
Module Export Dual.
Set Universe Polymorphism.
Local Open Scope morphism_scope.
Section opposite.
Definition opposite (C : PreCategory) : PreCategory
:= @Build_PreCategory'
C
(fun s d => morphism C d s)
(identity (C := C))
(fun _ _ _ m1 m2 => m2 o m1)
(fun _ _ _ _ _ _ _ => @associativity_sym _ _ _ _ _ _ _ _)
(fun _ _ _ _ _ _ _ => @associativity _ _ _ _ _ _ _ _)
(fun _ _ => @right_identity _ _ _)
(fun _ _ => @left_identity _ _ _)
(@identity_identity C)
_.
End opposite.
Module Export CategoryDualNotations.
Notation "C ^op" := (opposite C) (at level 3) : category_scope.
End CategoryDualNotations.
End Dual.
End Category.
End categories.
End HoTT.
End HoTT_DOT_categories_DOT_Category_DOT_Dual.
Module Export HoTT_DOT_categories_DOT_Functor_DOT_Composition_DOT_Core.
Module Export HoTT.
Module Export categories.
Module Export Functor.
Module Export Composition.
Module Export Core.
Set Universe Polymorphism.
Set Implicit Arguments.
Local Open Scope morphism_scope.
Section composition.
Variable C : PreCategory.
Variable D : PreCategory.
Variable E : PreCategory.
Variable G : Functor D E.
Variable F : Functor C D.
Local Notation c_object_of c := (G (F c)) (only parsing).
Local Notation c_morphism_of m := (morphism_of G (morphism_of F m)) (only parsing).
Let compose_composition_of' s d d'
(m1 : morphism C s d) (m2 : morphism C d d')
: c_morphism_of (m2 o m1) = c_morphism_of m2 o c_morphism_of m1.
admit.
Defined.
Definition compose_composition_of s d d' m1 m2
:= Eval cbv beta iota zeta delta
[compose_composition_of'] in
@compose_composition_of' s d d' m1 m2.
Let compose_identity_of' x
: c_morphism_of (identity x) = identity (c_object_of x).
admit.
Defined.
Definition compose_identity_of x
:= Eval cbv beta iota zeta delta
[compose_identity_of'] in
@compose_identity_of' x.
Definition compose : Functor C E
:= Build_Functor
C E
(fun c => G (F c))
(fun _ _ m => morphism_of G (morphism_of F m))
compose_composition_of
compose_identity_of.
End composition.
Module Export FunctorCompositionCoreNotations.
Infix "o" := compose : functor_scope.
End FunctorCompositionCoreNotations.
End Core.
End Composition.
End Functor.
End categories.
End HoTT.
End HoTT_DOT_categories_DOT_Functor_DOT_Composition_DOT_Core.
Module Export HoTT_DOT_categories_DOT_Functor_DOT_Dual.
Module Export HoTT.
Module Export categories.
Module Export Functor.
Module Export Dual.
Set Universe Polymorphism.
Set Implicit Arguments.
Section opposite.
Variable C : PreCategory.
Variable D : PreCategory.
Definition opposite (F : Functor C D) : Functor C^op D^op
:= Build_Functor (C^op) (D^op)
(object_of F)
(fun s d => morphism_of F (s := d) (d := s))
(fun d' d s m1 m2 => composition_of F s d d' m2 m1)
(identity_of F).
End opposite.
Module Export FunctorDualNotations.
Notation "F ^op" := (opposite F) : functor_scope.
End FunctorDualNotations.
End Dual.
End Functor.
End categories.
End HoTT.
End HoTT_DOT_categories_DOT_Functor_DOT_Dual.
Module Export HoTT_DOT_categories_DOT_Functor_DOT_Identity.
Module Export HoTT.
Module Export categories.
Module Export Functor.
Module Export Identity.
Set Universe Polymorphism.
Section identity.
Definition identity C : Functor C C
:= Build_Functor C C
(fun x => x)
(fun _ _ x => x)
(fun _ _ _ _ _ => idpath)
(fun _ => idpath).
End identity.
Module Export FunctorIdentityNotations.
Notation "1" := (identity _) : functor_scope.
End FunctorIdentityNotations.
End Identity.
End Functor.
End categories.
End HoTT.
End HoTT_DOT_categories_DOT_Functor_DOT_Identity.
Module Export HoTT_DOT_categories_DOT_NaturalTransformation_DOT_Core.
Module Export HoTT.
Module Export categories.
Module Export NaturalTransformation.
Module Export Core.
Set Universe Polymorphism.
Set Implicit Arguments.
Local Open Scope morphism_scope.
Section NaturalTransformation.
Variable C : PreCategory.
Variable D : PreCategory.
Variables F G : Functor C D.
