Quasi-isomorphism explained

In homological algebra, a branch of mathematics, a quasi-isomorphism or quism 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) (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.

See also

• Derived category

References

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