(Lax) monoidal functors #

A lax monoidal functor F between monoidal categories C and D is a functor between the underlying categories equipped with morphisms

ε : 𝟙_ D ⟶ F.obj (𝟙_ C) (called the unit morphism)
μ X Y : (F.obj X) ⊗ (F.obj Y) ⟶ F.obj (X ⊗ Y) (called the tensorator, or strength). satisfying various axioms.

A monoidal functor is a lax monoidal functor for which ε and μ are isomorphisms.

We show that the composition of (lax) monoidal functors gives a (lax) monoidal functor.

See also CategoryTheory.Monoidal.Functorial for a typeclass decorating an object-level function with the additional data of a monoidal functor. This is useful when stating that a pre-existing functor is monoidal.

See CategoryTheory.Monoidal.NaturalTransformation for monoidal natural transformations.

We show in CategoryTheory.Monoidal.Mon_ that lax monoidal functors take monoid objects to monoid objects.

References #

See https://stacks.math.columbia.edu/tag/0FFL.

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structure CategoryTheory.LaxMonoidalFunctor (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] extends CategoryTheory.Functor :

Type (max (max (max u₁ u₂) v₁) v₂)

A lax monoidal functor is a functor F : C ⥤ D between monoidal categories, equipped with morphisms ε : 𝟙 _D ⟶ F.obj (𝟙_ C) and μ X Y : F.obj X ⊗ F.obj Y ⟶ F.obj (X ⊗ Y), satisfying the appropriate coherences.

obj : C → D
map : {X Y : C} → (X ⟶ Y) → (self.obj X ⟶ self.obj Y)
map_id : ∀ (X : C), self.map (CategoryTheory.CategoryStruct.id X) = CategoryTheory.CategoryStruct.id (self.obj X)
map_comp : ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), self.map (CategoryTheory.CategoryStruct.comp f g) = CategoryTheory.CategoryStruct.comp (self.map f) (self.map g)
ε : 𝟙_ D ⟶ self.obj (𝟙_ C)
unit morphism
μ : (X Y : C) → CategoryTheory.MonoidalCategory.tensorObj (self.obj X) (self.obj Y) ⟶ self.obj (CategoryTheory.MonoidalCategory.tensorObj X Y)
tensorator
μ_natural_left : ∀ {X Y : C} (f : X ⟶ Y) (X' : C), CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (self.map f) (self.obj X')) (self.μ Y X') = CategoryTheory.CategoryStruct.comp (self.μ X X') (self.map (CategoryTheory.MonoidalCategory.whiskerRight f X'))
μ_natural_right : ∀ {X Y : C} (X' : C) (f : X ⟶ Y), CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X') (self.map f)) (self.μ X' Y) = CategoryTheory.CategoryStruct.comp (self.μ X' X) (self.map (CategoryTheory.MonoidalCategory.whiskerLeft X' f))
associativity : ∀ (X Y Z : C), CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (self.μ X Y) (self.obj Z)) (CategoryTheory.CategoryStruct.comp (self.μ (CategoryTheory.MonoidalCategory.tensorObj X Y) Z) (self.map (CategoryTheory.MonoidalCategory.associator X Y Z).hom)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator (self.obj X) (self.obj Y) (self.obj Z)).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X) (self.μ Y Z)) (self.μ X (CategoryTheory.MonoidalCategory.tensorObj Y Z)))
associativity of the tensorator
left_unitality : ∀ (X : C), (CategoryTheory.MonoidalCategory.leftUnitor (self.obj X)).hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight self.ε (self.obj X)) (CategoryTheory.CategoryStruct.comp (self.μ (𝟙_ C) X) (self.map (CategoryTheory.MonoidalCategory.leftUnitor X).hom))
right_unitality : ∀ (X : C), (CategoryTheory.MonoidalCategory.rightUnitor (self.obj X)).hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X) self.ε) (CategoryTheory.CategoryStruct.comp (self.μ X (𝟙_ C)) (self.map (CategoryTheory.MonoidalCategory.rightUnitor X).hom))

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theorem CategoryTheory.LaxMonoidalFunctor.μ_natural_left {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.LaxMonoidalFunctor C D) {X : C} {Y : C} (f : X ⟶ Y) (X' : C) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (self.map f) (self.obj X')) (self.μ Y X') = CategoryTheory.CategoryStruct.comp (self.μ X X') (self.map (CategoryTheory.MonoidalCategory.whiskerRight f X'))

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theorem CategoryTheory.LaxMonoidalFunctor.μ_natural_right {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.LaxMonoidalFunctor C D) {X : C} {Y : C} (X' : C) (f : X ⟶ Y) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X') (self.map f)) (self.μ X' Y) = CategoryTheory.CategoryStruct.comp (self.μ X' X) (self.map (CategoryTheory.MonoidalCategory.whiskerLeft X' f))

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theorem CategoryTheory.LaxMonoidalFunctor.associativity {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.LaxMonoidalFunctor C D) (X : C) (Y : C) (Z : C) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (self.μ X Y) (self.obj Z)) (CategoryTheory.CategoryStruct.comp (self.μ (CategoryTheory.MonoidalCategory.tensorObj X Y) Z) (self.map (CategoryTheory.MonoidalCategory.associator X Y Z).hom)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator (self.obj X) (self.obj Y) (self.obj Z)).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X) (self.μ Y Z)) (self.μ X (CategoryTheory.MonoidalCategory.tensorObj Y Z)))

associativity of the tensorator

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theorem CategoryTheory.LaxMonoidalFunctor.left_unitality {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.LaxMonoidalFunctor C D) (X : C) :

(CategoryTheory.MonoidalCategory.leftUnitor (self.obj X)).hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight self.ε (self.obj X)) (CategoryTheory.CategoryStruct.comp (self.μ (𝟙_ C) X) (self.map (CategoryTheory.MonoidalCategory.leftUnitor X).hom))

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theorem CategoryTheory.LaxMonoidalFunctor.right_unitality {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.LaxMonoidalFunctor C D) (X : C) :

(CategoryTheory.MonoidalCategory.rightUnitor (self.obj X)).hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X) self.ε) (CategoryTheory.CategoryStruct.comp (self.μ X (𝟙_ C)) (self.map (CategoryTheory.MonoidalCategory.rightUnitor X).hom))

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theorem CategoryTheory.LaxMonoidalFunctor.μ_natural_left_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.LaxMonoidalFunctor C D) {X : C} {Y : C} (f : X ⟶ Y) (X' : C) {Z : D} (h : self.obj (CategoryTheory.MonoidalCategory.tensorObj Y X') ⟶ Z) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (self.map f) (self.obj X')) (CategoryTheory.CategoryStruct.comp (self.μ Y X') h) = CategoryTheory.CategoryStruct.comp (self.μ X X') (CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.whiskerRight f X')) h)