Record NaturalTransformation :=
Build_NaturalTransformation' {
components_of :> forall c, morphism D (F c) (G c);
commutes : forall s d (m : morphism C s d),
components_of d o F _1 m = G _1 m o components_of s;
commutes_sym : forall s d (m : C.(morphism) s d),
G _1 m o components_of s = components_of d o F _1 m
}.
End NaturalTransformation.
End Core.
End NaturalTransformation.
End categories.
End HoTT.
End HoTT_DOT_categories_DOT_NaturalTransformation_DOT_Core.
Module Export HoTT_DOT_categories_DOT_NaturalTransformation_DOT_Dual.
Module Export HoTT.
Module Export categories.
Module Export NaturalTransformation.
Module Export Dual.
Set Universe Polymorphism.
Section opposite.
Variable C : PreCategory.
Variable D : PreCategory.
Definition opposite
(F G : Functor C D)
(T : NaturalTransformation F G)
: NaturalTransformation G^op F^op
:= Build_NaturalTransformation' (G^op) (F^op)
(components_of T)
(fun s d => commutes_sym T d s)
(fun s d => commutes T d s).
End opposite.
End Dual.
End NaturalTransformation.
End categories.
End HoTT.
End HoTT_DOT_categories_DOT_NaturalTransformation_DOT_Dual.
Module Export HoTT_DOT_categories_DOT_Category_DOT_Strict.
Module Export HoTT.
Module Export categories.
Module Export Category.
Module Export Strict.
Export Category.Core.
Set Universe Polymorphism.
End Strict.
End Category.
End categories.
End HoTT.
End HoTT_DOT_categories_DOT_Category_DOT_Strict.
Module Export HoTT.
Module Export categories.
Module Export Category.
Module Export Prod.
Set Universe Polymorphism.
Local Open Scope morphism_scope.
Section prod.
Variable C : PreCategory.
Variable D : PreCategory.
Definition prod : PreCategory.
refine (@Build_PreCategory
(C * D)%type
(fun s d => (morphism C (fst s) (fst d)
* morphism D (snd s) (snd d))%type)
(fun x => (identity (fst x), identity (snd x)))
(fun s d d' m2 m1 => (fst m2 o fst m1, snd m2 o snd m1))
_
_
_
_); admit.
Defined.
End prod.
Module Export CategoryProdNotations.
Infix "*" := prod : category_scope.
End CategoryProdNotations.
End Prod.
End Category.
End categories.
End HoTT.
Module Functor.
Module Export Prod.
Set Universe Polymorphism.
Set Implicit Arguments.
Local Open Scope morphism_scope.
Section proj.
Context {C : PreCategory}.
Context {D : PreCategory}.
Definition fst : Functor (C * D) C
:= Build_Functor (C * D) C
(@fst _ _)
(fun _ _ => @fst _ _)
(fun _ _ _ _ _ => idpath)
(fun _ => idpath).
Definition snd : Functor (C * D) D
:= Build_Functor (C * D) D
(@snd _ _)
(fun _ _ => @snd _ _)
(fun _ _ _ _ _ => idpath)
(fun _ => idpath).
End proj.
Section prod.
Variable C : PreCategory.
Variable D : PreCategory.
Variable D' : PreCategory.
Definition prod (F : Functor C D) (F' : Functor C D')
: Functor C (D * D')
:= Build_Functor
C (D * D')
(fun c => (F c, F' c))
(fun s d m => (F _1 m, F' _1 m))
(fun _ _ _ _ _ => path_prod' (composition_of F _ _ _ _ _)
(composition_of F' _ _ _ _ _))
(fun _ => path_prod' (identity_of F _) (identity_of F' _)).
End prod.
Local Infix "*" := prod : functor_scope.
Section pair.
Variable C : PreCategory.
Variable D : PreCategory.
Variable C' : PreCategory.
Variable D' : PreCategory.
Variable F : Functor C D.
Variable F' : Functor C' D'.
Definition pair : Functor (C * C') (D * D')
:= (F o fst) * (F' o snd).
End pair.
Module Export FunctorProdNotations.
Notation "( x , y , .. , z )" := (pair .. (pair x y) .. z) : functor_scope.
End FunctorProdNotations.
End Prod.
End Functor.
Module Export HoTT_DOT_categories_DOT_NaturalTransformation_DOT_Composition_DOT_Core.
Module Export HoTT.
Module categories.
Module Export NaturalTransformation.
Module Export Composition.
Module Export Core.
Set Universe Polymorphism.
Set Implicit Arguments.
Local Open Scope path_scope.
Local Open Scope morphism_scope.
Section composition.
