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Mathlib.CategoryTheory.Monoidal.Closed.Types

Cartesian closure of Type #

Show that Type u₁ is Cartesian closed, and C ⥤ Type u₁ is Cartesian closed for C a small category in Type u₁. Note this implies that the category of presheaves on a small category C is Cartesian closed.

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def CategoryTheory.Types.tensorProductAdjunction (X : Type v₁) :
MonoidalCategory.tensorLeft X ⊣ coyoneda.obj (Opposite.op X)

The adjunction tensorLeft.obj X ⊣ coyoneda.obj (Opposite.op X) for any X : Type v₁.

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Instances For
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    instance CategoryTheory.instIsLeftAdjointTensorLeft (X : Type v₁) :
    (MonoidalCategory.tensorLeft X).IsLeftAdjoint
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    @[implicit_reducible]
    instance CategoryTheory.instMonoidalClosedType :
    MonoidalClosed (Type v₁)
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    @[implicit_reducible]
    noncomputable instance CategoryTheory.instMonoidalClosedFunctorType {C : Type v₁} [SmallCategory C] :
    MonoidalClosed (Functor C (Type v₁))
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    @[implicit_reducible]
    noncomputable def CategoryTheory.cartesianClosedFunctorToTypes {C : Type u₁} [Category.{v₁, u₁} C] :
    MonoidalClosed (Functor C (Type (max u₁ v₁ u₂)))

    This is not a good instance because of the universe levels. Below is the instance where the target category is Type (max u₁ v₁).

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    Instances For
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      @[implicit_reducible]
      noncomputable instance CategoryTheory.instMonoidalClosedFunctorType_1 {C : Type u₁} [Category.{v₁, u₁} C] :
      MonoidalClosed (Functor C (Type (max u₁ v₁)))
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      @[implicit_reducible]
      noncomputable instance CategoryTheory.instMonoidalClosedFunctorTypeOfEssentiallySmall {C : Type u₁} [Category.{v₁, u₁} C] [EssentiallySmall.{v₁, v₁, u₁} C] :
      MonoidalClosed (Functor C (Type v₁))
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