Theorems · Definition · category theory

CochainComplex.mappingConeHomOfDegreewiseSplitXIso

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v, u_1} C] →
    [inst_1 : CategoryTheory.Preadditive C] →
      (S : CategoryTheory.ShortComplex (CochainComplex C ℤ)) →
        (σ : (n : ℤ) → (S.map (HomologicalComplex.eval C (ComplexShape.up ℤ) n)).Splitting) →
          [inst_2 : CategoryTheory.Limits.HasBinaryBiproducts C] →
            (p q : ℤ) →
              p + 1 = q → ((CochainComplex.mappingCone (CochainComplex.homOfDegreewiseSplit S σ)).X p ≅ S.X₂.X q)

The canonical isomorphism (mappingCone (homOfDegreewiseSplit S σ)).X p ≅ S.X₂.X q when p + 1 = q.

Defined in

Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit

Cited by

3 results in Mathlib

Foundations

Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Limits.HasBinaryBiproducts