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 compose_commutes s d (m : morphism C s d)
: CO d o morphism_of F m = morphism_of F'' m o CO s
:= (associativity _ _ _ _ _ _ _ _)
@ ap (fun x => _ o x) (commutes T _ _ m)
@ (associativity_sym _ _ _ _ _ _ _ _)
@ ap (fun x => x o _) (commutes T' _ _ m)
@ (associativity _ _ _ _ _ _ _ _).
Definition compose_commutes_sym s d (m : morphism C s d)
: morphism_of F'' m o CO s = CO d o morphism_of F m
:= (associativity_sym _ _ _ _ _ _ _ _)
@ ap (fun x => x o _) (commutes_sym T' _ _ m)
@ (associativity _ _ _ _ _ _ _ _)
@ ap (fun x => _ o x) (commutes_sym T _ _ m)
@ (associativity_sym _ _ _ _ _ _ _ _).
Definition compose
: NaturalTransformation F F''
:= Build_NaturalTransformation' F F''
(fun c => CO c)
compose_commutes
compose_commutes_sym.
End compose.
End composition.
Module Export NaturalTransformationCompositionCoreNotations.
Infix "o" := compose : natural_transformation_scope.
End NaturalTransformationCompositionCoreNotations.
End Core.
End Composition.
End NaturalTransformation.
End categories.
Set Universe Polymorphism.
Section path_natural_transformation.
Context `{Funext}.
Variable C : PreCategory.
Variable D : PreCategory.
Variables F G : Functor C D.
Global Instance trunc_natural_transformation
: IsHSet (NaturalTransformation F G).
admit.
Defined.
Section path.
Variables T U : NaturalTransformation F G.
Lemma path'_natural_transformation
: components_of T = components_of U
-> T = U.
admit.
Defined.
Lemma path_natural_transformation
: components_of T == components_of U
-> T = U.
Proof.
intros.
apply path'_natural_transformation.
apply path_forall; assumption.
Qed.
End path.
End path_natural_transformation.
Ltac path_natural_transformation :=
repeat match goal with
| _ => intro
| _ => apply path_natural_transformation; simpl
end.
Module Export Identity.
Set Universe Polymorphism.
Set Implicit Arguments.
Local Open Scope morphism_scope.
Local Open Scope path_scope.
Section identity.
Variable C : PreCategory.
Variable D : PreCategory.
Section generalized.
Variables F G : Functor C D.
Hypothesis HO : object_of F = object_of G.
Hypothesis HM : transport (fun GO => forall s d,
morphism C s d
-> morphism D (GO s) (GO d))
HO
(morphism_of F)
= morphism_of G.
Local Notation CO c := (transport (fun GO => morphism D (F c) (GO c))
HO
(identity (F c))).
Definition generalized_identity_commutes s d (m : morphism C s d)
: CO d o morphism_of F m = morphism_of G m o CO s.
Proof.
case HM.
case HO.
exact (left_identity _ _ _ _ @ (right_identity _ _ _ _)^).
Defined.
Definition generalized_identity_commutes_sym s d (m : morphism C s d)
: morphism_of G m o CO s = CO d o morphism_of F m.
admit.
Defined.
Definition generalized_identity
: NaturalTransformation F G
:= Build_NaturalTransformation'
F G
(fun c => CO c)
generalized_identity_commutes
generalized_identity_commutes_sym.
End generalized.
Definition identity (F : Functor C D)
: NaturalTransformation F F
:= Eval simpl in @generalized_identity F F 1 1.
End identity.
Module Export NaturalTransformationIdentityNotations.
Notation "1" := (identity _) : natural_transformation_scope.
End NaturalTransformationIdentityNotations.
End Identity.
Module Export Laws.
Import HoTT_DOT_categories_DOT_NaturalTransformation_DOT_Composition_DOT_Core.HoTT.categories.
Set Universe Polymorphism.
Local Open Scope natural_transformation_scope.
Section natural_transformation_identity.
Context `{fs : Funext}.
Variable C : PreCategory.
Variable D : PreCategory.
Lemma left_identity (F F' : Functor C D)
(T : NaturalTransformation F F')
: 1 o T = T.
Proof.
path_natural_transformation; auto with morphism.
Qed.
Lemma right_identity (F F' : Functor C D)
(T : NaturalTransformation F F')
: T o 1 = T.
Proof.
path_natural_transformation; auto with morphism.
Qed.
End natural_transformation_identity.
Section associativity.
Section nt.
Context `{fs : Funext}.
Definition associativity
C D F G H I
(V : @NaturalTransformation C D F G)
(U : @NaturalTransformation C D G H)
(T : @NaturalTransformation C D H I)
: (T o U) o V = T o (U o V).
Proof.
path_natural_transformation.
apply associativity.
Qed.
End nt.
End associativity.
End Laws.
Module Export FunctorCategory.
Module Export Core.
