Theorems · Theorem · category theory

CategoryTheory.Localization.Monoidal.associator_hom_app

∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} D] (L : CategoryTheory.Functor C D)
  (W : CategoryTheory.MorphismProperty C) [inst_2 : CategoryTheory.MonoidalCategory C] [inst_3 : W.IsMonoidal]
  [inst_4 : L.IsLocalization W] {unit : D} (ε : L.obj (CategoryTheory.MonoidalCategoryStruct.tensorUnit C) ≅ unit)
  (X₁ X₂ X₃ : C),
  (CategoryTheory.MonoidalCategoryStruct.associator
        ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj X₁)
        ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj X₂)
        ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj X₃)).hom =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.MonoidalCategoryStruct.tensorHom (CategoryTheory.Localization.Monoidal.μ L W ε X₁ X₂).hom
        (CategoryTheory.CategoryStruct.id ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj X₃)))
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Localization.Monoidal.μ L W ε (CategoryTheory.MonoidalCategoryStruct.tensorObj X₁ X₂) X₃).hom
        (CategoryTheory.CategoryStruct.comp
          ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).map
            (CategoryTheory.MonoidalCategoryStruct.associator X₁ X₂ X₃).hom)
          (CategoryTheory.CategoryStruct.comp
            (CategoryTheory.Localization.Monoidal.μ L W ε X₁
                (CategoryTheory.MonoidalCategoryStruct.tensorObj X₂ X₃)).inv
            (CategoryTheory.MonoidalCategoryStruct.tensorHom
              (CategoryTheory.CategoryStruct.id
                ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj X₁))
              (CategoryTheory.Localization.Monoidal.μ L W ε X₂ X₃).inv))))

Defined in

Mathlib.CategoryTheory.Localization.Monoidal.Basic

Cited by

2 results in Mathlib

Foundations

Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.MorphismProperty.IsMonoidalCategoryTheory.Functor.IsLocalization