Documentation

Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric

The symmetric monoidal structure on Module R. #

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

(implementation) the braiding for R-modules

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    @[simp]
    theorem ModuleCat.MonoidalCategory.braiding_naturality {R : Type u} [CommRing R] {X₁ : ModuleCat R} {X₂ : ModuleCat R} {Y₁ : ModuleCat R} {Y₂ : ModuleCat R} (f : X₁ ⟶ Y₁) (g : X₂ ⟶ Y₂) :
    CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.tensorHom f g) (Y₁.braiding Y₂).hom = CategoryTheory.CategoryStruct.comp (X₁.braiding X₂).hom (CategoryTheory.MonoidalCategory.tensorHom g f)
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    @[simp]
    theorem ModuleCat.MonoidalCategory.braiding_naturality_left {R : Type u} [CommRing R] {X : ModuleCat R} {Y : ModuleCat R} (f : X ⟶ Y) (Z : ModuleCat R) :
    CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight f Z) (Y.braiding Z).hom = CategoryTheory.CategoryStruct.comp (X.braiding Z).hom (CategoryTheory.MonoidalCategory.whiskerLeft Z f)
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    @[simp]
    theorem ModuleCat.MonoidalCategory.braiding_naturality_right {R : Type u} [CommRing R] (X : ModuleCat R) {Y : ModuleCat R} {Z : ModuleCat R} (f : Y ⟶ Z) :
    CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft X f) (X.braiding Z).hom = CategoryTheory.CategoryStruct.comp (X.braiding Y).hom (CategoryTheory.MonoidalCategory.whiskerRight f X)
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    theorem ModuleCat.MonoidalCategory.hexagon_forward {R : Type u} [CommRing R] (X : ModuleCat R) (Y : ModuleCat R) (Z : ModuleCat R) :
    CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator X Y Z).hom (CategoryTheory.CategoryStruct.comp (X.braiding (CategoryTheory.MonoidalCategory.tensorObj Y Z)).hom (CategoryTheory.MonoidalCategory.associator Y Z X).hom) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight (X.braiding Y).hom Z) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator Y X Z).hom (CategoryTheory.MonoidalCategory.whiskerLeft Y (X.braiding Z).hom))
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    @[simp]
    theorem ModuleCat.MonoidalCategory.hexagon_reverse {R : Type u} [CommRing R] (X : ModuleCat R) (Y : ModuleCat R) (Z : ModuleCat R) :
    CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator X Y Z).inv (CategoryTheory.CategoryStruct.comp ((CategoryTheory.MonoidalCategory.tensorObj X Y).braiding Z).hom (CategoryTheory.MonoidalCategory.associator Z X Y).inv) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft X (Y.braiding Z).hom) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator X Z Y).inv (CategoryTheory.MonoidalCategory.whiskerRight (X.braiding Z).hom Y))
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    noncomputable instance ModuleCat.MonoidalCategory.symmetricCategory {R : Type u} [CommRing R] :
    CategoryTheory.SymmetricCategory (ModuleCat R)

    The symmetric monoidal structure on Module R.

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    @[simp]
    theorem ModuleCat.MonoidalCategory.braiding_hom_apply {R : Type u} [CommRing R] {M : ModuleCat R} {N : ModuleCat R} (m : ↑M) (n : ↑N) :
    (β_ M N).hom (m ⊗ₜ[R] n) = n ⊗ₜ[R] m
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    @[simp]
    theorem ModuleCat.MonoidalCategory.braiding_inv_apply {R : Type u} [CommRing R] {M : ModuleCat R} {N : ModuleCat R} (m : ↑M) (n : ↑N) :
    (β_ M N).inv (n ⊗ₜ[R] m) = m ⊗ₜ[R] n
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    theorem ModuleCat.MonoidalCategory.tensor_μ_eq_tensorTensorTensorComm {R : Type u} [CommRing R] {A : ModuleCat R} {B : ModuleCat R} {C : ModuleCat R} {D : ModuleCat R} :
    CategoryTheory.tensor_μ A B C D = ↑(TensorProduct.tensorTensorTensorComm R ↑A ↑B ↑C ↑D)
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    @[simp]
    theorem ModuleCat.MonoidalCategory.tensor_μ_apply {R : Type u} [CommRing R] {A : ModuleCat R} {B : ModuleCat R} {C : ModuleCat R} {D : ModuleCat R} (x : ↑A) (y : ↑B) (z : ↑C) (w : ↑D) :
    (CategoryTheory.tensor_μ A B C D) ((x ⊗ₜ[R] y) ⊗ₜ[R] z ⊗ₜ[R] w) = (x ⊗ₜ[R] z) ⊗ₜ[R] y ⊗ₜ[R] w