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Theorems · Definition · category theory

CategoryTheory.ShortComplex.Splitting.r

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : CategoryTheory.Preadditive C] → {S : CategoryTheory.ShortComplex C} → S.Splitting → (S.X₂ ⟶ S.X₁)

a retraction of S.f

Defined in
Mathlib.Algebra.Homology.ShortComplex.Exact
Cited by
24 results in Mathlib
Foundations
Depth 19 from the axioms · uses propext
Assumes
CategoryTheory.CategoryCategoryTheory.Preadditive
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

CategoryTheory.ShortComplex.Splitting.id · cited by 5Splitting.idCategoryTheory.ShortComplex.Splitting.leftHomologyData · cited by 5Splitting.leftHomologyDataCategoryTheory.ShortComplex.Splitting.f_r · cited by 4Splitting.f_rCochainComplex.mappingConeHomOfDegreewiseSplitXIso · cited by 3CochainComplex.mappingCon…CategoryTheory.ShortComplex.Splitting.isoBinaryBiproduct · cited by 2Splitting.isoBinaryBiprod…CochainComplex.cocycleOfDegreewiseSplit · cited by 2CochainComplex.cocycleOfD…CategoryTheory.ShortComplex.Splitting.map · cited by 2Splitting.mapCategoryTheory.ShortComplex.Splitting.ofIso · cited by 2Splitting.ofIsoCategoryTheory.ShortComplex.Splitting.op · cited by 2Splitting.opCategoryTheory.ShortComplex.Splitting.s_r · cited by 2Splitting.s_rCategoryTheory.ShortComplex.Splitting.splitMono_f · cited by 2Splitting.splitMono_fCategoryTheory.ShortComplex.Splitting.unop · cited by 2Splitting.unopCategoryTheory.ShortComplex.Splitting.g_s · cited by 1Splitting.g_sCategoryTheory.ShortComplex.Splitting.g_s_assoc · cited by 1Splitting.g_s_assocCochainComplex.shift_f_comp_mappingConeHomOfDegreewiseSplitIso_inv · cited by 1CochainComplex.shift_f_co…CategoryTheory.Category · cited by 32673CategoryTheory.CategoryQuiver.Hom · cited by 32603Quiver.HomCategoryTheory.Preadditive · cited by 3309CategoryTheory.PreadditiveCategoryTheory.ShortComplex · cited by 1850CategoryTheory.ShortCompl…CategoryTheory.ShortComplex.X₂ · cited by 1115ShortComplex.X₂CategoryTheory.ShortComplex.X₁ · cited by 889ShortComplex.X₁CategoryTheory.ShortComplex.Splitting · cited by 62ShortComplex.SplittingSplitting.rCITED BYCITES

Cites7

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Cited by33

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