Theorems · Definition · category theory

CategoryTheory.ShortComplex.Exact.lift

{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 : A ⟶ S.X₂) → CategoryTheory.CategoryStruct.comp k S.g = 0 → [CategoryTheory.Mono S.f] → A ⟶ S.X₁

If a short complex S in a balanced category is exact and such that S.f is a mono, then a morphism k : A ⟶ S.X₂ such that k ≫ S.g = 0 lifts to a morphism A ⟶ S.X₁.

Defined in

Mathlib.Algebra.Homology.ShortComplex.Exact

Cited by

8 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.BalancedCategoryTheory.Mono