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theorem CategoryTheory.LaxMonoidalFunctor.μ_natural_right_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.LaxMonoidalFunctor C D) {X : C} {Y : C} (X' : C) (f : X ⟶ Y) {Z : D} (h : self.obj (CategoryTheory.MonoidalCategory.tensorObj X' Y) ⟶ Z) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X') (self.map f)) (CategoryTheory.CategoryStruct.comp (self.μ X' Y) h) = CategoryTheory.CategoryStruct.comp (self.μ X' X) (CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.whiskerLeft X' f)) h)

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theorem CategoryTheory.LaxMonoidalFunctor.associativity_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.LaxMonoidalFunctor C D) (X : C) (Y : C) (Z : C) {Z : D} (h : self.obj (CategoryTheory.MonoidalCategory.tensorObj X (CategoryTheory.MonoidalCategory.tensorObj Y Z✝)) ⟶ Z) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (self.μ X Y) (self.obj Z✝)) (CategoryTheory.CategoryStruct.comp (self.μ (CategoryTheory.MonoidalCategory.tensorObj X Y) Z✝) (CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.associator X Y Z✝).hom) h)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator (self.obj X) (self.obj Y) (self.obj Z✝)).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X) (self.μ Y Z✝)) (CategoryTheory.CategoryStruct.comp (self.μ X (CategoryTheory.MonoidalCategory.tensorObj Y Z✝)) h))

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theorem CategoryTheory.LaxMonoidalFunctor.μ_natural_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.LaxMonoidalFunctor C D) {X : C} {Y : C} {X' : C} {Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') {Z : D} (h : F.obj (CategoryTheory.MonoidalCategory.tensorObj Y Y') ⟶ Z) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.tensorHom (F.map f) (F.map g)) (CategoryTheory.CategoryStruct.comp (F.μ Y Y') h) = CategoryTheory.CategoryStruct.comp (F.μ X X') (CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategory.tensorHom f g)) h)

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theorem CategoryTheory.LaxMonoidalFunctor.μ_natural {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.LaxMonoidalFunctor C D) {X : C} {Y : C} {X' : C} {Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.tensorHom (F.map f) (F.map g)) (F.μ Y Y') = CategoryTheory.CategoryStruct.comp (F.μ X X') (F.map (CategoryTheory.MonoidalCategory.tensorHom f g))

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∀ (a : C), (CategoryTheory.LaxMonoidalFunctor.ofTensorHom F ε μ μ_natural associativity left_unitality right_unitality).obj a = F.obj a

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(CategoryTheory.LaxMonoidalFunctor.ofTensorHom F ε μ μ_natural associativity left_unitality right_unitality).μ X Y = μ X Y

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∀ {X Y : C} (a : X ⟶ Y), (CategoryTheory.LaxMonoidalFunctor.ofTensorHom F ε μ μ_natural associativity left_unitality right_unitality).map a = F.map a

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(CategoryTheory.LaxMonoidalFunctor.ofTensorHom F ε μ μ_natural associativity left_unitality right_unitality).ε = ε

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CategoryTheory.LaxMonoidalFunctor C D

A constructor for lax monoidal functors whose axioms are described by tensorHom instead of whiskerLeft and whiskerRight.

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theorem CategoryTheory.LaxMonoidalFunctor.left_unitality_inv_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.LaxMonoidalFunctor C D) (X : C) {Z : D} (h : F.obj (CategoryTheory.MonoidalCategory.tensorObj (𝟙_ C) X) ⟶ Z) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.leftUnitor (F.obj X)).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight F.ε (F.obj X)) (CategoryTheory.CategoryStruct.comp (F.μ (𝟙_ C) X) h)) = CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategory.leftUnitor X).inv) h

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theorem CategoryTheory.LaxMonoidalFunctor.left_unitality_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.LaxMonoidalFunctor C D) (X : C) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.leftUnitor (F.obj X)).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight F.ε (F.obj X)) (F.μ (𝟙_ C) X)) = F.map (CategoryTheory.MonoidalCategory.leftUnitor X).inv

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theorem CategoryTheory.LaxMonoidalFunctor.right_unitality_inv_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.LaxMonoidalFunctor C D) (X : C) {Z : D} (h : F.obj (CategoryTheory.MonoidalCategory.tensorObj X (𝟙_ C)) ⟶ Z) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.rightUnitor (F.obj X)).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (F.obj X) F.ε) (CategoryTheory.CategoryStruct.comp (F.μ X (𝟙_ C)) h)) = CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategory.rightUnitor X).inv) h

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theorem CategoryTheory.LaxMonoidalFunctor.right_unitality_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.LaxMonoidalFunctor C D) (X : C) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.rightUnitor (F.obj X)).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (F.obj X) F.ε) (F.μ X (𝟙_ C))) = F.map (CategoryTheory.MonoidalCategory.rightUnitor X).inv

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theorem CategoryTheory.LaxMonoidalFunctor.associativity_inv_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.LaxMonoidalFunctor C D) (X : C) (Y : C) (Z : C) {Z : D} (h : F.obj (CategoryTheory.MonoidalCategory.tensorObj (CategoryTheory.MonoidalCategory.tensorObj X Y) Z✝) ⟶ Z) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (F.obj X) (F.μ Y Z✝)) (CategoryTheory.CategoryStruct.comp (F.μ X (CategoryTheory.MonoidalCategory.tensorObj Y Z✝)) (CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategory.associator X Y Z✝).inv) h)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator (F.obj X) (F.obj Y) (F.obj Z✝)).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (F.μ X Y) (F.obj Z✝)) (CategoryTheory.CategoryStruct.comp (F.μ (CategoryTheory.MonoidalCategory.tensorObj X Y) Z✝) h))

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theorem CategoryTheory.LaxMonoidalFunctor.associativity_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.LaxMonoidalFunctor C D) (X : C) (Y : C) (Z : C) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (F.obj X) (F.μ Y Z)) (CategoryTheory.CategoryStruct.comp (F.μ X (CategoryTheory.MonoidalCategory.tensorObj Y Z)) (F.map (CategoryTheory.MonoidalCategory.associator X Y Z).inv)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator (F.obj X) (F.obj Y) (F.obj Z)).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (F.μ X Y) (F.obj Z)) (F.μ (CategoryTheory.MonoidalCategory.tensorObj X Y) Z))

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structure CategoryTheory.OplaxMonoidalFunctor (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] extends CategoryTheory.Functor :

Type (max (max (max u₁ u₂) v₁) v₂)

A oplax monoidal functor is a functor F : C ⥤ D between monoidal categories, equipped with morphisms η : F.obj (𝟙_ C) ⟶ 𝟙 _D and δ X Y : F.obj (X ⊗ Y) ⟶ F.obj X ⊗ F.obj Y, satisfying the appropriate coherences.

