Theorems · Theorem · category theory

CategoryTheory.Limits.preservesBiproduct_of_preservesCoproduct

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Preadditive C] {D : Type u'}
  [inst_2 : CategoryTheory.Category.{v', u'} D] [inst_3 : CategoryTheory.Preadditive D] (F : CategoryTheory.Functor C D)
  [inst_4 : F.PreservesZeroMorphisms] {J : Type u_1} [Finite J] {f : J → C}
  [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Discrete.functor f) F],
  CategoryTheory.Limits.PreservesBiproduct f F

A functor between preadditive categories that preserves (zero morphisms and) finite coproducts preserves finite biproducts.

Defined in

Mathlib.CategoryTheory.Preadditive.Biproducts

Cited by

1 results in Mathlib

Foundations

Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Functor.PreservesZeroMorphismsFiniteCategoryTheory.Limits.PreservesColimit