In
homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 t ...
es (respectively, cochain complexes) such that the induced morphisms
:
of
homology
Homology may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
* Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chrom ...
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 is ...
s 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 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.
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 pr ...
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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