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Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic

The monoidal category structure on R-modules #

Mostly this uses existing machinery in LinearAlgebra.TensorProduct. We just need to provide a few small missing pieces to build the MonoidalCategory instance. The SymmetricCategory instance is in Algebra.Category.ModuleCat.Monoidal.Symmetric to reduce imports.

Note the universe level of the modules must be at least the universe level of the ring, so that we have a monoidal unit. For now, we simplify by insisting both universe levels are the same.

We construct the monoidal closed structure on ModuleCat R in Algebra.Category.ModuleCat.Monoidal.Closed.

If you're happy using the bundled ModuleCat R, it may be possible to mostly use this as an interface and not need to interact much with the implementation details.

noncomputable def ModuleCat.MonoidalCategory.tensorObj {R : Type u} [CommRing R] (M : ModuleCat R) (N : ModuleCat R) :

(implementation) tensor product of R-modules

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    (implementation) tensor product of morphisms R-modules

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      (implementation) left whiskering for R-modules

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        (implementation) right whiskering for R-modules

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          theorem ModuleCat.MonoidalCategory.tensor_comp {R : Type u} [CommRing R] {X₁ : ModuleCat R} {Y₁ : ModuleCat R} {Z₁ : ModuleCat R} {X₂ : ModuleCat R} {Y₂ : ModuleCat R} {Z₂ : ModuleCat R} (f₁ : X₁ Y₁) (f₂ : X₂ Y₂) (g₁ : Y₁ Z₁) (g₂ : Y₂ Z₂) :
          noncomputable def ModuleCat.MonoidalCategory.leftUnitor {R : Type u} [CommRing R] (M : ModuleCat R) :

          (implementation) the left unitor for R-modules

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            noncomputable def ModuleCat.MonoidalCategory.rightUnitor {R : Type u} [CommRing R] (M : ModuleCat R) :
            ModuleCat.of R (TensorProduct R (↑M) R) M

            (implementation) the right unitor for R-modules

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

              The associator_naturality and pentagon lemmas below are very slow to elaborate.

              We give them some help by expressing the lemmas first non-categorically, then using convert _aux using 1 to have the elaborator work as little as possible.

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              noncomputable instance ModuleCat.instCommRingCarrierTensorUnit {R : Type u} [CommRing R] :
              CommRing (𝟙_ (ModuleCat R))

              Remind ourselves that the monoidal unit, being just R, is still a commutative ring.

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              @[simp]
              theorem ModuleCat.MonoidalCategory.hom_apply {R : Type u} [CommRing R] {K : ModuleCat R} {L : ModuleCat R} {M : ModuleCat R} {N : ModuleCat R} (f : K L) (g : M N) (k : K) (m : M) :
              @[simp]
              theorem ModuleCat.MonoidalCategory.whiskerLeft_apply {R : Type u} [CommRing R] (L : ModuleCat R) {M : ModuleCat R} {N : ModuleCat R} (f : M N) (l : L) (m : M) :
              @[simp]
              theorem ModuleCat.MonoidalCategory.whiskerRight_apply {R : Type u} [CommRing R] {L : ModuleCat R} {M : ModuleCat R} (f : L M) (N : ModuleCat R) (l : L) (n : N) :
              @[simp]
              theorem ModuleCat.MonoidalCategory.associator_hom_apply {R : Type u} [CommRing R] {M : ModuleCat R} {N : ModuleCat R} {K : ModuleCat R} (m : M) (n : N) (k : K) :
              @[simp]
              theorem ModuleCat.MonoidalCategory.associator_inv_apply {R : Type u} [CommRing R] {M : ModuleCat R} {N : ModuleCat R} {K : ModuleCat R} (m : M) (n : N) (k : K) :
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              • =