Functor categories have chosen finite products

If C is a category with chosen finite products, then so is J ⥤ C.

@[reducible, inline]

abbrev CategoryTheory.Functor.chosenTerminal (J : Type u_1) (C : Type u_2) [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] : Functor J C

The chosen terminal object in J ⥤ C.

Equations

• CategoryTheory.Functor.chosenTerminal J C = (CategoryTheory.Functor.const J).obj (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)

def CategoryTheory.Functor.chosenTerminalIsTerminal (J : Type u_1) (C : Type u_2) [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] : Limits.IsTerminal (chosenTerminal J C)

The chosen terminal object in J ⥤ C is terminal.

Equations

• One or more equations did not get rendered due to their size.

def CategoryTheory.Functor.chosenProd {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ F₂ : Functor J C) : Functor J C

The chosen binary product on J ⥤ C.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CategoryTheory.Functor.chosenProd_obj {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ F₂ : Functor J C) (j : J) : (F₁.chosenProd F₂).obj j = MonoidalCategoryStruct.tensorObj (F₁.obj j) (F₂.obj j)

@[simp]

theorem CategoryTheory.Functor.chosenProd_map {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ F₂ : Functor J C) {X✝ Y✝ : J} (φ : X✝ ⟶ Y✝) : (F₁.chosenProd F₂).map φ = MonoidalCategoryStruct.tensorHom (F₁.map φ) (F₂.map φ)

def CategoryTheory.Functor.chosenProd.fst {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ F₂ : Functor J C) : F₁.chosenProd F₂ ⟶ F₁

The first projection chosenProd F₁ F₂ ⟶ F₁.

Equations

• CategoryTheory.Functor.chosenProd.fst F₁ F₂ = { app := fun (x : J) => CategoryTheory.CartesianMonoidalCategory.fst (F₁.obj x) (F₂.obj x), naturality := ⋯ }

def CategoryTheory.Functor.chosenProd.snd {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ F₂ : Functor J C) : F₁.chosenProd F₂ ⟶ F₂

The second projection chosenProd F₁ F₂ ⟶ F₂.

Equations

• CategoryTheory.Functor.chosenProd.snd F₁ F₂ = { app := fun (x : J) => CategoryTheory.CartesianMonoidalCategory.snd (F₁.obj x) (F₂.obj x), naturality := ⋯ }

def CategoryTheory.Functor.chosenProd.isLimit {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ F₂ : Functor J C) : Limits.IsLimit (Limits.BinaryFan.mk (fst F₁ F₂) (snd F₁ F₂))

Functor.chosenProd F₁ F₂ is a binary product of F₁ and F₂.

Equations

• One or more equations did not get rendered due to their size.

instance CategoryTheory.Functor.cartesianMonoidalCategory (J : Type u_1) (C : Type u_2) [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] : CartesianMonoidalCategory (Functor J C)

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CategoryTheory.Functor.Monoidal.tensorObj_obj {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ F₂ : Functor J C) (j : J) : (MonoidalCategoryStruct.tensorObj F₁ F₂).obj j = MonoidalCategoryStruct.tensorObj (F₁.obj j) (F₂.obj j)

@[simp]

theorem CategoryTheory.Functor.Monoidal.tensorObj_map {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ F₂ : Functor J C) {j j' : J} (f : j ⟶ j') : (MonoidalCategoryStruct.tensorObj F₁ F₂).map f = MonoidalCategoryStruct.tensorHom (F₁.map f) (F₂.map f)

@[simp]

theorem CategoryTheory.Functor.Monoidal.fst_app {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ F₂ : Functor J C) (j : J) : (CartesianMonoidalCategory.fst F₁ F₂).app j = CartesianMonoidalCategory.fst (F₁.obj j) (F₂.obj j)

@[simp]

theorem CategoryTheory.Functor.Monoidal.snd_app {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ F₂ : Functor J C) (j : J) : (CartesianMonoidalCategory.snd F₁ F₂).app j = CartesianMonoidalCategory.snd (F₁.obj j) (F₂.obj j)

@[simp]

theorem CategoryTheory.Functor.Monoidal.leftUnitor_hom_app {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F : Functor J C) (j : J) : (MonoidalCategoryStruct.leftUnitor F).hom.app j = (MonoidalCategoryStruct.leftUnitor (F.obj j)).hom

@[simp]

theorem CategoryTheory.Functor.Monoidal.leftUnitor_inv_app {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F : Functor J C) (j : J) : (MonoidalCategoryStruct.leftUnitor F).inv.app j = (MonoidalCategoryStruct.leftUnitor (F.obj j)).inv

