Theorems · Inductive type · category theory
CategoryTheory.ShortComplex.Splitting
{C : Type u_1} →
[inst : CategoryTheory.Category.{v_1, u_1} C] →
[inst_1 : CategoryTheory.Preadditive C] → CategoryTheory.ShortComplex C → Type v_1A splitting for a short complex S consists of the data of a retraction r : X₂ ⟶ X₁
of S.f and section s : X₃ ⟶ X₂ of S.g which satisfy r ≫ S.f + S.g ≫ s = 𝟙 _
- Cited by
- 62 results in Mathlib
- Foundations
- Depth 18 from the axioms · uses propext
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CategoryTheory.Categorystatement · cited by 32,673
- CategoryTheory.Preadditivestatement · cited by 3,309
- CategoryTheory.ShortComplexstatement · cited by 1,850
Cited by98
Results whose statement or proof uses this declaration.
- CategoryTheory.ShortComplex.Splitting.rstatement and proof · cited by 24
- CategoryTheory.ShortComplex.Splitting.sstatement and proof · cited by 24
- CochainComplex.homOfDegreewiseSplitstatement and proof · cited by 10
- CochainComplex.mappingConeHomOfDegreewiseSplitIsostatement and proof · cited by 6
- CochainComplex.triangleOfDegreewiseSplitstatement and proof · cited by 6
- CategoryTheory.ShortComplex.Splitting.idstatement and proof · cited by 5
- CategoryTheory.ShortComplex.Splitting.leftHomologyDatastatement and proof · cited by 5
- CategoryTheory.ShortComplex.Splitting.rightHomologyDatastatement and proof · cited by 5
- CategoryTheory.ShortComplex.Splitting.f_rstatement and proof · cited by 4
- CategoryTheory.ShortComplex.Splitting.homologyDatastatement and proof · cited by 4
- CategoryTheory.ShortComplex.Splitting.s_gstatement and proof · cited by 4
- CochainComplex.mappingCone.triangleRotateShortComplexSplittingstatement · cited by 4