quasi-isomorphism

Two chain complexes can look completely different term by term and yet carry exactly the same homological information. A quasi-isomorphism is the map that certifies this: a chain map that is invisible to homology, in the sense that it induces an isomorphism on every homology group. From the point of view of homology, a quasi-isomorphism is as good as an actual isomorphism — even though it usually is not one.

Precisely, a chain map f : A -> B is a quasi-isomorphism if the induced maps H_n(f) : H_n(A) -> H_n(B) are isomorphisms for all n. Equivalently, by the mapping cone criterion, f is a quasi-isomorphism if and only if Cone(f) is acyclic. Every chain homotopy equivalence is a quasi-isomorphism, but the converse fails: a complex of projectives may be quasi-isomorphic to a single module concentrated in one degree without being homotopy equivalent to it.

Quasi-isomorphisms are precisely the maps one wants to invert. The derived category is constructed by formally adjoining inverses to all quasi-isomorphisms, so that a module and any of its resolutions become genuinely isomorphic objects. This is the conceptual home of derived functors: rather than choosing a resolution by hand, one works in a category where the choice no longer matters because quasi-isomorphic complexes are identified.