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.

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)

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.

Equations

• CategoryTheory.Monoidal.FunctorCategory.tensorHom α β = { app := fun (X : C) => CategoryTheory.MonoidalCategoryStruct.tensorHom (α.app X) (β.app X), naturality := ⋯ }

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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)

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.)

Equations

• CategoryTheory.Monoidal.FunctorCategory.whiskerLeft F β = { app := fun (X : C) => CategoryTheory.MonoidalCategoryStruct.whiskerLeft (F.obj X) (β.app X), naturality := ⋯ }

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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)

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.)

Equations

• CategoryTheory.Monoidal.FunctorCategory.whiskerRight α F' = { app := fun (X : C) => CategoryTheory.MonoidalCategoryStruct.whiskerRight (α.app X) (F'.obj X), naturality := ⋯ }

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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)

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

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.

Equations

• CategoryTheory.Monoidal.functorCategorySymmetric = { toBraidedCategory := CategoryTheory.Monoidal.functorCategoryBraided, symmetry := ⋯ }