Documentation

Mathlib.CategoryTheory.Monoidal.Mod_

The category of module objects over a monoid object. #

A morphism of module objects.

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
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

The identity morphism on a module object.

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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.

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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

A monoid object as a module over itself.

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The forgetful functor from module objects to the ambient category.

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

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

An action of a monoid object M on an object X is the data of a map smul : M ⊗ XX that satisfies unitality and associativity with multiplication.

See MulAction for the non-categorical version.

Instances

    The action map

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    • One or more equations did not get rendered due to their size.
    @[reducible, inline]

    The action of a monoid object on itself.

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    Construct an object of Mod_ (Mon_.mk' M) from an object X : C and a Mod_Class M X instance.

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    @[simp]
    theorem Mod_.mk'_X {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (M : C) [Mon_Class M] (X : C) [Mod_Class M X] :
    (mk' M X).X = X