Mathlib.CategoryTheory.Monoidal.Mod

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class ModObj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] (M : C) [MonObj M] (X : D) :

Type v₂

Given an action of a monoidal category C on a category D, an action of a monoid object M in C on an object X in D is the data of a map smul : M ⊙ₗ X ⟶ X that satisfies unitality and associativity with multiplication.

See MulAction for the non-categorical version.

smul : CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionObj M X ⟶ X
The action map
one_smul' : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomLeft MonObj.one X) smul = (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionUnitIso X).hom
The identity acts trivially.
mul_smul' : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomLeft MonObj.mul X) smul = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionAssocIso M M X).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomRight M smul) smul)
The action map is compatible with multiplication.

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@[deprecated ModObj (since := "2025-09-14")]

def Mod_Class {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] (M : C) [MonObj M] (X : D) :

Type v₂

Alias of ModObj.

Given an action of a monoidal category C on a category D, an action of a monoid object M in C on an object X in D is the data of a map smul : M ⊙ₗ X ⟶ X that satisfies unitality and associativity with multiplication.

See MulAction for the non-categorical version.

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@Mod_Class = @ModObj

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theorem ModObj.one_smul'_assoc {C : Type u₁} {inst✝ : CategoryTheory.Category.{v₁, u₁} C} {inst✝¹ : CategoryTheory.MonoidalCategory C} {D : Type u₂} {inst✝² : CategoryTheory.Category.{v₂, u₂} D} {inst✝³ : CategoryTheory.MonoidalCategory.MonoidalLeftAction C D} {M : C} {inst✝⁴ : MonObj M} (X : D) [self : ModObj M X] {Z : D} (h : X ⟶ Z) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomLeft MonObj.one X) (CategoryTheory.CategoryStruct.comp smul h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionUnitIso X).hom h

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theorem ModObj.mul_smul'_assoc {C : Type u₁} {inst✝ : CategoryTheory.Category.{v₁, u₁} C} {inst✝¹ : CategoryTheory.MonoidalCategory C} {D : Type u₂} {inst✝² : CategoryTheory.Category.{v₂, u₂} D} {inst✝³ : CategoryTheory.MonoidalCategory.MonoidalLeftAction C D} {M : C} {inst✝⁴ : MonObj M} (X : D) [self : ModObj M X] {Z : D} (h : X ⟶ Z) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomLeft MonObj.mul X) (CategoryTheory.CategoryStruct.comp smul h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionAssocIso M M X).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomRight M smul) (CategoryTheory.CategoryStruct.comp smul h))

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def MonObj.termγ :

Lean.ParserDescr

The action map

Equations

MonObj.termγ = Lean.ParserDescr.node `MonObj.termγ 1024 (Lean.ParserDescr.symbol "γ")

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def MonObj.«termγ[_]» :

Lean.ParserDescr

The action map

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

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def MonObj.«termγ[_,_]» :

Lean.ParserDescr

The action map

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

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@[simp]

theorem ModObj.one_smul {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {M : C} [MonObj M] (X : D) [ModObj M X] :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomLeft MonObj.one X) smul = (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionUnitIso X).hom

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@[simp]

theorem ModObj.one_smul_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {M : C} [MonObj M] (X : D) [ModObj M X] {Z : D} (h : X ⟶ Z) :

CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomLeft MonObj.one X) (CategoryTheory.CategoryStruct.comp smul h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionUnitIso X).hom h

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@[simp]

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@[reducible, inline]

abbrev ModObj.regular {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (M : C) [MonObj M] :

ModObj M M

The action of a monoid object on itself.

Equations

ModObj.regular M = { smul := MonObj.mul, one_smul' := ⋯, mul_smul' := ⋯ }

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@[simp]

theorem ModObj.smul_eq_mul {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (M : C) [MonObj M] :

smul = MonObj.mul

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instance ModObj.instTensorUnit {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] (X : D) :

ModObj (CategoryTheory.MonoidalCategoryStruct.tensorUnit C) X

If C acts monoidally on D, then every object of D is canonically a module over the trivial monoid.

