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 proceeds on the basis that the
objects of
D(
A) should be
chain complexes in
A, with two such chain complexes considered
isomorphic when there is a chain map that induces an isomorphism on the level of
homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of
hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated
spectral sequences.