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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.

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) :
(algebraMap R A.X) r = A.one r

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

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    @[simp]
    theorem ModuleCat.MonModuleEquivalenceAlgebra.functor_obj_carrier {R : Type u} [CommRing R] (A : Mon_ (ModuleCat R)) :
    (ModuleCat.MonModuleEquivalenceAlgebra.functor.obj A) = A.X
    @[simp]
    theorem ModuleCat.MonModuleEquivalenceAlgebra.functor_map_apply {R : Type u} [CommRing R] {A : Mon_ (ModuleCat R)} {B : Mon_ (ModuleCat R)} (f : A B) (a : A.X) :
    (ModuleCat.MonModuleEquivalenceAlgebra.functor.map f) a = f.hom a

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

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

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        @[simp]
        theorem ModuleCat.MonModuleEquivalenceAlgebra.inverse_obj {R : Type u} [CommRing R] (A : AlgebraCat R) :
        ModuleCat.MonModuleEquivalenceAlgebra.inverse.obj A = ModuleCat.MonModuleEquivalenceAlgebra.inverseObj A
        @[simp]
        theorem ModuleCat.MonModuleEquivalenceAlgebra.inverse_map_hom {R : Type u} [CommRing R] :
        ∀ {X Y : AlgebraCat R} (f : X Y), (ModuleCat.MonModuleEquivalenceAlgebra.inverse.map f).hom = AlgHom.toLinearMap f

        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] :
          ModuleCat.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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