Equations

ModObj.instTensorUnit X = { smul := (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionUnitIso X).hom, one_smul' := ⋯, mul_smul' := ⋯ }

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@[simp]

theorem ModObj.smul_def {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] (X : D) :

smul = (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionUnitIso X).hom

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theorem ModObj.ext {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : C} [MonObj M] {X : C} (h₁ h₂ : ModObj M X) (H : smul = smul) :

h₁ = h₂

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theorem ModObj.ext_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : C} [MonObj M] {X : C} {h₁ h₂ : ModObj M X} :

h₁ = h₂ ↔ smul = smul

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class IsMod_Hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] (A : C) [MonObj A] {M N : D} [ModObj A M] [ModObj A N] (f : M ⟶ N) :

Prop

A morphism in D is a morphism of A-module objects if it commutes with the action maps

smul_hom : CategoryTheory.CategoryStruct.comp ModObj.smul f = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomRight A f) ModObj.smul

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@[simp]

theorem IsMod_Hom.smul_hom_assoc {C : Type u₁} {inst✝ : CategoryTheory.Category.{v₁, u₁} C} {inst✝¹ : CategoryTheory.MonoidalCategory C} {D : Type u₂} {inst✝² : CategoryTheory.Category.{v₂, u₂} D} {inst✝³ : CategoryTheory.MonoidalCategory.MonoidalLeftAction C D} {A : C} {inst✝⁴ : MonObj A} {M N : D} {inst✝⁵ : ModObj A M} {inst✝⁶ : ModObj A N} {f : M ⟶ N} [self : IsMod_Hom A f] {Z : D} (h : N ⟶ Z) :

CategoryTheory.CategoryStruct.comp ModObj.smul (CategoryTheory.CategoryStruct.comp f h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomRight A f) (CategoryTheory.CategoryStruct.comp ModObj.smul h)

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instance instIsMod_HomId {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] (A : C) [MonObj A] {M : D} [ModObj A M] :

IsMod_Hom A (CategoryTheory.CategoryStruct.id M)

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instance instIsMod_HomComp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] (A : C) [MonObj A] {M N O : D} [ModObj A M] [ModObj A N] [ModObj A O] (f : M ⟶ N) (g : N ⟶ O) [IsMod_Hom A f] [IsMod_Hom A g] :

IsMod_Hom A (CategoryTheory.CategoryStruct.comp f g)

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instance instIsMod_HomInvOfHom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] (A : C) [MonObj A] {M N : D} [ModObj A M] [ModObj A N] (f : M ≅ N) [IsMod_Hom A f.hom] :

IsMod_Hom A f.inv

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structure Mod_ {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] (A : C) [MonObj A] :

Type (max u₂ v₂)

A module object for a monoid object in a monoidal category acting on the ambient category.

X : D
The underlying object in the ambient category
mod : ModObj A self.X

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structure Mod_.Hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A : C} [MonObj A] (M N : Mod_ D A) :

Type v₂

A morphism of module objects.

hom : M.X ⟶ N.X
The underlying morphism
isMod_Hom : IsMod_Hom A self.hom

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theorem Mod_.Hom.ext_iff {C : Type u₁} {inst✝ : CategoryTheory.Category.{v₁, u₁} C} {inst✝¹ : CategoryTheory.MonoidalCategory C} {D : Type u₂} {inst✝² : CategoryTheory.Category.{v₂, u₂} D} {inst✝³ : CategoryTheory.MonoidalCategory.MonoidalLeftAction C D} {A : C} {inst✝⁴ : MonObj A} {M N : Mod_ D A} {x y : M.Hom N} :

x = y ↔ x.hom = y.hom

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theorem Mod_.Hom.ext {C : Type u₁} {inst✝ : CategoryTheory.Category.{v₁, u₁} C} {inst✝¹ : CategoryTheory.MonoidalCategory C} {D : Type u₂} {inst✝² : CategoryTheory.Category.{v₂, u₂} D} {inst✝³ : CategoryTheory.MonoidalCategory.MonoidalLeftAction C D} {A : C} {inst✝⁴ : MonObj A} {M N : Mod_ D A} {x y : M.Hom N} (hom : x.hom = y.hom) :

x = y

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def Mod_.Hom.mk' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A : C} [MonObj A] {M N : Mod_ D A} (f : M.X ⟶ N.X) (smul_hom : CategoryTheory.CategoryStruct.comp ModObj.smul f = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomRight A f) ModObj.smul := by cat_disch) :

M.Hom N

An alternative constructor for Hom, taking a morphism without a [isMod_Hom] instance, as well as the relevant equality to put such an instance.

Equations

Mod_.Hom.mk' f smul_hom = { hom := f, isMod_Hom := ⋯ }

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@[simp]

theorem Mod_.Hom.mk'_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A : C} [MonObj A] {M N : Mod_ D A} (f : M.X ⟶ N.X) (smul_hom : CategoryTheory.CategoryStruct.comp ModObj.smul f = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomRight A f) ModObj.smul := by cat_disch) :

(mk' f smul_hom).hom = f

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def Mod_.Hom.mk'' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A : C} [MonObj A] {M N : D} [ModObj A M] [ModObj A N] (f : M ⟶ N) (smul_hom : CategoryTheory.CategoryStruct.comp ModObj.smul f = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomRight A f) ModObj.smul := by cat_disch) :

{ X := M, mod := inst✝ }.Hom { X := N, mod := inst✝¹ }

An alternative constructor for Hom, taking a morphism without a [isMod_Hom] instance, between objects with a ModObj instance (rather than bundled as Mod_), as well as the relevant equality to put such an instance.

