Theorems · Theorem · category theory

CategoryTheory.Functor.exact_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).ShortExact,
      ∀ (S : CategoryTheory.ShortComplex C), S.Exact → (S.map F).Exact, F.PreservesHomology,
      CategoryTheory.Limits.PreservesFiniteLimits 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, i.e. if 0 ⟶ A ⟶ B ⟶ C ⟶ 0 is exact then 0 ⟶ F(A) ⟶ F(B) ⟶ F(C) ⟶ 0 is exact. - F preserves exact sequences, i.e. if A ⟶ B ⟶ C is exact then F(A) ⟶ F(B) ⟶ F(C) is exact. - F preserves homology. - F preserves both finite limits and finite colimits.

Defined in

Mathlib.Algebra.Homology.ShortComplex.ExactFunctor

Cited by

3 results in Mathlib

Foundations

Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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