Theorems · Definition · category theory

CategoryTheory.ShortComplex.ShortExact.splittingOfInjective

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : CategoryTheory.Preadditive C] →
      {S : CategoryTheory.ShortComplex C} →
        S.ShortExact → [CategoryTheory.Injective S.X₁] → [CategoryTheory.Balanced C] → S.Splitting

A choice of splitting for a short exact short complex S in a balanced category such that S.X₁ is injective.

Defined in: Mathlib.Algebra.Homology.ShortComplex.ShortExact
Cited by: 1 results in Mathlib
Foundations: Depth 52 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Preadditive CategoryTheory.Injective CategoryTheory.Balanced

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CategoryTheory.Categorystatement and proof · cited by 32,673
CategoryTheory.CategoryStruct.idproof · cited by 6,235
CategoryTheory.Preadditivestatement and proof · cited by 3,309
CategoryTheory.ShortComplexstatement and proof · cited by 1,850
CategoryTheory.Monoproof · cited by 893
CategoryTheory.ShortComplex.X₁statement and proof · cited by 889
CategoryTheory.ShortComplex.fproof · cited by 653
CategoryTheory.ShortComplex.ShortExactstatement and proof · cited by 232
CategoryTheory.Injectivestatement and proof · cited by 70
CategoryTheory.ShortComplex.Splittingstatement · cited by 62
CategoryTheory.Balancedstatement and proof · cited by 61
CategoryTheory.Injective.factorThruproof · cited by 12

Cited by1

Results whose statement or proof uses this declaration.

CochainComplex.injective_opcyclesproof · cited by 1