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Mathlib.CategoryTheory.Monoidal.FunctorCategory

Monoidal structure on C ⥤ D when D is monoidal. #

When C is any category, and D is a monoidal category, there is a natural "pointwise" monoidal structure on C ⥤ D.

The initial intended application is tensor product of presheaves.

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theorem CategoryTheory.Monoidal.FunctorCategory.tensorObj_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.Functor C D) (G : CategoryTheory.Functor C D) :
∀ {X Y : C} (f : X ⟶ Y), (CategoryTheory.Monoidal.FunctorCategory.tensorObj F G).toPrefunctor.map f = CategoryTheory.MonoidalCategory.tensorHom (F.toPrefunctor.map f) (G.toPrefunctor.map f)
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theorem CategoryTheory.Monoidal.FunctorCategory.tensorObj_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.Functor C D) (G : CategoryTheory.Functor C D) (X : C) :
(CategoryTheory.Monoidal.FunctorCategory.tensorObj F G).toPrefunctor.obj X = CategoryTheory.MonoidalCategory.tensorObj (F.toPrefunctor.obj X) (G.toPrefunctor.obj X)
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def CategoryTheory.Monoidal.FunctorCategory.tensorObj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.Functor C D) (G : CategoryTheory.Functor C D) :
CategoryTheory.Functor C D

(An auxiliary definition for functorCategoryMonoidal.) Tensor product of functors C ⥤ D, when D is monoidal.

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    theorem CategoryTheory.Monoidal.FunctorCategory.tensorHom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} {F' : CategoryTheory.Functor C D} {G' : CategoryTheory.Functor C D} (α : F ⟶ G) (β : F' ⟶ G') (X : C) :
    (CategoryTheory.Monoidal.FunctorCategory.tensorHom α β).app X = CategoryTheory.MonoidalCategory.tensorHom (α.app X) (β.app X)
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    def CategoryTheory.Monoidal.FunctorCategory.tensorHom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} {F' : CategoryTheory.Functor C D} {G' : CategoryTheory.Functor C D} (α : F ⟶ G) (β : F' ⟶ G') :
    CategoryTheory.Monoidal.FunctorCategory.tensorObj F F' ⟶ CategoryTheory.Monoidal.FunctorCategory.tensorObj G G'

    (An auxiliary definition for functorCategoryMonoidal.) Tensor product of natural transformations into D, when D is monoidal.

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      theorem CategoryTheory.Monoidal.FunctorCategory.whiskerLeft_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F' : CategoryTheory.Functor C D} {G' : CategoryTheory.Functor C D} (F : CategoryTheory.Functor C D) (β : F' ⟶ G') (X : C) :
      (CategoryTheory.Monoidal.FunctorCategory.whiskerLeft F β).app X = CategoryTheory.MonoidalCategory.whiskerLeft (F.toPrefunctor.obj X) (β.app X)
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      def CategoryTheory.Monoidal.FunctorCategory.whiskerLeft {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F' : CategoryTheory.Functor C D} {G' : CategoryTheory.Functor C D} (F : CategoryTheory.Functor C D) (β : F' ⟶ G') :
      CategoryTheory.Monoidal.FunctorCategory.tensorObj F F' ⟶ CategoryTheory.Monoidal.FunctorCategory.tensorObj F G'

      (An auxiliary definition for functorCategoryMonoidal.)

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        theorem CategoryTheory.Monoidal.FunctorCategory.whiskerRight_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} (α : F ⟶ G) (F' : CategoryTheory.Functor C D) (X : C) :
        (CategoryTheory.Monoidal.FunctorCategory.whiskerRight α F').app X = CategoryTheory.MonoidalCategory.whiskerRight (α.app X) (F'.toPrefunctor.obj X)
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        def CategoryTheory.Monoidal.FunctorCategory.whiskerRight {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} (α : F ⟶ G) (F' : CategoryTheory.Functor C D) :
        CategoryTheory.Monoidal.FunctorCategory.tensorObj F F' ⟶ CategoryTheory.Monoidal.FunctorCategory.tensorObj G F'

        (An auxiliary definition for functorCategoryMonoidal.)

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          instance CategoryTheory.Monoidal.functorCategoryMonoidalStruct {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] :
          CategoryTheory.MonoidalCategoryStruct (CategoryTheory.Functor C D)

          When C is any category, and D is a monoidal category, the functor category C ⥤ D has a natural pointwise monoidal structure, where (F ⊗ G).obj X = F.obj X ⊗ G.obj X.