Import HoTT_DOT_categories_DOT_NaturalTransformation_DOT_Composition_DOT_Core.HoTT.categories.
Set Universe Polymorphism.
Section functor_category.
Context `{Funext}.
Variable C : PreCategory.
Variable D : PreCategory.
Definition functor_category : PreCategory
:= @Build_PreCategory (Functor C D)
(@NaturalTransformation C D)
(@identity C D)
(@compose C D)
(@associativity _ C D)
(@left_identity _ C D)
(@right_identity _ C D)
_.
End functor_category.
Module Export FunctorCategoryCoreNotations.
Notation "C -> D" := (functor_category C D) : category_scope.
End FunctorCategoryCoreNotations.
End Core.
End FunctorCategory.
Module Export Morphisms.
Set Universe Polymorphism.
Set Implicit Arguments.
Definition NaturalIsomorphism `{Funext} (C D : PreCategory) F G :=
@Isomorphic (C -> D) F G.
Module Export FunctorCategoryMorphismsNotations.
Infix "<~=~>" := NaturalIsomorphism : natural_transformation_scope.
End FunctorCategoryMorphismsNotations.
End Morphisms.
Module Export HSet.
Record hSet := BuildhSet {setT:> Type; iss :> IsHSet setT}.
Global Existing Instance iss.
End HSet.
Module Export Core.
Set Universe Polymorphism.
Notation cat_of obj :=
(@Build_PreCategory obj
(fun x y => x -> y)
(fun _ x => x)
(fun _ _ _ f g => f o g)%core
(fun _ _ _ _ _ _ _ => idpath)
(fun _ _ _ => idpath)
(fun _ _ _ => idpath)
_).
Definition set_cat `{Funext} : PreCategory := cat_of hSet.
Set Universe Polymorphism.
Local Open Scope morphism_scope.
Section hom_functor.
Context `{Funext}.
Variable C : PreCategory.
Local Notation obj_of c'c :=
(BuildhSet
(morphism
C
(fst (c'c : object (C^op * C)))
(snd (c'c : object (C^op * C))))
_).
Let hom_functor_morphism_of s's d'd (hf : morphism (C^op * C) s's d'd)
: morphism set_cat (obj_of s's) (obj_of d'd)
:= fun g => snd hf o g o fst hf.
Definition hom_functor : Functor (C^op * C) set_cat.
refine (Build_Functor (C^op * C) set_cat
(fun c'c => obj_of c'c)
hom_functor_morphism_of
_
_);
subst hom_functor_morphism_of;
simpl; admit.
Defined.
End hom_functor.
Set Universe Polymorphism.
Import Category.Dual Functor.Dual.
Import Category.Prod Functor.Prod.
Import Functor.Composition.Core.
Import Functor.Identity.
Set Universe Polymorphism.
Local Open Scope functor_scope.
Local Open Scope natural_transformation_scope.
Section Adjunction.
Context `{Funext}.
Variable C : PreCategory.
Variable D : PreCategory.
Variable F : Functor C D.
Variable G : Functor D C.
Let Adjunction_Type :=
Eval simpl in (hom_functor D) o (F^op, 1) <~=~> (hom_functor C) o (1, G).
Record AdjunctionHom :=
{
mate_of :
@NaturalIsomorphism H
(Prod.prod (Category.Dual.opposite C) D)
(@set_cat H)
(@compose (Prod.prod (Category.Dual.opposite C) D)
(Prod.prod (Category.Dual.opposite D) D)
(@set_cat H) (@hom_functor H D)
(@pair (Category.Dual.opposite C)
(Category.Dual.opposite D) D D
(@opposite C D F) (identity D)))
(@compose (Prod.prod (Category.Dual.opposite C) D)
(Prod.prod (Category.Dual.opposite C) C)
(@set_cat H) (@hom_functor H C)
(@pair (Category.Dual.opposite C)
(Category.Dual.opposite C) D C
(identity (Category.Dual.opposite C)) G))
}.
End Adjunction.
(* Error: Illegal application:
The term "NaturalIsomorphism" of type
"forall (H : Funext) (C D : PreCategory),
(C -> D)%category -> (C -> D)%category -> Type"
cannot be applied to the terms
"H" : "Funext"
"(C ^op * D)%category" : "PreCategory"
"set_cat" : "PreCategory"
"hom_functor D o (F ^op, 1)" : "Functor (C ^op * D) set_cat"
"hom_functor C o (1, G)" : "Functor (C ^op * D) set_cat"
The 5th term has type "Functor (C ^op * D) set_cat"
which should be coercible to "object (C ^op * D -> set_cat)".
*)
End Core.
End HoTT.
End HoTT_DOT_categories_DOT_NaturalTransformation_DOT_Composition_DOT_Core.
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