Theorems · Theorem · category theory

CategoryTheory.Limits.biprod.total

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Preadditive C] {X Y : C}
  [inst_2 : CategoryTheory.Limits.HasBinaryBiproduct X Y],
  CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.fst CategoryTheory.Limits.biprod.inl +
      CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.biprod.snd CategoryTheory.Limits.biprod.inr =
    CategoryTheory.CategoryStruct.id (X ⊞ Y)

In any preadditive category, any binary biproduct satisfies biprod.fst ≫ biprod.inl + biprod.snd ≫ biprod.inr = 𝟙 (X ⊞ Y).

Defined in

Mathlib.CategoryTheory.Preadditive.Biproducts

Cited by

4 results in Mathlib

Foundations

Depth 29 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Limits.HasBinaryBiproduct