Theorems · Theorem · category theory

CategoryTheory.Functor.preservesFiniteLimits_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.Mono (F.map S.f),
      ∀ (S : CategoryTheory.ShortComplex C),
        S.Exact ∧ CategoryTheory.Mono S.f → (S.map F).Exact ∧ CategoryTheory.Mono (F.map S.f),
      ∀ ⦃X Y : C⦄ (f : X ⟶ Y), CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.parallelPair f 0) F,
      CategoryTheory.Limits.PreservesFiniteLimits F].TFAE

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

Defined in

Mathlib.Algebra.Homology.ShortComplex.ExactFunctor

Cited by

5 results in Mathlib

Foundations

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

Assumes

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