Theorems · Theorem · category theory

CategoryTheory.MonoidalCategory.DayConvolution.hexagon_reverse

∀ {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 :
    ∀ (v : V) (d : C),
      CategoryTheory.Limits.PreservesColimitsOfShape
        (CategoryTheory.CostructuredArrow (CategoryTheory.MonoidalCategory.tensor C) d)
        (CategoryTheory.MonoidalCategory.tensorLeft v)]
  [inst_7 :
    ∀ (v : V) (d : C),
      CategoryTheory.Limits.PreservesColimitsOfShape
        (CategoryTheory.CostructuredArrow (CategoryTheory.MonoidalCategory.tensor C) d)
        (CategoryTheory.MonoidalCategory.tensorRight v)]
  (H : CategoryTheory.Functor C V) [inst_8 : CategoryTheory.MonoidalCategory.DayConvolution F G]
  [inst_9 : CategoryTheory.MonoidalCategory.DayConvolution G H]
  [inst_10 :
    CategoryTheory.MonoidalCategory.DayConvolution F (CategoryTheory.MonoidalCategory.DayConvolution.convolution G H)]
  [inst_11 :
    CategoryTheory.MonoidalCategory.DayConvolution (CategoryTheory.MonoidalCategory.DayConvolution.convolution F G) H]
  [inst_12 :
    CategoryTheory.MonoidalCategory.DayConvolution H (CategoryTheory.MonoidalCategory.DayConvolution.convolution F G)]
  [inst_13 : CategoryTheory.MonoidalCategory.DayConvolution H F]
  [inst_14 :
    CategoryTheory.MonoidalCategory.DayConvolution (CategoryTheory.MonoidalCategory.DayConvolution.convolution H F) G]
  [inst_15 : CategoryTheory.MonoidalCategory.DayConvolution H G]
  [inst_16 :
    CategoryTheory.MonoidalCategory.DayConvolution F (CategoryTheory.MonoidalCategory.DayConvolution.convolution H G)]
  [inst_17 : CategoryTheory.MonoidalCategory.DayConvolution F H]
  [inst_18 :
    CategoryTheory.MonoidalCategory.DayConvolution (CategoryTheory.MonoidalCategory.DayConvolution.convolution F H) G],
  CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.DayConvolution.associator F G H).inv
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.MonoidalCategory.DayConvolution.braiding
            (CategoryTheory.MonoidalCategory.DayConvolution.convolution F G) H).hom
        (CategoryTheory.MonoidalCategory.DayConvolution.associator H F G).inv) =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.MonoidalCategory.DayConvolution.map (CategoryTheory.CategoryStruct.id F)
        (CategoryTheory.MonoidalCategory.DayConvolution.braiding G H).hom)
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.DayConvolution.associator F H G).inv
        (CategoryTheory.MonoidalCategory.DayConvolution.map
          (CategoryTheory.MonoidalCategory.DayConvolution.braiding F H).hom (CategoryTheory.CategoryStruct.id G)))

Defined in

Mathlib.CategoryTheory.Monoidal.DayConvolution.Braided

Cited by

0 results in Mathlib

Foundations

Depth 59 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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