Theorems · Theorem · category theory

CategoryTheory.MonoidalCategory.DayConvolution.triangle

∀ {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.MonoidalCategory V]
  [inst_4 :
    ∀ (v : V) (d : C),
      CategoryTheory.Limits.PreservesColimitsOfShape
        (CategoryTheory.CostructuredArrow (CategoryTheory.MonoidalCategory.tensor C) d)
        (CategoryTheory.MonoidalCategory.tensorLeft v)]
  [inst_5 :
    ∀ (v : V) (d : C),
      CategoryTheory.Limits.PreservesColimitsOfShape
        (CategoryTheory.CostructuredArrow (CategoryTheory.MonoidalCategory.tensor C) d)
        (CategoryTheory.MonoidalCategory.tensorRight v)]
  [inst_6 :
    ∀ (v : V) (d : C),
      CategoryTheory.Limits.PreservesColimitsOfShape
        (CategoryTheory.CostructuredArrow
          (CategoryTheory.Functor.fromPUnit (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)) d)
        (CategoryTheory.MonoidalCategory.tensorLeft v)]
  [inst_7 :
    ∀ (v : V) (d : C),
      CategoryTheory.Limits.PreservesColimitsOfShape
        (CategoryTheory.CostructuredArrow
          (CategoryTheory.Functor.fromPUnit (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)) d)
        (CategoryTheory.MonoidalCategory.tensorRight v)]
  [∀ (v : V) (d : C × C),
      CategoryTheory.Limits.PreservesColimitsOfShape
        (CategoryTheory.CostructuredArrow
          ((CategoryTheory.Functor.id C).prod
            (CategoryTheory.Functor.fromPUnit (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)))
          d)
        (CategoryTheory.MonoidalCategory.tensorRight v)]
  (F G U : CategoryTheory.Functor C V) [inst_9 : CategoryTheory.MonoidalCategory.DayConvolutionUnit U]
  [inst_10 : CategoryTheory.MonoidalCategory.DayConvolution F U]
  [inst_11 : CategoryTheory.MonoidalCategory.DayConvolution U G]
  [inst_12 :
    CategoryTheory.MonoidalCategory.DayConvolution F (CategoryTheory.MonoidalCategory.DayConvolution.convolution U G)]
  [inst_13 :
    CategoryTheory.MonoidalCategory.DayConvolution (CategoryTheory.MonoidalCategory.DayConvolution.convolution F U) G]
  [inst_14 : CategoryTheory.MonoidalCategory.DayConvolution F G],
  CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.DayConvolution.associator F U G).hom
      (CategoryTheory.MonoidalCategory.DayConvolution.map (CategoryTheory.CategoryStruct.id F)
        (CategoryTheory.MonoidalCategory.DayConvolutionUnit.leftUnitor U G).hom) =
    CategoryTheory.MonoidalCategory.DayConvolution.map
      (CategoryTheory.MonoidalCategory.DayConvolutionUnit.rightUnitor U F).hom (CategoryTheory.CategoryStruct.id G)

Defined in

Mathlib.CategoryTheory.Monoidal.DayConvolution

Cited by

0 results in Mathlib

Foundations

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

Assumes

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