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 Module R in Algebra.Category.ModuleCat.Monoidal.Closed.

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

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

(implementation) tensor product of R-modules

def ModuleCat.MonoidalCategory.tensorHom {R : Type u} [CommRing R] {M : ModuleCat R} {N : ModuleCat R} {M' : ModuleCat R} {N' : ModuleCat R} (f : M ⟶ N) (g : M' ⟶ N') : ModuleCat.MonoidalCategory.tensorObj M M' ⟶ ModuleCat.MonoidalCategory.tensorObj N N'

(implementation) tensor product of morphisms R-modules

def ModuleCat.MonoidalCategory.whiskerLeft {R : Type u} [CommRing R] (M : ModuleCat R) {N₁ : ModuleCat R} {N₂ : ModuleCat R} (f : N₁ ⟶ N₂) : ModuleCat.MonoidalCategory.tensorObj M N₁ ⟶ ModuleCat.MonoidalCategory.tensorObj M N₂

(implementation) left whiskering for R-modules

def ModuleCat.MonoidalCategory.whiskerRight {R : Type u} [CommRing R] {M₁ : ModuleCat R} {M₂ : ModuleCat R} (f : M₁ ⟶ M₂) (N : ModuleCat R) : ModuleCat.MonoidalCategory.tensorObj M₁ N ⟶ ModuleCat.MonoidalCategory.tensorObj M₂ N

(implementation) right whiskering for R-modules

theorem ModuleCat.MonoidalCategory.tensor_id {R : Type u} [CommRing R] (M : ModuleCat R) (N : ModuleCat R) : ModuleCat.MonoidalCategory.tensorHom (CategoryTheory.CategoryStruct.id M) (CategoryTheory.CategoryStruct.id N) = CategoryTheory.CategoryStruct.id (ModuleCat.of R (TensorProduct R ↑M ↑N))

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) : ModuleCat.MonoidalCategory.tensorHom (CategoryTheory.CategoryStruct.comp f₁ g₁) (CategoryTheory.CategoryStruct.comp f₂ g₂) = CategoryTheory.CategoryStruct.comp (ModuleCat.MonoidalCategory.tensorHom f₁ f₂) (ModuleCat.MonoidalCategory.tensorHom g₁ g₂)

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

(implementation) the associator for R-modules

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.

theorem ModuleCat.MonoidalCategory.associator_naturality {R : Type u} [CommRing R] {X₁ : ModuleCat R} {X₂ : ModuleCat R} {X₃ : ModuleCat R} {Y₁ : ModuleCat R} {Y₂ : ModuleCat R} {Y₃ : ModuleCat R} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (f₃ : X₃ ⟶ Y₃) : CategoryTheory.CategoryStruct.comp (ModuleCat.MonoidalCategory.tensorHom (ModuleCat.MonoidalCategory.tensorHom f₁ f₂) f₃) (ModuleCat.MonoidalCategory.associator Y₁ Y₂ Y₃).hom = CategoryTheory.CategoryStruct.comp (ModuleCat.MonoidalCategory.associator X₁ X₂ X₃).hom (ModuleCat.MonoidalCategory.tensorHom f₁ (ModuleCat.MonoidalCategory.tensorHom f₂ f₃))

theorem ModuleCat.MonoidalCategory.pentagon {R : Type u} [CommRing R] (W : ModuleCat R) (X : ModuleCat R) (Y : ModuleCat R) (Z : ModuleCat R) : CategoryTheory.CategoryStruct.comp (ModuleCat.MonoidalCategory.tensorHom (ModuleCat.MonoidalCategory.associator W X Y).hom (CategoryTheory.CategoryStruct.id Z)) (CategoryTheory.CategoryStruct.comp (ModuleCat.MonoidalCategory.associator W (ModuleCat.MonoidalCategory.tensorObj X Y) Z).hom (ModuleCat.MonoidalCategory.tensorHom (CategoryTheory.CategoryStruct.id W) (ModuleCat.MonoidalCategory.associator X Y Z).hom)) = CategoryTheory.CategoryStruct.comp (ModuleCat.MonoidalCategory.associator (ModuleCat.MonoidalCategory.tensorObj W X) Y Z).hom (ModuleCat.MonoidalCategory.associator W X (ModuleCat.MonoidalCategory.tensorObj Y Z)).hom

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

(implementation) the left unitor for R-modules

theorem ModuleCat.MonoidalCategory.leftUnitor_naturality {R : Type u} [CommRing R] {M : ModuleCat R} {N : ModuleCat R} (f : M ⟶ N) : CategoryTheory.CategoryStruct.comp (ModuleCat.MonoidalCategory.tensorHom (CategoryTheory.CategoryStruct.id (ModuleCat.of R R)) f) (ModuleCat.MonoidalCategory.leftUnitor N).hom = CategoryTheory.CategoryStruct.comp (ModuleCat.MonoidalCategory.leftUnitor M).hom f

