Theorems · Definition · category theory

CategoryTheory.MonoidalCategory.externalProductSwap

(J₁ : Type u₁) →
  (J₂ : Type u₂) →
    (C : Type u₃) →
      [inst : CategoryTheory.Category.{v₁, u₁} J₁] →
        [inst_1 : CategoryTheory.Category.{v₂, u₂} J₂] →
          [inst_2 : CategoryTheory.Category.{v₃, u₃} C] →
            [inst_3 : CategoryTheory.MonoidalCategory C] →
              [CategoryTheory.BraidedCategory C] →
                (CategoryTheory.MonoidalCategory.externalProductBifunctor J₁ J₂ C).comp
                    ((CategoryTheory.Functor.whiskeringLeft (J₂ × J₁) (J₁ × J₂) C).obj
                      (CategoryTheory.Prod.swap J₂ J₁)) ≅
                  (CategoryTheory.Prod.swap (CategoryTheory.Functor J₁ C) (CategoryTheory.Functor J₂ C)).comp
                    (CategoryTheory.MonoidalCategory.externalProductBifunctor J₂ J₁ C)

When C is braided, there is an isomorphism Prod.swap _ _ ⋙ F₁ ⊠ F₂ ≅ F₂ ⊠ F₁, natural in both F₁ and F₂. Note that (externalProductSwap _ _ _).app (F₁, F₂) : Prod.swap _ _ ⋙ F₁ ⊠ F₂ ≅ F₂ ⊠ F₁ type checks.

Defined in

Mathlib.CategoryTheory.Monoidal.ExternalProduct.Basic

Cited by

2 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.BraidedCategory