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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) :
(algebraMap R ↑A.X) r = 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_apply {R : Type u} [CommRing R] {A : Mon_ (ModuleCat R)} {B : Mon_ (ModuleCat R)} (f : A ⟶ B) (a : ↑A.X) :
    (ModuleCat.MonModuleEquivalenceAlgebra.functor.toPrefunctor.map f) a = f.hom a
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    @[simp]
    theorem ModuleCat.MonModuleEquivalenceAlgebra.functor_obj_carrier {R : Type u} [CommRing R] (A : Mon_ (ModuleCat R)) :
    ↑(ModuleCat.MonModuleEquivalenceAlgebra.functor.toPrefunctor.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_mul {R : Type u} [CommRing R] (A : AlgebraCat R) :
      (ModuleCat.MonModuleEquivalenceAlgebra.inverseObj A).mul = LinearMap.mul' R ↑A
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      @[simp]
      theorem ModuleCat.MonModuleEquivalenceAlgebra.inverseObj_X {R : Type u} [CommRing R] (A : AlgebraCat R) :
      (ModuleCat.MonModuleEquivalenceAlgebra.inverseObj A).X = ModuleCat.of R ↑A
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      @[simp]
      theorem ModuleCat.MonModuleEquivalenceAlgebra.inverseObj_one {R : Type u} [CommRing R] (A : AlgebraCat R) :
      (ModuleCat.MonModuleEquivalenceAlgebra.inverseObj A).one = Algebra.linearMap R ↑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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        theorem ModuleCat.MonModuleEquivalenceAlgebra.inverse_map_hom {R : Type u} [CommRing R] :
        ∀ {X Y : AlgebraCat R} (f : X ⟶ Y), (ModuleCat.MonModuleEquivalenceAlgebra.inverse.toPrefunctor.map f).hom = AlgHom.toLinearMap f
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        @[simp]
        theorem ModuleCat.MonModuleEquivalenceAlgebra.inverse_obj {R : Type u} [CommRing R] (A : AlgebraCat R) :
        ModuleCat.MonModuleEquivalenceAlgebra.inverse.toPrefunctor.obj A = ModuleCat.MonModuleEquivalenceAlgebra.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] :
          CategoryTheory.Functor.comp ModuleCat.MonModuleEquivalenceAlgebra.functor (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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