Theorems · Theorem · category theory

CategoryTheory.Localization.Monoidal.braidingNatIso_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)
  [inst_5 : CategoryTheory.BraidedCategory C] (X Y : C),
  ((CategoryTheory.Localization.Monoidal.braidingNatIso L W ε).hom.app
          ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj X)).app
      ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).obj Y) =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Functor.LaxMonoidal.μ (CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε) X Y)
      (CategoryTheory.CategoryStruct.comp
        ((CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε).map (β_ X Y).hom)
        (CategoryTheory.Functor.OplaxMonoidal.δ (CategoryTheory.Localization.Monoidal.toMonoidalCategory L W ε) Y X))

Defined in

Mathlib.CategoryTheory.Localization.Monoidal.Braided

Cited by

3 results in Mathlib

Foundations

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

Assumes

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