obj : C → D
map : {X Y : C} → (X ⟶ Y) → (self.obj X ⟶ self.obj Y)
map_id : ∀ (X : C), self.map (CategoryTheory.CategoryStruct.id X) = CategoryTheory.CategoryStruct.id (self.obj X)
map_comp : ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), self.map (CategoryTheory.CategoryStruct.comp f g) = CategoryTheory.CategoryStruct.comp (self.map f) (self.map g)
η : self.obj (𝟙_ C) ⟶ 𝟙_ D
counit morphism
δ : (X Y : C) → self.obj (CategoryTheory.MonoidalCategory.tensorObj X Y) ⟶ CategoryTheory.MonoidalCategory.tensorObj (self.obj X) (self.obj Y)
cotensorator
δ_natural_left : ∀ {X Y : C} (f : X ⟶ Y) (X' : C), CategoryTheory.CategoryStruct.comp (self.δ X X') (CategoryTheory.MonoidalCategory.whiskerRight (self.map f) (self.obj X')) = CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.whiskerRight f X')) (self.δ Y X')
δ_natural_right : ∀ {X Y : C} (X' : C) (f : X ⟶ Y), CategoryTheory.CategoryStruct.comp (self.δ X' X) (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X') (self.map f)) = CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.whiskerLeft X' f)) (self.δ X' Y)
associativity : ∀ (X Y Z : C), CategoryTheory.CategoryStruct.comp (self.δ (CategoryTheory.MonoidalCategory.tensorObj X Y) Z) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (self.δ X Y) (self.obj Z)) (CategoryTheory.MonoidalCategory.associator (self.obj X) (self.obj Y) (self.obj Z)).hom) = CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.associator X Y Z).hom) (CategoryTheory.CategoryStruct.comp (self.δ X (CategoryTheory.MonoidalCategory.tensorObj Y Z)) (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X) (self.δ Y Z)))
associativity of the tensorator
left_unitality : ∀ (X : C), (CategoryTheory.MonoidalCategory.leftUnitor (self.obj X)).inv = CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.leftUnitor X).inv) (CategoryTheory.CategoryStruct.comp (self.δ (𝟙_ C) X) (CategoryTheory.MonoidalCategory.whiskerRight self.η (self.obj X)))
right_unitality : ∀ (X : C), (CategoryTheory.MonoidalCategory.rightUnitor (self.obj X)).inv = CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.rightUnitor X).inv) (CategoryTheory.CategoryStruct.comp (self.δ X (𝟙_ C)) (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X) self.η))

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theorem CategoryTheory.OplaxMonoidalFunctor.δ_natural_left {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.OplaxMonoidalFunctor C D) {X : C} {Y : C} (f : X ⟶ Y) (X' : C) :

CategoryTheory.CategoryStruct.comp (self.δ X X') (CategoryTheory.MonoidalCategory.whiskerRight (self.map f) (self.obj X')) = CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.whiskerRight f X')) (self.δ Y X')

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theorem CategoryTheory.OplaxMonoidalFunctor.δ_natural_right {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.OplaxMonoidalFunctor C D) {X : C} {Y : C} (X' : C) (f : X ⟶ Y) :

CategoryTheory.CategoryStruct.comp (self.δ X' X) (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X') (self.map f)) = CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.whiskerLeft X' f)) (self.δ X' Y)

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theorem CategoryTheory.OplaxMonoidalFunctor.associativity {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.OplaxMonoidalFunctor C D) (X : C) (Y : C) (Z : C) :

CategoryTheory.CategoryStruct.comp (self.δ (CategoryTheory.MonoidalCategory.tensorObj X Y) Z) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (self.δ X Y) (self.obj Z)) (CategoryTheory.MonoidalCategory.associator (self.obj X) (self.obj Y) (self.obj Z)).hom) = CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.associator X Y Z).hom) (CategoryTheory.CategoryStruct.comp (self.δ X (CategoryTheory.MonoidalCategory.tensorObj Y Z)) (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X) (self.δ Y Z)))

associativity of the tensorator

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theorem CategoryTheory.OplaxMonoidalFunctor.left_unitality {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.OplaxMonoidalFunctor C D) (X : C) :

(CategoryTheory.MonoidalCategory.leftUnitor (self.obj X)).inv = CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.leftUnitor X).inv) (CategoryTheory.CategoryStruct.comp (self.δ (𝟙_ C) X) (CategoryTheory.MonoidalCategory.whiskerRight self.η (self.obj X)))

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theorem CategoryTheory.OplaxMonoidalFunctor.right_unitality {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.OplaxMonoidalFunctor C D) (X : C) :

(CategoryTheory.MonoidalCategory.rightUnitor (self.obj X)).inv = CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.rightUnitor X).inv) (CategoryTheory.CategoryStruct.comp (self.δ X (𝟙_ C)) (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X) self.η))

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theorem CategoryTheory.OplaxMonoidalFunctor.δ_natural_left_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.OplaxMonoidalFunctor C D) {X : C} {Y : C} (f : X ⟶ Y) (X' : C) {Z : D} (h : CategoryTheory.MonoidalCategory.tensorObj (self.obj Y) (self.obj X') ⟶ Z) :

CategoryTheory.CategoryStruct.comp (self.δ X X') (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (self.map f) (self.obj X')) h) = CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.whiskerRight f X')) (CategoryTheory.CategoryStruct.comp (self.δ Y X') h)

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@[simp]

theorem CategoryTheory.OplaxMonoidalFunctor.δ_natural_right_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.OplaxMonoidalFunctor C D) {X : C} {Y : C} (X' : C) (f : X ⟶ Y) {Z : D} (h : CategoryTheory.MonoidalCategory.tensorObj (self.obj X') (self.obj Y) ⟶ Z) :

CategoryTheory.CategoryStruct.comp (self.δ X' X) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X') (self.map f)) h) = CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.whiskerLeft X' f)) (CategoryTheory.CategoryStruct.comp (self.δ X' Y) h)

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theorem CategoryTheory.OplaxMonoidalFunctor.associativity_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.OplaxMonoidalFunctor C D) (X : C) (Y : C) (Z : C) {Z : D} (h : CategoryTheory.MonoidalCategory.tensorObj (self.obj X) (CategoryTheory.MonoidalCategory.tensorObj (self.obj Y) (self.obj Z✝)) ⟶ Z) :

CategoryTheory.CategoryStruct.comp (self.δ (CategoryTheory.MonoidalCategory.tensorObj X Y) Z✝) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (self.δ X Y) (self.obj Z✝)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator (self.obj X) (self.obj Y) (self.obj Z✝)).hom h)) = CategoryTheory.CategoryStruct.comp (self.map (CategoryTheory.MonoidalCategory.associator X Y Z✝).hom) (CategoryTheory.CategoryStruct.comp (self.δ X (CategoryTheory.MonoidalCategory.tensorObj Y Z✝)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X) (self.δ Y Z✝)) h))

