Quasi-isomorphism

In homological algebra, a branch of mathematics, a quasi-isomorphism is a morphism "A" → "B" of chain complexes (respectively, cochain complexes) such that the induced morphisms

:

of homology groups (respectively, of cohomology groups) are isomorphisms for all "n" ≥ 0.

Applications

In the theory of model categories, quasi-isomorphisms are sometimes used as the class of weak equivalences when the objects of the category are chain or cochain complexes. This results in a homology-local theory, in the sense of Bousfield localization in homotopy theory.

Quasi-isomorphisms play the fundamental role in defining the derived category of an abelian category.

References

*Gelfand, Manin. "Methods of Homological Algebra", 2nd ed. Springer, 2000.