Theorems · Definition · category theory

CategoryTheory.ShortComplex.Exact.desc

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : CategoryTheory.Preadditive C] →
      {S : CategoryTheory.ShortComplex C} →
        [CategoryTheory.Balanced C] →
          S.Exact →
            {A : C} →
              (k : S.X₂ ⟶ A) → CategoryTheory.CategoryStruct.comp S.f k = 0 → [CategoryTheory.Epi S.g] → S.X₃ ⟶ A

If a short complex S in a balanced category is exact and such that S.g is an epi, then a morphism k : S.X₂ ⟶ A such that S.f ≫ k = 0 descends to a morphism S.X₃ ⟶ A.

Defined in

Mathlib.Algebra.Homology.ShortComplex.Exact

Cited by

6 results in Mathlib

Foundations

Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.BalancedCategoryTheory.Epi