Theorems · Theorem · category theory

CategoryTheory.MonoidalCategory.tensor_associativity

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.MonoidalCategory C]
  [inst_2 : CategoryTheory.BraidedCategory C] (X₁ X₂ Y₁ Y₂ Z₁ Z₂ : C),
  CategoryTheory.CategoryStruct.comp
      (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.MonoidalCategory.tensorμ X₁ X₂ Y₁ Y₂)
        (CategoryTheory.MonoidalCategoryStruct.tensorObj Z₁ Z₂))
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.MonoidalCategory.tensorμ (CategoryTheory.MonoidalCategoryStruct.tensorObj X₁ Y₁)
          (CategoryTheory.MonoidalCategoryStruct.tensorObj X₂ Y₂) Z₁ Z₂)
        (CategoryTheory.MonoidalCategoryStruct.tensorHom (CategoryTheory.MonoidalCategoryStruct.associator X₁ Y₁ Z₁).hom
          (CategoryTheory.MonoidalCategoryStruct.associator X₂ Y₂ Z₂).hom)) =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.MonoidalCategoryStruct.associator (CategoryTheory.MonoidalCategoryStruct.tensorObj X₁ X₂)
          (CategoryTheory.MonoidalCategoryStruct.tensorObj Y₁ Y₂)
          (CategoryTheory.MonoidalCategoryStruct.tensorObj Z₁ Z₂)).hom
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.MonoidalCategoryStruct.whiskerLeft (CategoryTheory.MonoidalCategoryStruct.tensorObj X₁ X₂)
          (CategoryTheory.MonoidalCategory.tensorμ Y₁ Y₂ Z₁ Z₂))
        (CategoryTheory.MonoidalCategory.tensorμ X₁ X₂ (CategoryTheory.MonoidalCategoryStruct.tensorObj Y₁ Z₁)
          (CategoryTheory.MonoidalCategoryStruct.tensorObj Y₂ Z₂)))

Defined in

Mathlib.CategoryTheory.Monoidal.Braided.Basic

Cited by

3 results in Mathlib

Foundations

Depth 23 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.BraidedCategory