The symmetric monoidal structure on Module R.

noncomputable def ModuleCat.braiding {R : Type u} [CommRing R] (M N : ModuleCat R) : CategoryTheory.MonoidalCategoryStruct.tensorObj M N ≅ CategoryTheory.MonoidalCategoryStruct.tensorObj N M

(implementation) the braiding for R-modules

Equations

• M.braiding N = (TensorProduct.comm R ↑M ↑N).toModuleIso

@[simp]

theorem ModuleCat.MonoidalCategory.braiding_naturality {R : Type u} [CommRing R] {X₁ X₂ Y₁ Y₂ : ModuleCat R} (f : X₁ ⟶ Y₁) (g : X₂ ⟶ Y₂) : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom f g) (Y₁.braiding Y₂).hom = CategoryTheory.CategoryStruct.comp (X₁.braiding X₂).hom (CategoryTheory.MonoidalCategoryStruct.tensorHom g f)

@[simp]

theorem ModuleCat.MonoidalCategory.braiding_naturality_left {R : Type u} [CommRing R] {X Y : ModuleCat R} (f : X ⟶ Y) (Z : ModuleCat R) : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight f Z) (Y.braiding Z).hom = CategoryTheory.CategoryStruct.comp (X.braiding Z).hom (CategoryTheory.MonoidalCategoryStruct.whiskerLeft Z f)

@[simp]

theorem ModuleCat.MonoidalCategory.braiding_naturality_right {R : Type u} [CommRing R] (X : ModuleCat R) {Y Z : ModuleCat R} (f : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X f) (X.braiding Z).hom = CategoryTheory.CategoryStruct.comp (X.braiding Y).hom (CategoryTheory.MonoidalCategoryStruct.whiskerRight f X)

@[simp]

theorem ModuleCat.MonoidalCategory.hexagon_forward {R : Type u} [CommRing R] (X Y Z : ModuleCat R) : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.associator X Y Z).hom (CategoryTheory.CategoryStruct.comp (X.braiding (CategoryTheory.MonoidalCategoryStruct.tensorObj Y Z)).hom (CategoryTheory.MonoidalCategoryStruct.associator Y Z X).hom) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight (X.braiding Y).hom Z) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.associator Y X Z).hom (CategoryTheory.MonoidalCategoryStruct.whiskerLeft Y (X.braiding Z).hom))

@[simp]

theorem ModuleCat.MonoidalCategory.hexagon_reverse {R : Type u} [CommRing R] (X Y Z : ModuleCat R) : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.associator X Y Z).inv (CategoryTheory.CategoryStruct.comp ((CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).braiding Z).hom (CategoryTheory.MonoidalCategoryStruct.associator Z X Y).inv) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X (Y.braiding Z).hom) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.associator X Z Y).inv (CategoryTheory.MonoidalCategoryStruct.whiskerRight (X.braiding Z).hom Y))

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

The symmetric monoidal structure on Module R.

Equations

• ModuleCat.MonoidalCategory.symmetricCategory = CategoryTheory.SymmetricCategory.mk ⋯

@[simp]

theorem ModuleCat.MonoidalCategory.braiding_hom_apply {R : Type u} [CommRing R] {M N : ModuleCat R} (m : ↑M) (n : ↑N) : (β_ M N).hom.hom (m ⊗ₜ[R] n) = n ⊗ₜ[R] m

@[simp]

theorem ModuleCat.MonoidalCategory.braiding_inv_apply {R : Type u} [CommRing R] {M N : ModuleCat R} (m : ↑M) (n : ↑N) : (β_ M N).inv.hom (n ⊗ₜ[R] m) = m ⊗ₜ[R] n

theorem ModuleCat.MonoidalCategory.tensorμ_eq_tensorTensorTensorComm {R : Type u} [CommRing R] {A B C D : ModuleCat R} : CategoryTheory.MonoidalCategory.tensorμ A B C D = ofHom ↑(TensorProduct.tensorTensorTensorComm R ↑A ↑B ↑C ↑D)

@[simp]

theorem ModuleCat.MonoidalCategory.tensorμ_apply {R : Type u} [CommRing R] {A B C D : ModuleCat R} (x : ↑A) (y : ↑B) (z : ↑C) (w : ↑D) : (CategoryTheory.MonoidalCategory.tensorμ A B C D).hom ((x ⊗ₜ[R] y) ⊗ₜ[R] z ⊗ₜ[R] w) = (x ⊗ₜ[R] z) ⊗ₜ[R] y ⊗ₜ[R] w