Theorems · Definition · category theory

CategoryTheory.MonoidalCategory.DayConvolution.associatorCorepresentingIso

{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] →
            (F G H : CategoryTheory.Functor C V) →
              ((CategoryTheory.Functor.whiskeringLeft (C × C) C V).obj (CategoryTheory.MonoidalCategory.tensor C)).comp
                  (((CategoryTheory.Functor.whiskeringLeft ((C × C) × C) (C × C) V).obj
                        ((CategoryTheory.MonoidalCategory.tensor C).prod (CategoryTheory.Functor.id C))).comp
                    (CategoryTheory.coyoneda.obj
                      (Opposite.op
                        (CategoryTheory.MonoidalCategory.externalProduct
                          (CategoryTheory.MonoidalCategory.externalProduct F G) H)))) ≅
                ((CategoryTheory.Functor.whiskeringLeft (C × C) C V).obj
                      (CategoryTheory.MonoidalCategory.tensor C)).comp
                  (((CategoryTheory.Functor.whiskeringLeft (C × C × C) (C × C) V).obj
                        ((CategoryTheory.Functor.id C).prod (CategoryTheory.MonoidalCategory.tensor C))).comp
                    (CategoryTheory.coyoneda.obj
                      (Opposite.op
                        (CategoryTheory.MonoidalCategory.externalProduct F
                          (CategoryTheory.MonoidalCategory.externalProduct G H)))))

The isomorphism of functors between ((F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ -) and (F ⊠ G ⊠ H ⟶ (𝟭 C).prod (tensor C) ⋙ tensor C ⋙ -) that corresponds to the associator isomorphism for Day convolution through corepresentableBy₂ and corepresentableBy₂.

Defined in

Mathlib.CategoryTheory.Monoidal.DayConvolution

Cited by

4 results in Mathlib

Foundations

Depth 36 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.MonoidalCategory