a convenient setting for sheaf cohomology. Perhaps the biggest advance was the formulation of the Riemann–Hilbert correspondence in dimensions greater than 1 in derived terms, around 1980. The Sato school adopted the language of derived categories, and the subsequent history of D-modules was of a theory expressed in those terms. A parallel development was the category of spectra in homotopy theory. The homotopy category of spectra and the derived category of a ring are both examples of triangulated categories. Definition Let be an abelian category. (Examples include the category of modules over a ring and the category of sheaves of abelian groups on a topological space.) The derived category is defined by a universal property with respect to the category of cochain complexes with terms in . The objects of are of the form where each Xi is an object of and each of the composites is zero. The ith cohomology group of the complex is . If and are two objects in this category, then a morphism is defined to be a family of morphisms such that . Such a morphism induces morphisms on cohomology groups , and is called a quasi-isomorphism if each of these morphisms is an isomorphism in . The universal property of the derived category is that it is a localization of the category of complexes with respect to quasi-isomorphisms. Specifically, the derived category is a category, together with a functor , having the following universal property: Suppose is another category (not necessarily abelian) and is a functor such that, whenever is a quasi-isomorphism in , its image is an isomorphism in ; then factors through . Any two categories having this universal property are equivalent. Relation to the homotopy category If and are two morphisms in , then a chain homotopy or simply homotopy is a collection of morphisms such that for every i. It is straightforward to show that two homotopic morphisms induce identical morphisms on cohomology groups. We say that is a chain homotopy equivalence if there exists such that and are chain homotopic to the identity morphisms on and , respectively. The homotopy category of cochain complexes is the category with the same objects as but whose morphisms are equivalence classes of morphisms of complexes with respect to the relation of chain homotopy equivalence. There is a natural functor which is the identity on objects and which sends each morphism to