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Mathlib.CategoryTheory.Monoidal.Internal.Module

Mon_ (ModuleCat R) ≌ AlgebraCat R #

The category of internal monoid objects in ModuleCat R is equivalent to the category of "native" bundled R-algebras.

Moreover, this equivalence is compatible with the forgetful functors to ModuleCat R.

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noncomputable instance ModuleCat.MonModuleEquivalenceAlgebra.Ring_of_Mon_ {R : Type u} [CommRing R] (A : Mon_ (ModuleCat R)) :
Ring ↑A.X
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noncomputable instance ModuleCat.MonModuleEquivalenceAlgebra.Algebra_of_Mon_ {R : Type u} [CommRing R] (A : Mon_ (ModuleCat R)) :
Algebra R ↑A.X
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@[simp]
theorem ModuleCat.MonModuleEquivalenceAlgebra.algebraMap {R : Type u} [CommRing R] (A : Mon_ (ModuleCat R)) (r : R) :
(_root_.algebraMap R ↑A.X) r = (CategoryTheory.ConcreteCategory.hom A.one) r
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noncomputable def ModuleCat.MonModuleEquivalenceAlgebra.functor {R : Type u} [CommRing R] :
CategoryTheory.Functor (Mon_ (ModuleCat R)) (AlgebraCat R)

Converting a monoid object in ModuleCat R to a bundled algebra.

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    @[simp]
    theorem ModuleCat.MonModuleEquivalenceAlgebra.functor_map_hom_apply {R : Type u} [CommRing R] {x✝ x✝¹ : Mon_ (ModuleCat R)} (f : x✝ ⟶ x✝¹) (a : ↑x✝.X) :
    (AlgebraCat.Hom.hom (functor.map f)) a = (CategoryTheory.ConcreteCategory.hom f.hom) a
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    @[simp]
    theorem ModuleCat.MonModuleEquivalenceAlgebra.functor_obj_carrier {R : Type u} [CommRing R] (A : Mon_ (ModuleCat R)) :
    ↑(functor.obj A) = ↑A.X
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    noncomputable def ModuleCat.MonModuleEquivalenceAlgebra.inverseObj {R : Type u} [CommRing R] (A : AlgebraCat R) :
    Mon_ (ModuleCat R)

    Converting a bundled algebra to a monoid object in ModuleCat R.

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      theorem ModuleCat.MonModuleEquivalenceAlgebra.inverseObj_one {R : Type u} [CommRing R] (A : AlgebraCat R) :
      (inverseObj A).one = ofHom (Algebra.linearMap R ↑A)
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      @[simp]
      theorem ModuleCat.MonModuleEquivalenceAlgebra.inverseObj_mul {R : Type u} [CommRing R] (A : AlgebraCat R) :
      (inverseObj A).mul = ofHom (LinearMap.mul' R ↑A)
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      @[simp]
      theorem ModuleCat.MonModuleEquivalenceAlgebra.inverseObj_X_isModule {R : Type u} [CommRing R] (A : AlgebraCat R) :
      (inverseObj A).X.isModule = Algebra.toModule
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      @[simp]
      theorem ModuleCat.MonModuleEquivalenceAlgebra.inverseObj_X_isAddCommGroup {R : Type u} [CommRing R] (A : AlgebraCat R) :
      (inverseObj A).X.isAddCommGroup = Ring.toAddCommGroup
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      @[simp]
      theorem ModuleCat.MonModuleEquivalenceAlgebra.inverseObj_X_carrier {R : Type u} [CommRing R] (A : AlgebraCat R) :
      ↑(inverseObj A).X = ↑A
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      noncomputable def ModuleCat.MonModuleEquivalenceAlgebra.inverse {R : Type u} [CommRing R] :
      CategoryTheory.Functor (AlgebraCat R) (Mon_ (ModuleCat R))

      Converting a bundled algebra to a monoid object in ModuleCat R.

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        @[simp]
        theorem ModuleCat.MonModuleEquivalenceAlgebra.inverse_map_hom {R : Type u} [CommRing R] {X✝ Y✝ : AlgebraCat R} (f : X✝ ⟶ Y✝) :
        (inverse.map f).hom = ofHom (AlgebraCat.Hom.hom f).toLinearMap
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        @[simp]
        theorem ModuleCat.MonModuleEquivalenceAlgebra.inverse_obj {R : Type u} [CommRing R] (A : AlgebraCat R) :
        inverse.obj A = inverseObj A
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        noncomputable def ModuleCat.monModuleEquivalenceAlgebra {R : Type u} [CommRing R] :
        Mon_ (ModuleCat R) ≌ AlgebraCat R

        The category of internal monoid objects in ModuleCat R is equivalent to the category of "native" bundled R-algebras.

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          noncomputable def ModuleCat.monModuleEquivalenceAlgebraForget {R : Type u} [CommRing R] :
          MonModuleEquivalenceAlgebra.functor.comp (CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R)) ≅ Mon_.forget (ModuleCat R)

          The equivalence Mon_ (ModuleCat R) ≌ AlgebraCat R is naturally compatible with the forgetful functors to ModuleCat R.

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