The category of module objects over a monoid object.

structure Mod_ {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Mon_ C) : Type (max u₁ v₁)

A module object for a monoid object, all internal to some monoidal category.

• X : C

  The underlying object in the ambient monoidal category

• act : CategoryTheory.MonoidalCategoryStruct.tensorObj A.X self.X ⟶ self.X

  The action morphism of the module object

• one_act : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight A.one self.X) self.act = (CategoryTheory.MonoidalCategoryStruct.leftUnitor self.X).hom
• assoc : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight A.mul self.X) self.act = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.associator A.X A.X self.X).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft A.X self.act) self.act)

@[simp]

theorem Mod_.one_act_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Mon_ C} (self : Mod_ A) {Z : C} (h : self.X ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight A.one self.X) (CategoryTheory.CategoryStruct.comp self.act h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.leftUnitor self.X).hom h

@[simp]

theorem Mod_.assoc_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Mon_ C} (self : Mod_ A) {Z : C} (h : self.X ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight A.mul self.X) (CategoryTheory.CategoryStruct.comp self.act h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.associator A.X A.X self.X).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft A.X self.act) (CategoryTheory.CategoryStruct.comp self.act h))

theorem Mod_.assoc_flip {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Mon_ C} (M : Mod_ A) : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft A.X M.act) M.act = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.associator A.X A.X M.X).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight A.mul M.X) M.act)

structure Mod_.Hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Mon_ C} (M N : Mod_ A) : Type v₁

A morphism of module objects.

• hom : M.X ⟶ N.X

  The underlying morphism

• act_hom : CategoryTheory.CategoryStruct.comp M.act self.hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft A.X self.hom) N.act

theorem Mod_.Hom.ext_iff {C : Type u₁} {inst✝ : CategoryTheory.Category.{v₁, u₁} C} {inst✝¹ : CategoryTheory.MonoidalCategory C} {A : Mon_ C} {M N : Mod_ A} {x y : M.Hom N} : x = y ↔ x.hom = y.hom

theorem Mod_.Hom.ext {C : Type u₁} {inst✝ : CategoryTheory.Category.{v₁, u₁} C} {inst✝¹ : CategoryTheory.MonoidalCategory C} {A : Mon_ C} {M N : Mod_ A} {x y : M.Hom N} (hom : x.hom = y.hom) : x = y

@[simp]

theorem Mod_.Hom.act_hom_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Mon_ C} {M N : Mod_ A} (self : M.Hom N) {Z : C} (h : N.X ⟶ Z) : CategoryTheory.CategoryStruct.comp M.act (CategoryTheory.CategoryStruct.comp self.hom h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft A.X self.hom) (CategoryTheory.CategoryStruct.comp N.act h)

def Mod_.id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Mon_ C} (M : Mod_ A) : M.Hom M

The identity morphism on a module object.

Equations

• M.id = { hom := CategoryTheory.CategoryStruct.id M.X, act_hom := ⋯ }

@[simp]

theorem Mod_.id_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Mon_ C} (M : Mod_ A) : M.id.hom = CategoryTheory.CategoryStruct.id M.X

instance Mod_.homInhabited {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Mon_ C} (M : Mod_ A) : Inhabited (M.Hom M)

Equations

• M.homInhabited = { default := M.id }

def Mod_.comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Mon_ C} {M N O : Mod_ A} (f : M.Hom N) (g : N.Hom O) : M.Hom O

Composition of module object morphisms.

Equations

• Mod_.comp f g = { hom := CategoryTheory.CategoryStruct.comp f.hom g.hom, act_hom := ⋯ }

@[simp]

theorem Mod_.comp_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Mon_ C} {M N O : Mod_ A} (f : M.Hom N) (g : N.Hom O) : (comp f g).hom = CategoryTheory.CategoryStruct.comp f.hom g.hom

instance Mod_.instCategory {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Mon_ C} : CategoryTheory.Category.{v₁, max u₁ v₁} (Mod_ A)

Equations

• Mod_.instCategory = { Hom := fun (M N : Mod_ A) => M.Hom N, id := Mod_.id, comp := fun {X Y Z : Mod_ A} (f : X.Hom Y) (g : Y.Hom Z) => Mod_.comp f g, id_comp := ⋯, comp_id := ⋯, assoc := ⋯ }

theorem Mod_.hom_ext {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Mon_ C} {M N : Mod_ A} (f₁ f₂ : M ⟶ N) (h : f₁.hom = f₂.hom) : f₁ = f₂

theorem Mod_.hom_ext_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Mon_ C} {M N : Mod_ A} {f₁ f₂ : M ⟶ N} : f₁ = f₂ ↔ f₁.hom = f₂.hom

@[simp]

theorem Mod_.id_hom' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Mon_ C} (M : Mod_ A) : (CategoryTheory.CategoryStruct.id M).hom = CategoryTheory.CategoryStruct.id M.X

@[simp]

theorem Mod_.comp_hom' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Mon_ C} {M N K : Mod_ A} (f : M ⟶ N) (g : N ⟶ K) : (CategoryTheory.CategoryStruct.comp f g).hom = CategoryTheory.CategoryStruct.comp f.hom g.hom

def Mod_.regular {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Mon_ C) : Mod_ A

A monoid object as a module over itself.

Equations

• Mod_.regular A = { X := A.X, act := A.mul, one_act := ⋯, assoc := ⋯ }

@[simp]

theorem Mod_.regular_X {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Mon_ C) : (regular A).X = A.X

@[simp]

theorem Mod_.regular_act {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Mon_ C) : (regular A).act = A.mul

instance Mod_.instInhabited {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Mon_ C) : Inhabited (Mod_ A)

Equations

• Mod_.instInhabited A = { default := Mod_.regular A }

def Mod_.forget {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Mon_ C) : CategoryTheory.Functor (Mod_ A) C

The forgetful functor from module objects to the ambient category.

Equations

• Mod_.forget A = { obj := fun (A_1 : Mod_ A) => A_1.X, map := fun {X Y : Mod_ A} (f : X ⟶ Y) => f.hom, map_id := ⋯, map_comp := ⋯ }

def Mod_.comap {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A B : Mon_ C} (f : A ⟶ B) : CategoryTheory.Functor (Mod_ B) (Mod_ A)

A morphism of monoid objects induces a "restriction" or "comap" functor between the categories of module objects.

Equations

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

@[simp]

theorem Mod_.comap_obj_X {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A B : Mon_ C} (f : A ⟶ B) (M : Mod_ B) : ((comap f).obj M).X = M.X

@[simp]

theorem Mod_.comap_obj_act {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A B : Mon_ C} (f : A ⟶ B) (M : Mod_ B) : ((comap f).obj M).act = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight f.hom M.X) M.act

@[simp]

theorem Mod_.comap_map_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A B : Mon_ C} (f : A ⟶ B) {X✝ Y✝ : Mod_ B} (g : X✝ ⟶ Y✝) : ((comap f).map g).hom = g.hom