Theorems · Theorem · category theory

CategoryTheory.Functor.preservesFiniteColimits_tfae

∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} D] [inst_2 : CategoryTheory.Abelian C]
  [inst_3 : CategoryTheory.Abelian D] (F : CategoryTheory.Functor C D) [inst_4 : F.Additive],
  [∀ (S : CategoryTheory.ShortComplex C), S.ShortExact → (S.map F).Exact ∧ CategoryTheory.Epi (F.map S.g),
      ∀ (S : CategoryTheory.ShortComplex C),
        S.Exact ∧ CategoryTheory.Epi S.g → (S.map F).Exact ∧ CategoryTheory.Epi (F.map S.g),
      ∀ ⦃X Y : C⦄ (f : X ⟶ Y), CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair f 0) F,
      CategoryTheory.Limits.PreservesFiniteColimits F].TFAE

For an additive functor F : C ⥤ D between abelian categories, the following are equivalent: - F preserves short exact sequences on the right-hand side, i.e. if 0 ⟶ A ⟶ B ⟶ C ⟶ 0 is exact then F(A) ⟶ F(B) ⟶ F(C) ⟶ 0 is exact. - F preserves exact sequences on the right-hand side, i.e. if A ⟶ B ⟶ C is exact where B ⟶ C is epi, then F(A) ⟶ F(B) ⟶ F(C) ⟶ 0 is exact and F(B) ⟶ F(C) is epi as well. - F preserves cokernels. - F preserves finite colimits.

Defined in

Mathlib.Algebra.Homology.ShortComplex.ExactFunctor

Cited by

4 results in Mathlib

Foundations

Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.AbelianCategoryTheory.Functor.Additive