Quasi-isomorphism

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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

$H_{n}(A_{\bullet })\to H_{n}(B_{\bullet })\ ({\text{respectively, }}H^{n}(A^{\bullet })\to H^{n}(B^{\bullet }))\$

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

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.

References

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

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