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          theorem CategoryTheory.Monoidal.tensorUnit_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {X : C} :
          (𝟙_ (CategoryTheory.Functor C D)).toPrefunctor.obj X = 𝟙_ D
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          theorem CategoryTheory.Monoidal.tensorUnit_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {X : C} {Y : C} {f : X ⟶ Y} :
          (𝟙_ (CategoryTheory.Functor C D)).toPrefunctor.map f = CategoryTheory.CategoryStruct.id (𝟙_ D)
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          theorem CategoryTheory.Monoidal.tensorObj_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} {X : C} :
          (CategoryTheory.MonoidalCategory.tensorObj F G).toPrefunctor.obj X = CategoryTheory.MonoidalCategory.tensorObj (F.toPrefunctor.obj X) (G.toPrefunctor.obj X)
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          theorem CategoryTheory.Monoidal.tensorObj_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} {X : C} {Y : C} {f : X ⟶ Y} :
          (CategoryTheory.MonoidalCategory.tensorObj F G).toPrefunctor.map f = CategoryTheory.MonoidalCategory.tensorHom (F.toPrefunctor.map f) (G.toPrefunctor.map f)
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          theorem CategoryTheory.Monoidal.tensorHom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} {F' : CategoryTheory.Functor C D} {G' : CategoryTheory.Functor C D} {α : F ⟶ G} {β : F' ⟶ G'} {X : C} :
          (CategoryTheory.MonoidalCategory.tensorHom α β).app X = CategoryTheory.MonoidalCategory.tensorHom (α.app X) (β.app X)
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          theorem CategoryTheory.Monoidal.whiskerLeft_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} {F' : CategoryTheory.Functor C D} {G' : CategoryTheory.Functor C D} {β : F' ⟶ G'} {X : C} :
          (CategoryTheory.MonoidalCategory.whiskerLeft F β).app X = CategoryTheory.MonoidalCategory.whiskerLeft (F.toPrefunctor.obj X) (β.app X)
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          theorem CategoryTheory.Monoidal.whiskerRight_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} {F' : CategoryTheory.Functor C D} {α : F ⟶ G} {X : C} :
          (CategoryTheory.MonoidalCategory.whiskerRight α F').app X = CategoryTheory.MonoidalCategory.whiskerRight (α.app X) (F'.toPrefunctor.obj X)
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          theorem CategoryTheory.Monoidal.leftUnitor_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} {X : C} :
          (CategoryTheory.MonoidalCategory.leftUnitor F).hom.app X = (CategoryTheory.MonoidalCategory.leftUnitor (F.toPrefunctor.obj X)).hom
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          theorem CategoryTheory.Monoidal.leftUnitor_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} {X : C} :
          (CategoryTheory.MonoidalCategory.leftUnitor F).inv.app X = (CategoryTheory.MonoidalCategory.leftUnitor (F.toPrefunctor.obj X)).inv
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          theorem CategoryTheory.Monoidal.rightUnitor_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} {X : C} :
          (CategoryTheory.MonoidalCategory.rightUnitor F).hom.app X = (CategoryTheory.MonoidalCategory.rightUnitor (F.toPrefunctor.obj X)).hom
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          theorem CategoryTheory.Monoidal.rightUnitor_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} {X : C} :
          (CategoryTheory.MonoidalCategory.rightUnitor F).inv.app X = (CategoryTheory.MonoidalCategory.rightUnitor (F.toPrefunctor.obj X)).inv
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          theorem CategoryTheory.Monoidal.associator_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} {H : CategoryTheory.Functor C D} {X : C} :
          (CategoryTheory.MonoidalCategory.associator F G H).hom.app X = (CategoryTheory.MonoidalCategory.associator (F.toPrefunctor.obj X) (G.toPrefunctor.obj X) (H.toPrefunctor.obj X)).hom
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          theorem CategoryTheory.Monoidal.associator_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} {H : CategoryTheory.Functor C D} {X : C} :
          (CategoryTheory.MonoidalCategory.associator F G H).inv.app X = (CategoryTheory.MonoidalCategory.associator (F.toPrefunctor.obj X) (G.toPrefunctor.obj X) (H.toPrefunctor.obj X)).inv
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          instance CategoryTheory.Monoidal.functorCategoryMonoidal {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] :
          CategoryTheory.MonoidalCategory (CategoryTheory.Functor C D)

          When C is any category, and D is a monoidal category, the functor category C ⥤ D has a natural pointwise monoidal structure, where (F ⊗ G).obj X = F.obj X ⊗ G.obj X.

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          • CategoryTheory.Monoidal.functorCategoryMonoidal = CategoryTheory.MonoidalCategory.mk
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          instance CategoryTheory.Monoidal.functorCategoryBraided {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] :
          CategoryTheory.BraidedCategory (CategoryTheory.Functor C D)

          When C is any category, and D is a braided monoidal category, the natural pointwise monoidal structure on the functor category C ⥤ D is also braided.

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          instance CategoryTheory.Monoidal.functorCategorySymmetric {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.SymmetricCategory D] :
          CategoryTheory.SymmetricCategory (CategoryTheory.Functor C D)

          When C is any category, and D is a symmetric monoidal category, the natural pointwise monoidal structure on the functor category C ⥤ D is also symmetric.

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          • CategoryTheory.Monoidal.functorCategorySymmetric = CategoryTheory.SymmetricCategory.mk