Editing Derived functor (section)

== Generalization ==

The more modern (and more general) approach to derived functors uses the language of [[derived category|derived categories]].

In 1968 [[Daniel Quillen|Quillen]] developed the theory of [[model category|model categories]], which give an abstract category-theoretic system of fibrations, cofibrations and weak equivalences. Typically one is interested in the underlying [[homotopy category]] obtained by localizing against the weak equivalences. A [[Quillen adjunction]] is an adjunction between model categories that descends to an adjunction between the homotopy categories. For example, the category of topological spaces and the category of simplicial sets both admit Quillen model structures whose [[simplicial set|nerve and realization]] adjunction gives a Quillen adjunction that is in fact an equivalence of homotopy categories. Particular objects in a model structure have “nice properties” (concerning the existence of lifts against particular morphisms), the “fibrant” and “cofibrant” objects, and every object is weakly equivalent to a fibrant-cofibrant “resolution.”

Although originally developed to handle the category of topological spaces Quillen model structures appear in numerous places in mathematics; in particular the category of chain complexes from any Abelian category (modules, sheaves of modules on a topological space or [[scheme (mathematics)|scheme]], etc.) admit a model structure whose weak equivalences are those morphisms between chain complexes preserving homology. Often we have a functor between two such model categories (e.g. the global sections functor sending a complex of Abelian sheaves to the obvious complex of Abelian groups) that preserves weak equivalences ''within the subcategory of “good” (fibrant or cofibrant) objects''. By first taking a fibrant or cofibrant resolution of an object and then applying that functor, we have successfully extended it to the whole category in such a way that weak equivalences are always preserved (and hence it descends to a functor from the homotopy category). This is the “derived functor.” The “derived functors” of sheaf cohomology, for example, are the homologies of the output of this derived functor. Applying these to a sheaf of Abelian groups interpreted in the obvious way as a complex concentrated in homology, they measure the failure of the global sections functor to preserve weak equivalences of such, its failure of “exactness.” General theory of model structures shows the uniqueness of this construction (that it does not depend of choice of fibrant or cofibrant resolution, etc.)