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@[simp]

theorem CategoryTheory.OplaxMonoidalFunctor.δ_natural_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.OplaxMonoidalFunctor C D) {X : C} {Y : C} {X' : C} {Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') {Z : D} (h : CategoryTheory.MonoidalCategory.tensorObj (F.obj Y) (F.obj Y') ⟶ Z) :

CategoryTheory.CategoryStruct.comp (F.δ X X') (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.tensorHom (F.map f) (F.map g)) h) = CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategory.tensorHom f g)) (CategoryTheory.CategoryStruct.comp (F.δ Y Y') h)

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@[simp]

theorem CategoryTheory.OplaxMonoidalFunctor.δ_natural {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.OplaxMonoidalFunctor C D) {X : C} {Y : C} {X' : C} {Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :

CategoryTheory.CategoryStruct.comp (F.δ X X') (CategoryTheory.MonoidalCategory.tensorHom (F.map f) (F.map g)) = CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategory.tensorHom f g)) (F.δ Y Y')

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theorem CategoryTheory.OplaxMonoidalFunctor.left_unitality_hom_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.OplaxMonoidalFunctor C D) (X : C) {Z : D} (h : F.obj X ⟶ Z) :

CategoryTheory.CategoryStruct.comp (F.δ (𝟙_ C) X) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight F.η (F.obj X)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.leftUnitor (F.obj X)).hom h)) = CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategory.leftUnitor X).hom) h

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theorem CategoryTheory.OplaxMonoidalFunctor.left_unitality_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.OplaxMonoidalFunctor C D) (X : C) :

CategoryTheory.CategoryStruct.comp (F.δ (𝟙_ C) X) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight F.η (F.obj X)) (CategoryTheory.MonoidalCategory.leftUnitor (F.obj X)).hom) = F.map (CategoryTheory.MonoidalCategory.leftUnitor X).hom

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@[simp]

theorem CategoryTheory.OplaxMonoidalFunctor.right_unitality_hom_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.OplaxMonoidalFunctor C D) (X : C) {Z : D} (h : F.obj X ⟶ Z) :

CategoryTheory.CategoryStruct.comp (F.δ X (𝟙_ C)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (F.obj X) F.η) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.rightUnitor (F.obj X)).hom h)) = CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategory.rightUnitor X).hom) h

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@[simp]

theorem CategoryTheory.OplaxMonoidalFunctor.right_unitality_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.OplaxMonoidalFunctor C D) (X : C) :

CategoryTheory.CategoryStruct.comp (F.δ X (𝟙_ C)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (F.obj X) F.η) (CategoryTheory.MonoidalCategory.rightUnitor (F.obj X)).hom) = F.map (CategoryTheory.MonoidalCategory.rightUnitor X).hom

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@[simp]

theorem CategoryTheory.OplaxMonoidalFunctor.associativity_inv_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.OplaxMonoidalFunctor C D) (X : C) (Y : C) (Z : C) {Z : D} (h : CategoryTheory.MonoidalCategory.tensorObj (CategoryTheory.MonoidalCategory.tensorObj (F.obj X) (F.obj Y)) (F.obj Z✝) ⟶ Z) :

CategoryTheory.CategoryStruct.comp (F.δ X (CategoryTheory.MonoidalCategory.tensorObj Y Z✝)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (F.obj X) (F.δ Y Z✝)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator (F.obj X) (F.obj Y) (F.obj Z✝)).inv h)) = CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategory.associator X Y Z✝).inv) (CategoryTheory.CategoryStruct.comp (F.δ (CategoryTheory.MonoidalCategory.tensorObj X Y) Z✝) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (F.δ X Y) (F.obj Z✝)) h))

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@[simp]

theorem CategoryTheory.OplaxMonoidalFunctor.associativity_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.OplaxMonoidalFunctor C D) (X : C) (Y : C) (Z : C) :

CategoryTheory.CategoryStruct.comp (F.δ X (CategoryTheory.MonoidalCategory.tensorObj Y Z)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (F.obj X) (F.δ Y Z)) (CategoryTheory.MonoidalCategory.associator (F.obj X) (F.obj Y) (F.obj Z)).inv) = CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategory.associator X Y Z).inv) (CategoryTheory.CategoryStruct.comp (F.δ (CategoryTheory.MonoidalCategory.tensorObj X Y) Z) (CategoryTheory.MonoidalCategory.whiskerRight (F.δ X Y) (F.obj Z)))

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structure CategoryTheory.MonoidalFunctor (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] extends CategoryTheory.LaxMonoidalFunctor :

Type (max (max (max u₁ u₂) v₁) v₂)

A monoidal functor is a lax monoidal functor for which the tensorator and unitor are isomorphisms.

See https://stacks.math.columbia.edu/tag/0FFL.

obj : C → D
map : {X Y : C} → (X ⟶ Y) → (self.obj X ⟶ self.obj Y)
map_id : ∀ (X : C), self.map (CategoryTheory.CategoryStruct.id X) = CategoryTheory.CategoryStruct.id (self.obj X)
map_comp : ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), self.map (CategoryTheory.CategoryStruct.comp f g) = CategoryTheory.CategoryStruct.comp (self.map f) (self.map g)
ε : 𝟙_ D ⟶ self.obj (𝟙_ C)
μ : (X Y : C) → CategoryTheory.MonoidalCategory.tensorObj (self.obj X) (self.obj Y) ⟶ self.obj (CategoryTheory.MonoidalCategory.tensorObj X Y)
μ_natural_left : ∀ {X Y : C} (f : X ⟶ Y) (X' : C), CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (self.map f) (self.obj X')) (self.μ Y X') = CategoryTheory.CategoryStruct.comp (self.μ X X') (self.map (CategoryTheory.MonoidalCategory.whiskerRight f X'))
μ_natural_right : ∀ {X Y : C} (X' : C) (f : X ⟶ Y), CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X') (self.map f)) (self.μ X' Y) = CategoryTheory.CategoryStruct.comp (self.μ X' X) (self.map (CategoryTheory.MonoidalCategory.whiskerLeft X' f))
associativity : ∀ (X Y Z : C), CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (self.μ X Y) (self.obj Z)) (CategoryTheory.CategoryStruct.comp (self.μ (CategoryTheory.MonoidalCategory.tensorObj X Y) Z) (self.map (CategoryTheory.MonoidalCategory.associator X Y Z).hom)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator (self.obj X) (self.obj Y) (self.obj Z)).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X) (self.μ Y Z)) (self.μ X (CategoryTheory.MonoidalCategory.tensorObj Y Z)))
left_unitality : ∀ (X : C), (CategoryTheory.MonoidalCategory.leftUnitor (self.obj X)).hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight self.ε (self.obj X)) (CategoryTheory.CategoryStruct.comp (self.μ (𝟙_ C) X) (self.map (CategoryTheory.MonoidalCategory.leftUnitor X).hom))
right_unitality : ∀ (X : C), (CategoryTheory.MonoidalCategory.rightUnitor (self.obj X)).hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (self.obj X) self.ε) (CategoryTheory.CategoryStruct.comp (self.μ X (𝟙_ C)) (self.map (CategoryTheory.MonoidalCategory.rightUnitor X).hom))
ε_isIso : CategoryTheory.IsIso self.ε
μ_isIso : ∀ (X Y : C), CategoryTheory.IsIso (self.μ X Y)

