Theorems · Theorem · category theory

CategoryTheory.ShortComplex.exact_iff_exact_up_to_refinements

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Abelian C]
  (S : CategoryTheory.ShortComplex C),
  S.Exact ↔
    ∀ ⦃A : C⦄ (x₂ : A ⟶ S.X₂),
      CategoryTheory.CategoryStruct.comp x₂ S.g = 0 →
        ∃ A' π,
          ∃ (_ : CategoryTheory.Epi π),
            ∃ x₁, CategoryTheory.CategoryStruct.comp π x₂ = CategoryTheory.CategoryStruct.comp x₁ S.f

Defined in

Mathlib.CategoryTheory.Abelian.Refinements

Cited by

14 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Abelian