@[simp]

theorem CategoryTheory.Functor.Monoidal.rightUnitor_hom_app {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F : Functor J C) (j : J) : (MonoidalCategoryStruct.rightUnitor F).hom.app j = (MonoidalCategoryStruct.rightUnitor (F.obj j)).hom

@[simp]

theorem CategoryTheory.Functor.Monoidal.rightUnitor_inv_app {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F : Functor J C) (j : J) : (MonoidalCategoryStruct.rightUnitor F).inv.app j = (MonoidalCategoryStruct.rightUnitor (F.obj j)).inv

@[simp]

theorem CategoryTheory.Functor.Monoidal.tensorHom_app_fst {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] {F₁ F₁' F₂ F₂' : Functor J C} (f : F₁ ⟶ F₁') (g : F₂ ⟶ F₂') (j : J) : CategoryStruct.comp ((MonoidalCategoryStruct.tensorHom f g).app j) (CartesianMonoidalCategory.fst (F₁'.obj j) (F₂'.obj j)) = CategoryStruct.comp (CartesianMonoidalCategory.fst (F₁.obj j) (F₂.obj j)) (f.app j)

@[simp]

theorem CategoryTheory.Functor.Monoidal.tensorHom_app_fst_assoc {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] {F₁ F₁' F₂ F₂' : Functor J C} (f : F₁ ⟶ F₁') (g : F₂ ⟶ F₂') (j : J) {Z : C} (h : F₁'.obj j ⟶ Z) : CategoryStruct.comp ((MonoidalCategoryStruct.tensorHom f g).app j) (CategoryStruct.comp (CartesianMonoidalCategory.fst (F₁'.obj j) (F₂'.obj j)) h) = CategoryStruct.comp (CartesianMonoidalCategory.fst (F₁.obj j) (F₂.obj j)) (CategoryStruct.comp (f.app j) h)

@[simp]

theorem CategoryTheory.Functor.Monoidal.tensorHom_app_snd {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] {F₁ F₁' F₂ F₂' : Functor J C} (f : F₁ ⟶ F₁') (g : F₂ ⟶ F₂') (j : J) : CategoryStruct.comp ((MonoidalCategoryStruct.tensorHom f g).app j) (CartesianMonoidalCategory.snd (F₁'.obj j) (F₂'.obj j)) = CategoryStruct.comp (CartesianMonoidalCategory.snd (F₁.obj j) (F₂.obj j)) (g.app j)

@[simp]

theorem CategoryTheory.Functor.Monoidal.tensorHom_app_snd_assoc {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] {F₁ F₁' F₂ F₂' : Functor J C} (f : F₁ ⟶ F₁') (g : F₂ ⟶ F₂') (j : J) {Z : C} (h : F₂'.obj j ⟶ Z) : CategoryStruct.comp ((MonoidalCategoryStruct.tensorHom f g).app j) (CategoryStruct.comp (CartesianMonoidalCategory.snd (F₁'.obj j) (F₂'.obj j)) h) = CategoryStruct.comp (CartesianMonoidalCategory.snd (F₁.obj j) (F₂.obj j)) (CategoryStruct.comp (g.app j) h)

@[simp]

theorem CategoryTheory.Functor.Monoidal.whiskerLeft_app_fst {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ : Functor J C) {F₂ F₂' : Functor J C} (g : F₂ ⟶ F₂') (j : J) : CategoryStruct.comp ((MonoidalCategoryStruct.whiskerLeft F₁ g).app j) (CartesianMonoidalCategory.fst (F₁.obj j) (F₂'.obj j)) = CartesianMonoidalCategory.fst (F₁.obj j) (F₂.obj j)

@[simp]

theorem CategoryTheory.Functor.Monoidal.whiskerLeft_app_fst_assoc {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ : Functor J C) {F₂ F₂' : Functor J C} (g : F₂ ⟶ F₂') (j : J) {Z : C} (h : F₁.obj j ⟶ Z) : CategoryStruct.comp ((MonoidalCategoryStruct.whiskerLeft F₁ g).app j) (CategoryStruct.comp (CartesianMonoidalCategory.fst (F₁.obj j) (F₂'.obj j)) h) = CategoryStruct.comp (CartesianMonoidalCategory.fst (F₁.obj j) (F₂.obj j)) h

@[simp]

theorem CategoryTheory.Functor.Monoidal.whiskerLeft_app_snd {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ : Functor J C) {F₂ F₂' : Functor J C} (g : F₂ ⟶ F₂') (j : J) : CategoryStruct.comp ((MonoidalCategoryStruct.whiskerLeft F₁ g).app j) (CartesianMonoidalCategory.snd (F₁.obj j) (F₂'.obj j)) = CategoryStruct.comp (CartesianMonoidalCategory.snd (F₁.obj j) (F₂.obj j)) (g.app j)