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theorem CategoryTheory.MonoidalFunctor.ε_isIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.MonoidalFunctor C D) :

CategoryTheory.IsIso self.ε

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theorem CategoryTheory.MonoidalFunctor.μ_isIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (self : CategoryTheory.MonoidalFunctor C D) (X : C) (Y : C) :

CategoryTheory.IsIso (self.μ X Y)

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noncomputable def CategoryTheory.MonoidalFunctor.εIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) :

𝟙_ D ≅ F.obj (𝟙_ C)

The unit morphism of a (strong) monoidal functor as an isomorphism.

Equations

F.εIso = CategoryTheory.asIso F.ε

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noncomputable def CategoryTheory.MonoidalFunctor.μIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) (Y : C) :

CategoryTheory.MonoidalCategory.tensorObj (F.obj X) (F.obj Y) ≅ F.obj (CategoryTheory.MonoidalCategory.tensorObj X Y)

The tensorator of a (strong) monoidal functor as an isomorphism.

Equations

F.μIso X Y = CategoryTheory.asIso (F.μ X Y)

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theorem CategoryTheory.MonoidalFunctor.toOplaxMonoidalFunctor_δ {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) (Y : C) :

F.toOplaxMonoidalFunctor.δ X Y = CategoryTheory.inv (F.μ X Y)

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theorem CategoryTheory.MonoidalFunctor.toOplaxMonoidalFunctor_toFunctor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) :

F.toOplaxMonoidalFunctor.toFunctor = F.toFunctor

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theorem CategoryTheory.MonoidalFunctor.toOplaxMonoidalFunctor_η {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) :

F.toOplaxMonoidalFunctor.η = CategoryTheory.inv F.ε

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noncomputable def CategoryTheory.MonoidalFunctor.toOplaxMonoidalFunctor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) :

CategoryTheory.OplaxMonoidalFunctor C D

The underlying oplax monoidal functor of a (strong) monoidal functor.

Equations

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

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theorem CategoryTheory.LaxMonoidalFunctor.id_μ (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C) (Y : C) :

(CategoryTheory.LaxMonoidalFunctor.id C).μ X Y = CategoryTheory.CategoryStruct.id (CategoryTheory.MonoidalCategory.tensorObj ((CategoryTheory.Functor.id C).obj X) ((CategoryTheory.Functor.id C).obj Y))

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theorem CategoryTheory.LaxMonoidalFunctor.id_toFunctor (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(CategoryTheory.LaxMonoidalFunctor.id C).toFunctor = CategoryTheory.Functor.id C

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theorem CategoryTheory.LaxMonoidalFunctor.id_ε (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(CategoryTheory.LaxMonoidalFunctor.id C).ε = CategoryTheory.CategoryStruct.id (𝟙_ C)

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def CategoryTheory.LaxMonoidalFunctor.id (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

CategoryTheory.LaxMonoidalFunctor C C

The identity lax monoidal functor.

Equations

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

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instance CategoryTheory.LaxMonoidalFunctor.instInhabited (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

Inhabited (CategoryTheory.LaxMonoidalFunctor C C)

Equations

CategoryTheory.LaxMonoidalFunctor.instInhabited C = { default := CategoryTheory.LaxMonoidalFunctor.id C }

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theorem CategoryTheory.OplaxMonoidalFunctor.id_δ (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C) (Y : C) :

(CategoryTheory.OplaxMonoidalFunctor.id C).δ X Y = CategoryTheory.CategoryStruct.id ((CategoryTheory.Functor.id C).obj (CategoryTheory.MonoidalCategory.tensorObj X Y))

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theorem CategoryTheory.OplaxMonoidalFunctor.id_toFunctor (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(CategoryTheory.OplaxMonoidalFunctor.id C).toFunctor = CategoryTheory.Functor.id C

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theorem CategoryTheory.OplaxMonoidalFunctor.id_η (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(CategoryTheory.OplaxMonoidalFunctor.id C).η = CategoryTheory.CategoryStruct.id ((CategoryTheory.Functor.id C).obj (𝟙_ C))

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def CategoryTheory.OplaxMonoidalFunctor.id (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

CategoryTheory.OplaxMonoidalFunctor C C

The identity lax monoidal functor.

Equations

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

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instance CategoryTheory.OplaxMonoidalFunctor.instInhabited (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

Inhabited (CategoryTheory.OplaxMonoidalFunctor C C)

Equations

CategoryTheory.OplaxMonoidalFunctor.instInhabited C = { default := CategoryTheory.OplaxMonoidalFunctor.id C }

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theorem CategoryTheory.MonoidalFunctor.map_tensor_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) {X : C} {Y : C} {X' : C} {Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') {Z : D} (h : F.obj (CategoryTheory.MonoidalCategory.tensorObj Y Y') ⟶ Z) :

CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategory.tensorHom f g)) h = CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (F.μ X X')) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.tensorHom (F.map f) (F.map g)) (CategoryTheory.CategoryStruct.comp (F.μ Y Y') h))

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theorem CategoryTheory.MonoidalFunctor.map_tensor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) {X : C} {Y : C} {X' : C} {Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :

F.map (CategoryTheory.MonoidalCategory.tensorHom f g) = CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (F.μ X X')) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.tensorHom (F.map f) (F.map g)) (F.μ Y Y'))

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theorem CategoryTheory.MonoidalFunctor.map_whiskerLeft_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) {Y : C} {Z : C} (f : Y ⟶ Z✝) {Z : D} (h : F.obj (CategoryTheory.MonoidalCategory.tensorObj X Z✝) ⟶ Z) :

CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategory.whiskerLeft X f)) h = CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (F.μ X Y)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (F.obj X) (F.map f)) (CategoryTheory.CategoryStruct.comp (F.μ X Z✝) h))

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theorem CategoryTheory.MonoidalFunctor.map_whiskerLeft {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) {Y : C} {Z : C} (f : Y ⟶ Z) :