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

theorem ModuleCat.MonoidalCategory.rightUnitor_naturality {R : Type u} [CommRing R] {M : ModuleCat R} {N : ModuleCat R} (f : M ⟶ N) : CategoryTheory.CategoryStruct.comp (ModuleCat.MonoidalCategory.tensorHom f (CategoryTheory.CategoryStruct.id (ModuleCat.of R R))) (ModuleCat.MonoidalCategory.rightUnitor N).hom = CategoryTheory.CategoryStruct.comp (ModuleCat.MonoidalCategory.rightUnitor M).hom f

theorem ModuleCat.MonoidalCategory.triangle {R : Type u} [CommRing R] (M : ModuleCat R) (N : ModuleCat R) : CategoryTheory.CategoryStruct.comp (ModuleCat.MonoidalCategory.associator M (ModuleCat.of R R) N).hom (ModuleCat.MonoidalCategory.tensorHom (CategoryTheory.CategoryStruct.id M) (ModuleCat.MonoidalCategory.leftUnitor N).hom) = ModuleCat.MonoidalCategory.tensorHom (ModuleCat.MonoidalCategory.rightUnitor M).hom (CategoryTheory.CategoryStruct.id N)

instance ModuleCat.monoidalCategory {R : Type u} [CommRing R] : CategoryTheory.MonoidalCategory (ModuleCat R)

instance ModuleCat.instCommRingCarrierToRingTensorUnitModuleCatModuleCategoryMonoidalCategory {R : Type u} [CommRing R] : CommRing ↑(CategoryTheory.MonoidalCategory.tensorUnit (ModuleCat R))

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

@[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) : ↑(CategoryTheory.MonoidalCategory.tensorHom f g) (k ⊗ₜ[R] m) = ↑f k ⊗ₜ[R] ↑g m

@[simp]

theorem ModuleCat.MonoidalCategory.leftUnitor_hom_apply {R : Type u} [CommRing R] {M : ModuleCat R} (r : R) (m : ↑M) : ↑(CategoryTheory.MonoidalCategory.leftUnitor M).hom (r ⊗ₜ[R] m) = r • m

@[simp]

theorem ModuleCat.MonoidalCategory.leftUnitor_inv_apply {R : Type u} [CommRing R] {M : ModuleCat R} (m : ↑M) : ↑(CategoryTheory.MonoidalCategory.leftUnitor M).inv m = 1 ⊗ₜ[R] m

@[simp]

theorem ModuleCat.MonoidalCategory.rightUnitor_hom_apply {R : Type u} [CommRing R] {M : ModuleCat R} (m : ↑M) (r : R) : ↑(CategoryTheory.MonoidalCategory.rightUnitor M).hom (m ⊗ₜ[R] r) = r • m

@[simp]

theorem ModuleCat.MonoidalCategory.rightUnitor_inv_apply {R : Type u} [CommRing R] {M : ModuleCat R} (m : ↑M) : ↑(CategoryTheory.MonoidalCategory.rightUnitor M).inv m = m ⊗ₜ[R] 1

@[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) : ↑(CategoryTheory.MonoidalCategory.associator M N K).hom ((m ⊗ₜ[R] n) ⊗ₜ[R] k) = m ⊗ₜ[R] n ⊗ₜ[R] 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) : ↑(CategoryTheory.MonoidalCategory.associator M N K).inv (m ⊗ₜ[R] n ⊗ₜ[R] k) = (m ⊗ₜ[R] n) ⊗ₜ[R] k

instance ModuleCat.instMonoidalPreadditiveModuleCatToRingModuleCategoryInstPreadditiveModuleCatModuleCategoryMonoidalCategory {R : Type u} [CommRing R] : CategoryTheory.MonoidalPreadditive (ModuleCat R)

instance ModuleCat.instMonoidalLinearToSemiringToCommSemiringModuleCatToRingModuleCategoryInstPreadditiveModuleCatModuleCategoryInstLinearToSemiringToCommSemiringModuleCatToRingModuleCategoryInstPreadditiveModuleCatModuleCategoryMonoidalCategoryInstMonoidalPreadditiveModuleCatToRingModuleCategoryInstPreadditiveModuleCatModuleCategoryMonoidalCategory {R : Type u} [CommRing R] : CategoryTheory.MonoidalLinear R (ModuleCat R)