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

          (An auxiliary definition for functorCategoryMonoidal.)

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

              (An auxiliary definition for functorCategoryMonoidal.)

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