F.map (CategoryTheory.MonoidalCategory.whiskerLeft X f) = CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (F.μ X Y)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (F.obj X) (F.map f)) (F.μ X Z))

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theorem CategoryTheory.MonoidalFunctor.map_whiskerRight_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) {X : C} {Y : C} (f : X ⟶ Y) (Z : C) {Z : D} (h : F.obj (CategoryTheory.MonoidalCategory.tensorObj Y Z✝) ⟶ Z) :

CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategory.whiskerRight f Z✝)) h = CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (F.μ X Z✝)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (F.map f) (F.obj Z✝)) (CategoryTheory.CategoryStruct.comp (F.μ Y Z✝) h))

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theorem CategoryTheory.MonoidalFunctor.map_whiskerRight {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) {X : C} {Y : C} (f : X ⟶ Y) (Z : C) :

F.map (CategoryTheory.MonoidalCategory.whiskerRight f Z) = CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (F.μ X Z)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (F.map f) (F.obj Z)) (F.μ Y Z))

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theorem CategoryTheory.MonoidalFunctor.map_leftUnitor_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) {Z : D} (h : F.obj X ⟶ Z) :

CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategory.leftUnitor X).hom) h = CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (F.μ (𝟙_ C) X)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (CategoryTheory.inv F.ε) (F.obj X)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.leftUnitor (F.obj X)).hom h))

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theorem CategoryTheory.MonoidalFunctor.map_leftUnitor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) :

F.map (CategoryTheory.MonoidalCategory.leftUnitor X).hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (F.μ (𝟙_ C) X)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (CategoryTheory.inv F.ε) (F.obj X)) (CategoryTheory.MonoidalCategory.leftUnitor (F.obj X)).hom)

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theorem CategoryTheory.MonoidalFunctor.map_rightUnitor_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) {Z : D} (h : F.obj X ⟶ Z) :

CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategory.rightUnitor X).hom) h = CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (F.μ X (𝟙_ C))) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (F.obj X) (CategoryTheory.inv F.ε)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.rightUnitor (F.obj X)).hom h))

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theorem CategoryTheory.MonoidalFunctor.map_rightUnitor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) :

F.map (CategoryTheory.MonoidalCategory.rightUnitor X).hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (F.μ X (𝟙_ C))) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (F.obj X) (CategoryTheory.inv F.ε)) (CategoryTheory.MonoidalCategory.rightUnitor (F.obj X)).hom)

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noncomputable def CategoryTheory.MonoidalFunctor.μNatIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) :

(F.prod F.toFunctor).comp (CategoryTheory.MonoidalCategory.tensor D) ≅ (CategoryTheory.MonoidalCategory.tensor C).comp F.toFunctor

The tensorator as a natural isomorphism.

Equations

F.μNatIso = CategoryTheory.NatIso.ofComponents (fun (X : C × C) => F.μIso X.1 X.2) ⋯

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@[simp]

theorem CategoryTheory.MonoidalFunctor.μIso_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) (Y : C) :

(F.μIso X Y).hom = F.μ X Y

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@[simp]

theorem CategoryTheory.MonoidalFunctor.μ_inv_hom_id_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) (Y : C) {Z : D} (h : F.obj (CategoryTheory.MonoidalCategory.tensorObj X Y) ⟶ Z) :

CategoryTheory.CategoryStruct.comp (F.μIso X Y).inv (CategoryTheory.CategoryStruct.comp (F.μ X Y) h) = h

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@[simp]

theorem CategoryTheory.MonoidalFunctor.μ_inv_hom_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) (Y : C) :

CategoryTheory.CategoryStruct.comp (F.μIso X Y).inv (F.μ X Y) = CategoryTheory.CategoryStruct.id (F.obj (CategoryTheory.MonoidalCategory.tensorObj X Y))

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@[simp]

theorem CategoryTheory.MonoidalFunctor.μ_hom_inv_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) (Y : C) :

CategoryTheory.CategoryStruct.comp (F.μ X Y) (F.μIso X Y).inv = CategoryTheory.CategoryStruct.id (CategoryTheory.MonoidalCategory.tensorObj (F.obj X) (F.obj Y))

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@[simp]

theorem CategoryTheory.MonoidalFunctor.εIso_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) :

F.εIso.hom = F.ε

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@[simp]

theorem CategoryTheory.MonoidalFunctor.ε_inv_hom_id_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) {Z : D} (h : F.obj (𝟙_ C) ⟶ Z) :

CategoryTheory.CategoryStruct.comp F.εIso.inv (CategoryTheory.CategoryStruct.comp F.ε h) = h

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@[simp]

theorem CategoryTheory.MonoidalFunctor.ε_inv_hom_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) :

CategoryTheory.CategoryStruct.comp F.εIso.inv F.ε = CategoryTheory.CategoryStruct.id (F.obj (𝟙_ C))

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@[simp]

theorem CategoryTheory.MonoidalFunctor.ε_hom_inv_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) :

CategoryTheory.CategoryStruct.comp F.ε F.εIso.inv = CategoryTheory.CategoryStruct.id (𝟙_ D)

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@[simp]

theorem CategoryTheory.MonoidalFunctor.commTensorLeft_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) (X : C) :

(F.commTensorLeft X✝).inv.app X = (F.μIso X✝ X).inv

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@[simp]

theorem CategoryTheory.MonoidalFunctor.commTensorLeft_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) (X : C) :

(F.commTensorLeft X✝).hom.app X = F.μ X✝ X

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noncomputable def CategoryTheory.MonoidalFunctor.commTensorLeft {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) :

F.comp (CategoryTheory.MonoidalCategory.tensorLeft (F.obj X)) ≅ (CategoryTheory.MonoidalCategory.tensorLeft X).comp F.toFunctor

Monoidal functors commute with left tensoring up to isomorphism

Equations

F.commTensorLeft X = CategoryTheory.NatIso.ofComponents (fun (Y : C) => F.μIso X Y) ⋯

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@[simp]

theorem CategoryTheory.MonoidalFunctor.commTensorRight_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) (X : C) :

(F.commTensorRight X✝).inv.app X = (F.μIso X X✝).inv

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@[simp]

theorem CategoryTheory.MonoidalFunctor.commTensorRight_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) (X : C) :

(F.commTensorRight X✝).hom.app X = F.μ X X✝

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noncomputable def CategoryTheory.MonoidalFunctor.commTensorRight {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) (X : C) :

F.comp (CategoryTheory.MonoidalCategory.tensorRight (F.obj X)) ≅ (CategoryTheory.MonoidalCategory.tensorRight X).comp F.toFunctor

Monoidal functors commute with right tensoring up to isomorphism

Equations

F.commTensorRight X = CategoryTheory.NatIso.ofComponents (fun (Y : C) => F.μIso Y X) ⋯

