Theorems · Definition · category theory

CategoryTheory.ShortComplex.Exact.descToInjective

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : CategoryTheory.Abelian C] →
      {S : CategoryTheory.ShortComplex C} →
        S.Exact →
          {J : C} →
            (f : S.X₂ ⟶ J) → [CategoryTheory.Injective J] → CategoryTheory.CategoryStruct.comp S.f f = 0 → (S.X₃ ⟶ J)

If S is an exact short complex and f : S.X₂ ⟶ J is a morphism to an injective object J such that S.f ≫ f = 0, this is a morphism φ : S.X₃ ⟶ J such that S.g ≫ φ = f.

Defined in

Mathlib.Algebra.Homology.ShortComplex.Exact

Cited by

4 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.Injective