Quelle HoTT_coq_101.v Sprache: Coq

Require Import TestSuite.admit.
Set Universe Polymorphism.
Set Implicit Arguments.
Generalizable All Variables.

Record SpecializedCategory (obj : Type) :=
  {
    Object :> _ := obj;
    Morphism : obj -> obj -> Type
  }.

Record > Category :=
  {
    CObject : Type;

    UnderlyingCategory :> @SpecializedCategory CObject
  }.

Record SpecializedFunctor `(C : @SpecializedCategory objC) `(D : @SpecializedCategory objD) :=
  {
    ObjectOf :> objC -> objD;
    MorphismOf : forall s d, C.(Morphism) s d -> D.(Morphism) (ObjectOf s) (ObjectOf d)
  }.

(* Replacing this with [Definition Functor (C D : Category) :=
SpecializedFunctor C D.] gets rid of the universe inconsistency. *)
Section Functor.
  Variable C D : Category.

  Definition Functor := SpecializedFunctor C D.
End Functor.

Record SpecializedNaturalTransformation `(C : @SpecializedCategory objC) `(D : @SpecializedCategory objD) (F G : SpecializedFunctor C D) :=
  {
    ComponentsOf :> forall c, D.(Morphism) (F c) (G c)
  }.

Definition FunctorProduct' `(F : Functor C D) : SpecializedFunctor C D.
admit.
Defined.

Definition TypeCat : @SpecializedCategory Type.
  admit.
Defined.

Definition CovariantHomFunctor `(C : @SpecializedCategory objC) : SpecializedFunctor C TypeCat.
  refine (Build_SpecializedFunctor C TypeCat
                                   (fun X : C => C.(Morphism) X X)
                                   _
         ); admit.
Defined.

Definition FunctorCategory `(C : @SpecializedCategory objC) `(D : @SpecializedCategory objD) : @SpecializedCategory (SpecializedFunctor C D).
  refine (@Build_SpecializedCategory _
                                     (SpecializedNaturalTransformation (C := C) (D := D))).
Defined.

Definition Yoneda `(C : @SpecializedCategory objC) : SpecializedFunctor C (FunctorCategory C TypeCat).
  match goal with
    | [ |- SpecializedFunctor ?C0 ?D0 ] =>
      refine (Build_SpecializedFunctor C0 D0
                                       (fun c => CovariantHomFunctor C)
                                       _
             )
  end;
  admit.
Defined.

Section FullyFaithful.
  Context `(C : @SpecializedCategory objC).
  Let TypeCatC := FunctorCategory C TypeCat.
  Let YC := (Yoneda C).
  Set Printing Universes.
  Check @FunctorProduct' C TypeCatC YC. (* Toplevel input, characters 0-37:
Error: Universe inconsistency. Cannot enforce Top.187 = Top.186 because
Top.186 <= Top.189 < Top.191 <= Top.187). *)
End FullyFaithful.

quality99%

¤ Dauer der Verarbeitung: 0.6 Sekunden ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.