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@[simp]

theorem CategoryTheory.MonoidalFunctor.id_toLaxMonoidalFunctor_toFunctor (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(CategoryTheory.MonoidalFunctor.id C).toFunctor = CategoryTheory.Functor.id C

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@[simp]

theorem CategoryTheory.MonoidalFunctor.id_toLaxMonoidalFunctor_ε (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(CategoryTheory.MonoidalFunctor.id C).ε = CategoryTheory.CategoryStruct.id (𝟙_ C)

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@[simp]

theorem CategoryTheory.MonoidalFunctor.id_toLaxMonoidalFunctor_μ (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C) (Y : C) :

(CategoryTheory.MonoidalFunctor.id C).μ X Y = CategoryTheory.CategoryStruct.id (CategoryTheory.MonoidalCategory.tensorObj ((CategoryTheory.Functor.id C).obj X) ((CategoryTheory.Functor.id C).obj Y))

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def CategoryTheory.MonoidalFunctor.id (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

CategoryTheory.MonoidalFunctor C C

The identity monoidal functor.

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

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instance CategoryTheory.MonoidalFunctor.instInhabited (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

Inhabited (CategoryTheory.MonoidalFunctor C C)

Equations

CategoryTheory.MonoidalFunctor.instInhabited C = { default := CategoryTheory.MonoidalFunctor.id C }

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@[simp]

theorem CategoryTheory.LaxMonoidalFunctor.comp_ε {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.LaxMonoidalFunctor C D) (G : CategoryTheory.LaxMonoidalFunctor D E) :

(F ⊗⋙ G).ε = CategoryTheory.CategoryStruct.comp G.ε (G.map F.ε)

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@[simp]

theorem CategoryTheory.LaxMonoidalFunctor.comp_toFunctor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.LaxMonoidalFunctor C D) (G : CategoryTheory.LaxMonoidalFunctor D E) :

(F ⊗⋙ G).toFunctor = F.comp G.toFunctor

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@[simp]

theorem CategoryTheory.LaxMonoidalFunctor.comp_μ {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.LaxMonoidalFunctor C D) (G : CategoryTheory.LaxMonoidalFunctor D E) (X : C) (Y : C) :

(F ⊗⋙ G).μ X Y = CategoryTheory.CategoryStruct.comp (G.μ (F.obj X) (F.obj Y)) (G.map (F.μ X Y))

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def CategoryTheory.LaxMonoidalFunctor.comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.LaxMonoidalFunctor C D) (G : CategoryTheory.LaxMonoidalFunctor D E) :

CategoryTheory.LaxMonoidalFunctor C E

The composition of two lax monoidal functors is again lax monoidal.

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

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def CategoryTheory.LaxMonoidalFunctor.«term_⊗⋙_» :

Lean.TrailingParserDescr

The composition of two lax monoidal functors is again lax monoidal.

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

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theorem CategoryTheory.OplaxMonoidalFunctor.comp_δ {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.OplaxMonoidalFunctor C D) (G : CategoryTheory.OplaxMonoidalFunctor D E) (X : C) (Y : C) :

(F ⊗⋙ G).δ X Y = CategoryTheory.CategoryStruct.comp (G.map (F.δ X Y)) (G.δ (F.obj X) (F.obj Y))

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@[simp]

theorem CategoryTheory.OplaxMonoidalFunctor.comp_toFunctor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.OplaxMonoidalFunctor C D) (G : CategoryTheory.OplaxMonoidalFunctor D E) :

(F ⊗⋙ G).toFunctor = F.comp G.toFunctor

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@[simp]

theorem CategoryTheory.OplaxMonoidalFunctor.comp_η {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.OplaxMonoidalFunctor C D) (G : CategoryTheory.OplaxMonoidalFunctor D E) :

(F ⊗⋙ G).η = CategoryTheory.CategoryStruct.comp (G.map F.η) G.η

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def CategoryTheory.OplaxMonoidalFunctor.comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.OplaxMonoidalFunctor C D) (G : CategoryTheory.OplaxMonoidalFunctor D E) :

CategoryTheory.OplaxMonoidalFunctor C E

The composition of two oplax monoidal functors is again oplax monoidal.

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

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def CategoryTheory.OplaxMonoidalFunctor.«term_⊗⋙_» :

Lean.TrailingParserDescr

The composition of two oplax monoidal functors is again oplax monoidal.

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

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@[simp]

theorem CategoryTheory.LaxMonoidalFunctor.prod_μ {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] {B : Type u₀} [CategoryTheory.Category.{v₀, u₀} B] [CategoryTheory.MonoidalCategory B] (F : CategoryTheory.LaxMonoidalFunctor B C) (G : CategoryTheory.LaxMonoidalFunctor D E) (X : B × D) (Y : B × D) :

(F.prod G).μ X Y = (F.μ X.1 Y.1, G.μ X.2 Y.2)

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@[simp]

theorem CategoryTheory.LaxMonoidalFunctor.prod_ε {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] {B : Type u₀} [CategoryTheory.Category.{v₀, u₀} B] [CategoryTheory.MonoidalCategory B] (F : CategoryTheory.LaxMonoidalFunctor B C) (G : CategoryTheory.LaxMonoidalFunctor D E) :

(F.prod G).ε = (F.ε, G.ε)

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@[simp]

theorem CategoryTheory.LaxMonoidalFunctor.prod_toFunctor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] {B : Type u₀} [CategoryTheory.Category.{v₀, u₀} B] [CategoryTheory.MonoidalCategory B] (F : CategoryTheory.LaxMonoidalFunctor B C) (G : CategoryTheory.LaxMonoidalFunctor D E) :

(F.prod G).toFunctor = F.prod G.toFunctor

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def CategoryTheory.LaxMonoidalFunctor.prod {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] {B : Type u₀} [CategoryTheory.Category.{v₀, u₀} B] [CategoryTheory.MonoidalCategory B] (F : CategoryTheory.LaxMonoidalFunctor B C) (G : CategoryTheory.LaxMonoidalFunctor D E) :

CategoryTheory.LaxMonoidalFunctor (B × D) (C × E)

The cartesian product of two lax monoidal functors is lax monoidal.

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@[simp]

theorem CategoryTheory.MonoidalFunctor.diag_toLaxMonoidalFunctor_ε (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(CategoryTheory.MonoidalFunctor.diag C).ε = CategoryTheory.CategoryStruct.id (𝟙_ (C × C))

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@[simp]

theorem CategoryTheory.MonoidalFunctor.diag_toLaxMonoidalFunctor_toFunctor (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(CategoryTheory.MonoidalFunctor.diag C).toFunctor = CategoryTheory.Functor.diag C

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@[simp]

theorem CategoryTheory.MonoidalFunctor.diag_toLaxMonoidalFunctor_μ (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C) (Y : C) :

(CategoryTheory.MonoidalFunctor.diag C).μ X Y = CategoryTheory.CategoryStruct.id (CategoryTheory.MonoidalCategory.tensorObj ((CategoryTheory.Functor.diag C).obj X) ((CategoryTheory.Functor.diag C).obj Y))

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def CategoryTheory.MonoidalFunctor.diag (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

CategoryTheory.MonoidalFunctor C (C × C)

The diagonal functor as a monoidal functor.

