Quasi-isomorphism

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

H_n(A_•, Z) → H_n(B_•, Z)

of homology groups or

Hⁿ(A^•, Z) → Hⁿ(B^•, Z)

of cohomology groups are isomorphisms for all n ≥ 0.

[edit] 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.

[edit] Reference

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