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
- <math>H_n(A_\bullet) \to H_n(B_\bullet)\ (\text{respectively, } H^n(A^\bullet) \to H^n(B^\bullet))\ </math>
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.
