Theorems · Definition · category theory

CategoryTheory.ShortComplex.Splitting.ofExactOfRetraction

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : CategoryTheory.Preadditive C] →
      [CategoryTheory.Balanced C] →
        (S : CategoryTheory.ShortComplex C) →
          S.Exact →
            (r : S.X₂ ⟶ S.X₁) →
              CategoryTheory.CategoryStruct.comp S.f r = CategoryTheory.CategoryStruct.id S.X₁ →
                CategoryTheory.Epi S.g → S.Splitting

This is the splitting of a short complex S in a balanced category induced by a retraction of the morphism S.f : S.X₁ ⟶ S.X₂

Defined in

Mathlib.Algebra.Homology.ShortComplex.Exact

Cited by

1 results in Mathlib

Foundations

Depth 51 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Balanced