# 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

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.