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Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed

The monoidal closed structure on Module R. #

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noncomputable def ModuleCat.monoidalClosedHomEquiv {R : Type u} [CommRing R] (M : ModuleCat R) (N : ModuleCat R) (P : ModuleCat R) :
((CategoryTheory.MonoidalCategory.tensorLeft M).obj N ⟶ P) ≃ (N ⟶ ((CategoryTheory.linearCoyoneda R (ModuleCat R)).obj (Opposite.op M)).obj P)

Auxiliary definition for the MonoidalClosed instance on Module R. (This is only a separate definition in order to speed up typechecking. )

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Instances For
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    noncomputable instance ModuleCat.instMonoidalClosedModuleCatToRingModuleCategoryMonoidalCategory {R : Type u} [CommRing R] :
    CategoryTheory.MonoidalClosed (ModuleCat R)
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    theorem ModuleCat.ihom_map_apply {R : Type u} [CommRing R] {M : ModuleCat R} {N : ModuleCat R} {P : ModuleCat R} (f : N ⟶ P) (g : ↑(ModuleCat.of R (M ⟶ N))) :
    ((CategoryTheory.ihom M).map f) g = CategoryTheory.CategoryStruct.comp g f
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    theorem ModuleCat.monoidalClosed_curry {R : Type u} [CommRing R] {M : ModuleCat R} {N : ModuleCat R} {P : ModuleCat R} (f : CategoryTheory.MonoidalCategory.tensorObj M N ⟶ P) (x : ↑M) (y : ↑N) :
    ((CategoryTheory.MonoidalClosed.curry f) y) x = f (x ⊗ₜ[R] y)
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    @[simp]
    theorem ModuleCat.monoidalClosed_uncurry {R : Type u} [CommRing R] {M : ModuleCat R} {N : ModuleCat R} {P : ModuleCat R} (f : N ⟶ (CategoryTheory.ihom M).obj P) (x : ↑M) (y : ↑N) :
    (CategoryTheory.MonoidalClosed.uncurry f) (x ⊗ₜ[R] y) = (f y) x
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    theorem ModuleCat.ihom_ev_app {R : Type u} [CommRing R] (M : ModuleCat R) (N : ModuleCat R) :
    (CategoryTheory.ihom.ev M).app N = (TensorProduct.uncurry R (↑M) (↑M →ₗ[R] ↑N) ↑N) (LinearMap.flip LinearMap.id)

    Describes the counit of the adjunction M ⊗ - ⊣ Hom(M, -). Given an R-module N this should give a map M ⊗ Hom(M, N) ⟶ N, so we flip the order of the arguments in the identity map Hom(M, N) ⟶ (M ⟶ N) and uncurry the resulting map M ⟶ Hom(M, N) ⟶ N.

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    theorem ModuleCat.ihom_coev_app {R : Type u} [CommRing R] (M : ModuleCat R) (N : ModuleCat R) :
    (CategoryTheory.ihom.coev M).app N = LinearMap.flip (TensorProduct.mk R ↑(Opposite.op M).unop ↑((CategoryTheory.Functor.id (ModuleCat R)).obj N))

    Describes the unit of the adjunction M ⊗ - ⊣ Hom(M, -). Given an R-module N this should define a map N ⟶ Hom(M, M ⊗ N), which is given by flipping the arguments in the natural R-bilinear map M ⟶ N ⟶ M ⊗ N.

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    theorem ModuleCat.monoidalClosed_pre_app {R : Type u} [CommRing R] {M : ModuleCat R} {N : ModuleCat R} (P : ModuleCat R) (f : N ⟶ M) :
    (CategoryTheory.MonoidalClosed.pre f).app P = LinearMap.lcomp R (↑P) f