Theorems · Theorem · category theory

CategoryTheory.leftDistrib_hom

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v, u_1} C] [inst_1 : CategoryTheory.MonoidalCategory C]
  [inst_2 : CategoryTheory.Limits.HasBinaryCoproducts C] [inst_3 : CategoryTheory.IsMonoidalLeftDistrib C] {X Y Z : C},
  (CategoryTheory.leftDistrib X Y Z).hom =
    CategoryTheory.Limits.coprod.desc
      (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X CategoryTheory.Limits.coprod.inl)
      (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X CategoryTheory.Limits.coprod.inr)

The forward direction of the left distributivity isomorphism is the cogap morphism coprod.desc (_ ◁ coprod.inl) (_ ◁ coprod.inr) : (X ⊗ Y) ⨿ (X ⊗ Z) ⟶ X ⊗ (Y ⨿ Z).

Defined in

Mathlib.CategoryTheory.Distributive.Monoidal

Cited by

3 results in Mathlib

Foundations

Depth 41 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.Limits.HasBinaryCoproductsCategoryTheory.IsMonoidalLeftDistrib