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

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def CategoryTheory.LaxMonoidalFunctor.prod' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.LaxMonoidalFunctor C D) (G : CategoryTheory.LaxMonoidalFunctor C E) :

CategoryTheory.LaxMonoidalFunctor C (D × E)

The cartesian product of two lax monoidal functors starting from the same monoidal category C is lax monoidal.

Equations

F.prod' G = (CategoryTheory.MonoidalFunctor.diag C).toLaxMonoidalFunctor ⊗⋙ F.prod G

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@[simp]

theorem CategoryTheory.LaxMonoidalFunctor.prod'_toFunctor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.LaxMonoidalFunctor C D) (G : CategoryTheory.LaxMonoidalFunctor C E) :

(F.prod' G).toFunctor = F.prod' G.toFunctor

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theorem CategoryTheory.LaxMonoidalFunctor.prod'_ε {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.LaxMonoidalFunctor C D) (G : CategoryTheory.LaxMonoidalFunctor C E) :

(F.prod' G).ε = (F.ε, G.ε)

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theorem CategoryTheory.LaxMonoidalFunctor.prod'_μ {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.LaxMonoidalFunctor C D) (G : CategoryTheory.LaxMonoidalFunctor C E) (X : C) (Y : C) :

(F.prod' G).μ X Y = (F.μ X Y, G.μ X Y)

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theorem CategoryTheory.MonoidalFunctor.comp_toLaxMonoidalFunctor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.MonoidalFunctor C D) (G : CategoryTheory.MonoidalFunctor D E) :

(F ⊗⋙ G).toLaxMonoidalFunctor = F.toLaxMonoidalFunctor ⊗⋙ G.toLaxMonoidalFunctor

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def CategoryTheory.MonoidalFunctor.comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.MonoidalFunctor C D) (G : CategoryTheory.MonoidalFunctor D E) :

CategoryTheory.MonoidalFunctor C E

The composition of two monoidal functors is again monoidal.

Equations

F ⊗⋙ G = let __src := F.toLaxMonoidalFunctor ⊗⋙ G.toLaxMonoidalFunctor; { toLaxMonoidalFunctor := __src, ε_isIso := ⋯, μ_isIso := ⋯ }

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def CategoryTheory.MonoidalFunctor.«term_⊗⋙_» :

Lean.TrailingParserDescr

The composition of two monoidal functors is again monoidal.

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

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theorem CategoryTheory.MonoidalFunctor.prod_toLaxMonoidalFunctor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] {B : Type u₀} [CategoryTheory.Category.{v₀, u₀} B] [CategoryTheory.MonoidalCategory B] (F : CategoryTheory.MonoidalFunctor B C) (G : CategoryTheory.MonoidalFunctor D E) :

(F.prod G).toLaxMonoidalFunctor = F.prod G.toLaxMonoidalFunctor

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def CategoryTheory.MonoidalFunctor.prod {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] {B : Type u₀} [CategoryTheory.Category.{v₀, u₀} B] [CategoryTheory.MonoidalCategory B] (F : CategoryTheory.MonoidalFunctor B C) (G : CategoryTheory.MonoidalFunctor D E) :

CategoryTheory.MonoidalFunctor (B × D) (C × E)

The cartesian product of two monoidal functors is monoidal.

Equations

F.prod G = let __src := F.prod G.toLaxMonoidalFunctor; { toLaxMonoidalFunctor := __src, ε_isIso := ⋯, μ_isIso := ⋯ }

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def CategoryTheory.MonoidalFunctor.prod' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.MonoidalFunctor C D) (G : CategoryTheory.MonoidalFunctor C E) :

CategoryTheory.MonoidalFunctor C (D × E)

The cartesian product of two monoidal functors starting from the same monoidal category C is monoidal.

Equations

F.prod' G = CategoryTheory.MonoidalFunctor.diag C ⊗⋙ F.prod G

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theorem CategoryTheory.MonoidalFunctor.prod'_toLaxMonoidalFunctor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] [CategoryTheory.MonoidalCategory E] (F : CategoryTheory.MonoidalFunctor C D) (G : CategoryTheory.MonoidalFunctor C E) :

(F.prod' G).toLaxMonoidalFunctor = F.prod' G.toLaxMonoidalFunctor

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noncomputable def CategoryTheory.monoidalAdjoint {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) {G : CategoryTheory.Functor D C} (h : F.toFunctor ⊣ G) :

CategoryTheory.LaxMonoidalFunctor D C

If we have a right adjoint functor G to a monoidal functor F, then G has a lax monoidal structure as well.

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

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instance CategoryTheory.instIsIsoεMonoidalAdjointOfIsEquivalence {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) {G : CategoryTheory.Functor D C} (h : F.toFunctor ⊣ G) [F.IsEquivalence] :

CategoryTheory.IsIso (CategoryTheory.monoidalAdjoint F h).ε

Equations

⋯ = ⋯

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instance CategoryTheory.instIsIsoμMonoidalAdjointOfIsEquivalence {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) {G : CategoryTheory.Functor D C} (h : F.toFunctor ⊣ G) (X : D) (Y : D) [F.IsEquivalence] :

CategoryTheory.IsIso ((CategoryTheory.monoidalAdjoint F h).μ X Y)

Equations

⋯ = ⋯

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theorem CategoryTheory.monoidalInverse_toLaxMonoidalFunctor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) {G : CategoryTheory.Functor D C} (h : F.toFunctor ⊣ G) [F.IsEquivalence] :

(CategoryTheory.monoidalInverse F h).toLaxMonoidalFunctor = CategoryTheory.monoidalAdjoint F h

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noncomputable def CategoryTheory.monoidalInverse {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : CategoryTheory.MonoidalFunctor C D) {G : CategoryTheory.Functor D C} (h : F.toFunctor ⊣ G) [F.IsEquivalence] :

CategoryTheory.MonoidalFunctor D C

If a monoidal functor F is an equivalence of categories then its inverse is also monoidal.

Equations

CategoryTheory.monoidalInverse F h = { toLaxMonoidalFunctor := CategoryTheory.monoidalAdjoint F h, ε_isIso := ⋯, μ_isIso := ⋯ }

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Documentation

Mathlib.CategoryTheory.Monoidal.Functor

(Lax) monoidal functors #

References #