Theorems · Theorem · category theory

CategoryTheory.MonoidalCategory.DayConvolutionUnit.corepresentableByLeft_homEquiv

∀ {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]
  (U : CategoryTheory.Functor C V) [inst_4 : CategoryTheory.MonoidalCategory.DayConvolutionUnit U]
  (F : CategoryTheory.Functor C V)
  [inst_5 :
    ∀ (v : V) (d : C),
      CategoryTheory.Limits.PreservesColimitsOfShape
        (CategoryTheory.CostructuredArrow
          (CategoryTheory.Functor.fromPUnit (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)) d)
        (CategoryTheory.MonoidalCategory.tensorRight v)]
  [inst_6 : CategoryTheory.MonoidalCategory.DayConvolution U F] {Y : CategoryTheory.Functor C V},
  (CategoryTheory.MonoidalCategory.DayConvolutionUnit.corepresentableByLeft U F).homEquiv =
    ((CategoryTheory.MonoidalCategory.DayConvolution.convolution U F).homEquivOfIsLeftKanExtension
          (CategoryTheory.MonoidalCategory.DayConvolution.unit U F) Y).trans
      ((CategoryTheory.MonoidalCategory.externalProduct U F).homEquivOfIsLeftKanExtension
        (CategoryTheory.MonoidalCategory.ExternalProduct.extensionUnitLeft U
          (CategoryTheory.MonoidalCategory.DayConvolutionUnit.φ U) F)
        ((CategoryTheory.MonoidalCategory.tensor C).comp Y))

Defined in

Mathlib.CategoryTheory.Monoidal.DayConvolution

Cited by

0 results in Mathlib

Foundations

Depth 56 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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