@[simp]

theorem CategoryTheory.Functor.Monoidal.whiskerLeft_app_snd_assoc {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ : Functor J C) {F₂ F₂' : Functor J C} (g : F₂ ⟶ F₂') (j : J) {Z : C} (h : F₂'.obj j ⟶ Z) : CategoryStruct.comp ((MonoidalCategoryStruct.whiskerLeft F₁ g).app j) (CategoryStruct.comp (CartesianMonoidalCategory.snd (F₁.obj j) (F₂'.obj j)) h) = CategoryStruct.comp (CartesianMonoidalCategory.snd (F₁.obj j) (F₂.obj j)) (CategoryStruct.comp (g.app j) h)

@[simp]

theorem CategoryTheory.Functor.Monoidal.whiskerRight_app_fst {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] {F₁ F₁' : Functor J C} (f : F₁ ⟶ F₁') (F₂ : Functor J C) (j : J) : CategoryStruct.comp ((MonoidalCategoryStruct.whiskerRight f F₂).app j) (CartesianMonoidalCategory.fst (F₁'.obj j) (F₂.obj j)) = CategoryStruct.comp (CartesianMonoidalCategory.fst (F₁.obj j) (F₂.obj j)) (f.app j)

@[simp]

theorem CategoryTheory.Functor.Monoidal.whiskerRight_app_fst_assoc {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] {F₁ F₁' : Functor J C} (f : F₁ ⟶ F₁') (F₂ : Functor J C) (j : J) {Z : C} (h : F₁'.obj j ⟶ Z) : CategoryStruct.comp ((MonoidalCategoryStruct.whiskerRight f F₂).app j) (CategoryStruct.comp (CartesianMonoidalCategory.fst (F₁'.obj j) (F₂.obj j)) h) = CategoryStruct.comp (CartesianMonoidalCategory.fst (F₁.obj j) (F₂.obj j)) (CategoryStruct.comp (f.app j) h)

@[simp]

theorem CategoryTheory.Functor.Monoidal.whiskerRight_app_snd {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] {F₁ F₁' : Functor J C} (f : F₁ ⟶ F₁') (F₂ : Functor J C) (j : J) : CategoryStruct.comp ((MonoidalCategoryStruct.whiskerRight f F₂).app j) (CartesianMonoidalCategory.snd (F₁'.obj j) (F₂.obj j)) = CartesianMonoidalCategory.snd (F₁.obj j) (F₂.obj j)

@[simp]

theorem CategoryTheory.Functor.Monoidal.whiskerRight_app_snd_assoc {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] {F₁ F₁' : Functor J C} (f : F₁ ⟶ F₁') (F₂ : Functor J C) (j : J) {Z : C} (h : F₂.obj j ⟶ Z) : CategoryStruct.comp ((MonoidalCategoryStruct.whiskerRight f F₂).app j) (CategoryStruct.comp (CartesianMonoidalCategory.snd (F₁'.obj j) (F₂.obj j)) h) = CategoryStruct.comp (CartesianMonoidalCategory.snd (F₁.obj j) (F₂.obj j)) h

@[simp]

theorem CategoryTheory.Functor.Monoidal.associator_hom_app {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ F₂ F₃ : Functor J C) (j : J) : (MonoidalCategoryStruct.associator F₁ F₂ F₃).hom.app j = (MonoidalCategoryStruct.associator (F₁.obj j) (F₂.obj j) (F₃.obj j)).hom

@[simp]

theorem CategoryTheory.Functor.Monoidal.associator_inv_app {J : Type u_1} {C : Type u_2} [Category.{u_3, u_1} J] [Category.{u_4, u_2} C] [CartesianMonoidalCategory C] (F₁ F₂ F₃ : Functor J C) (j : J) : (MonoidalCategoryStruct.associator F₁ F₂ F₃).inv.app j = (MonoidalCategoryStruct.associator (F₁.obj j) (F₂.obj j) (F₃.obj j)).inv

instance CategoryTheory.Functor.Monoidal.instPreservesColimitsOfShapeTensorLeftOfHasColimitsOfShape {J : Type u_1} {C : Type u_2} [Category.{u_6, u_1} J] [Category.{u_5, u_2} C] [CartesianMonoidalCategory C] {K : Type u_3} [Category.{u_4, u_3} K] [Limits.HasColimitsOfShape K C] [∀ (X : C), Limits.PreservesColimitsOfShape K (MonoidalCategory.tensorLeft X)] {F : Functor J C} : Limits.PreservesColimitsOfShape K (MonoidalCategory.tensorLeft F)