In
homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology ...
, a branch of
mathematics, a quasi-isomorphism or quism is a morphism ''A'' → ''B'' of
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
es (respectively, cochain complexes) such that the induced morphisms
:
of
homology groups (respectively, of cohomology groups) are
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
s for all ''n''.
In the theory of
model categories
In mathematics, particularly in homotopy theory, a model category is a category theory, category with distinguished classes of morphisms ('arrows') called 'weak equivalence (homotopy theory), weak equivalences', 'fibrations' and 'cofibrations' sati ...
, 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 category theory, a branch of mathematics, a (left) Bousfield localization of a model category replaces the model structure with another model structure with the same cofibrations but with more weak equivalences.
Bousfield localization is named ...
in
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topol ...
.
See also
*
Derived category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pro ...
References
*Gelfand, Sergei I.,
Manin, Yuri I. ''Methods of Homological Algebra'', 2nd ed. Springer, 2000.
Algebraic topology
Homological algebra
Equivalence (mathematics)
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