Theorems · Theorem · category theory

CategoryTheory.MonoidalCategory.DayConvolution.unit_app_braiding_hom_app_assoc

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {V : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} V]
  [inst_2 : CategoryTheory.MonoidalCategory C] [inst_3 : CategoryTheory.BraidedCategory C]
  [inst_4 : CategoryTheory.MonoidalCategory V] [inst_5 : CategoryTheory.BraidedCategory V]
  (F G : CategoryTheory.Functor C V) [inst_6 : CategoryTheory.MonoidalCategory.DayConvolution F G]
  [inst_7 : CategoryTheory.MonoidalCategory.DayConvolution G F] (x y : C) {Z : V}
  (h :
    (CategoryTheory.MonoidalCategory.DayConvolution.convolution G F).obj
        (CategoryTheory.MonoidalCategoryStruct.tensorObj x y) ⟶
      Z),
  CategoryTheory.CategoryStruct.comp ((CategoryTheory.MonoidalCategory.DayConvolution.unit F G).app (x, y))
      (CategoryTheory.CategoryStruct.comp
        ((CategoryTheory.MonoidalCategory.DayConvolution.braiding F G).hom.app
          (CategoryTheory.MonoidalCategoryStruct.tensorObj x y))
        h) =
    CategoryTheory.CategoryStruct.comp (β_ (F.obj x) (G.obj y)).hom
      (CategoryTheory.CategoryStruct.comp ((CategoryTheory.MonoidalCategory.DayConvolution.unit G F).app (y, x))
        (CategoryTheory.CategoryStruct.comp
          ((CategoryTheory.MonoidalCategory.DayConvolution.convolution G F).map (β_ y x).hom) h))

Defined in

Mathlib.CategoryTheory.Monoidal.DayConvolution.Braided

Cited by

5 results in Mathlib

Foundations

Depth 44 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.BraidedCategoryCategoryTheory.MonoidalCategoryCategoryTheory.BraidedCategoryCategoryTheory.MonoidalCategory.DayConvolutionCategoryTheory.MonoidalCategory.DayConvolution