Equations

Mod_.Hom.mk'' f smul_hom = { hom := f, isMod_Hom := ⋯ }

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theorem Mod_.Hom.mk''_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A : C} [MonObj A] {M N : D} [ModObj A M] [ModObj A N] (f : M ⟶ N) (smul_hom : CategoryTheory.CategoryStruct.comp ModObj.smul f = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomRight A f) ModObj.smul := by cat_disch) :

(mk'' f smul_hom).hom = f

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def Mod_.id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A : C} [MonObj A] (M : Mod_ D A) :

M.Hom M

The identity morphism on a module object.

Equations

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

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theorem Mod_.id_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A : C} [MonObj A] (M : Mod_ D A) :

M.id.hom = CategoryTheory.CategoryStruct.id M.X

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instance Mod_.homInhabited {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A : C} [MonObj A] (M : Mod_ D A) :

Inhabited (M.Hom M)

Equations

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

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def Mod_.comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A : C} [MonObj A] {M N O : Mod_ D 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, isMod_Hom := ⋯ }

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theorem Mod_.comp_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A : C} [MonObj A] {M N O : Mod_ D A} (f : M.Hom N) (g : N.Hom O) :

(comp f g).hom = CategoryTheory.CategoryStruct.comp f.hom g.hom

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instance Mod_.instCategory {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A : C} [MonObj A] :

CategoryTheory.Category.{v₂, max u₂ v₂} (Mod_ D A)

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

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theorem Mod_.hom_ext {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A : C} [MonObj A] {M N : Mod_ D A} (f₁ f₂ : M ⟶ N) (h : f₁.hom = f₂.hom) :

f₁ = f₂

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theorem Mod_.hom_ext_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A : C} [MonObj A] {M N : Mod_ D A} {f₁ f₂ : M ⟶ N} :

f₁ = f₂ ↔ f₁.hom = f₂.hom

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theorem Mod_.id_hom' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A : C} [MonObj A] (M : Mod_ D A) :

(CategoryTheory.CategoryStruct.id M).hom = CategoryTheory.CategoryStruct.id M.X

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theorem Mod_.comp_hom' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A : C} [MonObj A] {M N K : Mod_ D A} (f : M ⟶ N) (g : N ⟶ K) :

(CategoryTheory.CategoryStruct.comp f g).hom = CategoryTheory.CategoryStruct.comp f.hom g.hom

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

Mod_ C A

A monoid object as a module over itself.

Equations

Mod_.regular A = { X := A, mod := ModObj.regular A }

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

(regular A).X = A

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theorem Mod_.regular_mod_smul {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : C) [MonObj A] :

ModObj.smul = MonObj.mul

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

Inhabited (Mod_ C A)

Equations

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

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def Mod_.forget {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] (A : C) [MonObj A] :

CategoryTheory.Functor (Mod_ D A) D

The forgetful functor from module objects to the ambient category.

Equations

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

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theorem Mod_.forget_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] (A : C) [MonObj A] (A✝ : Mod_ D A) :

(forget A).obj A✝ = A✝.X

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theorem Mod_.forget_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] (A : C) [MonObj A] {X✝ Y✝ : Mod_ D A} (f : X✝ ⟶ Y✝) :

(forget A).map f = f.hom

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def Mod_.scalarRestriction {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A B : C} [MonObj A] [MonObj B] (f : A ⟶ B) [IsMonHom f] (M : D) [ModObj B M] :

ModObj A M

When M is a B-module in D and f : A ⟶ B is a morphism of internal monoid objects, M inherits an A-module structure via "restriction of scalars", i.e γ[A, M] = f.hom ⊵ₗ M ≫ γ[B, M].

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

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theorem Mod_.scalarRestriction_smul {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A B : C} [MonObj A] [MonObj B] (f : A ⟶ B) [IsMonHom f] (M : D) [ModObj B M] :

ModObj.smul = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.MonoidalLeftActionStruct.actionHomLeft f M) ModObj.smul

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theorem Mod_.scalarRestriction_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory.MonoidalLeftAction C D] {A B : C} [MonObj A] [MonObj B] (f : A ⟶ B) [IsMonHom f] (M N : D) [ModObj B M] [ModObj B N] (g : M ⟶ N) [IsMod_Hom B g] :

IsMod_Hom A g

If g : M ⟶ N is a B-linear morphisms of B-modules, then it induces an A-linear morphism when M and N have an A-module structure obtained by restricting scalars along a monoid morphism A ⟶ B.

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