Let be an abelian category, and let be the category of chain complexes in , with the morphisms being chain homotopy classes of maps. Call a morphism of chain complexes a quasi-isomorphism if it induces an isomorphism on homology groups of the complexes. For example, any chain homotopy is a quasi-isomorphism, but not conversely. Now let the derived category be the category obtained from by adding a formal inverse to every quasi-isomorphism (technically this called a localization of the category).

Derived categories seem somewhat obscure, but in fact, many mathematicians believe they are the appropriate place to do homological algebra. One of their great advantages is that the important functors of homological algebra which are left or right exact ($\mathrm{Hom}$ ,$N\otimes_k-$ , where $N$ is a fixed $k$ -module, the global sections functor $\Gamma$ , etc.) become exact on the level of derived functors (with an appropriately modified definition of exact).

See Methods of Homological Algebra, by Gelfand and Manin for more details.