Theorems · Theorem · category theory

CategoryTheory.Localization.Monoidal.map_hexagon_reverse_assoc

∀ {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)
  [inst_5 : CategoryTheory.BraidedCategory C] (X Y Z : C) {Z_1 : CategoryTheory.LocalizedMonoidal L W ε}
  (h :
    CategoryTheory.MonoidalCategoryStruct.tensorObj
        (CategoryTheory.MonoidalCategoryStruct.tensorObj
          ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Z)
          ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj X))
        ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Y) ⟶
      Z_1),
  CategoryTheory.CategoryStruct.comp
      (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.Localization.Monoidal.braidingNatIso L W ε).app
                (CategoryTheory.MonoidalCategoryStruct.tensorObj
                  ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj X)
                  ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Y))).app
            ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Z)).hom
        (CategoryTheory.CategoryStruct.comp
          (CategoryTheory.MonoidalCategoryStruct.associator
              ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Z)
              ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj X)
              ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Y)).inv
          h)) =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.MonoidalCategoryStruct.whiskerLeft
        ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj X)
        (((CategoryTheory.Localization.Monoidal.braidingNatIso L W ε).app
                ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Y)).app
            ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Z)).hom)
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.MonoidalCategoryStruct.associator
            ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj X)
            ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Z)
            ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Y)).inv
        (CategoryTheory.CategoryStruct.comp
          (CategoryTheory.MonoidalCategoryStruct.whiskerRight
            (((CategoryTheory.Localization.Monoidal.braidingNatIso L W ε).app
                    ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj X)).app
                ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Z)).hom
            ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Y))
          h))

Defined in

Mathlib.CategoryTheory.Localization.Monoidal.Braided

Cited by

0 results in Mathlib

Foundations

Depth 105 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.MorphismProperty.IsMonoidalCategoryTheory.Functor.IsLocalizationCategoryTheory.BraidedCategory