abHomologyIso — Mathlib

English

Reiterates that quasi-isomorphisms are stable under retracts in HomologicalComplex.

Русский

Снова утверждается стабильность квазиизоморфизмов под retracts в HomologicalComplex.

LaTeX

$$$\text{StableUnderRetracts}$$$

Lean4

/-- Given a short complex `S` of abelian groups, this is the isomorphism between
the abstract `S.homology` of the homology API and the more explicit
quotient of `AddMonoidHom.ker S.g` by the image of
`S.abToCycles : S.X₁ →+ AddMonoidHom.ker S.g`. -/
noncomputable def abHomologyIso :
    S.homology ≅ AddCommGrpCat.of ((AddMonoidHom.ker S.g.hom) ⧸ AddMonoidHom.range S.abToCycles) :=
  S.abLeftHomologyData.homologyIso

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