Theorems · Theorem · category theory

CategoryTheory.associator_inv

∀ {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)
  (X Y Z : C),
  (CategoryTheory.MonoidalCategoryStruct.associator
        ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj X)
        ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Y)
        ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Z)).inv =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.MonoidalCategoryStruct.whiskerLeft
        ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj X)
        (CategoryTheory.Functor.LaxMonoidal.μ (CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε) Y Z))
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Functor.LaxMonoidal.μ (CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε) X
          (CategoryTheory.MonoidalCategoryStruct.tensorObj Y Z))
        (CategoryTheory.CategoryStruct.comp
          ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).map
            (CategoryTheory.MonoidalCategoryStruct.associator X Y Z).inv)
          (CategoryTheory.CategoryStruct.comp
            (CategoryTheory.Functor.OplaxMonoidal.δ (CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε)
              (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y) Z)
            (CategoryTheory.MonoidalCategoryStruct.whiskerRight
              (CategoryTheory.Functor.OplaxMonoidal.δ (CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε) X
                Y)
              ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Z)))))

Defined in

Mathlib.CategoryTheory.Localization.Monoidal.Basic

Cited by

1 results in Mathlib

Foundations

Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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