identity functors. Compactly generated triangulated categories Let D be a triangulated category such that direct sums indexed by an arbitrary set (not necessarily finite) exist in D. An object X in D is called compact if the functor commutes with direct sums. Explicitly, this means that for every family of objects in D indexed by a set S, the natural homomorphism of abelian groups is an isomorphism. This is different from the general notion of a compact object in category theory, which involves all colimits rather than only coproducts. For example, a compact object in the stable homotopy category is a finite spectrum. A compact object in the derived category of a ring, or in the quasi-coherent derived category of a scheme, is a perfect complex. In the case of a smooth projective variety X over a field, the category Perf(X) of perfect complexes can also be viewed as the bounded derived category of coherent sheaves, . A triangulated category D is compactly generated if D has arbitrary (not necessarily finite) direct sums; There is a set S of compact objects in D such that for every nonzero object X in D, there is an object Y in S with a nonzero map for some integer n. Many naturally occurring "large" triangulated categories are compactly generated: The derived category of modules over a ring R is compactly generated by one object, the R-module R. The quasi-coherent derived category of a quasi-compact quasi-separated scheme is compactly generated by one object. The stable homotopy category is compactly generated by one object, the sphere spectrum . Amnon Neeman generalized the Brown representability theorem to any compactly generated triangulated category, as follows. Let D be a compactly generated triangulated category, a cohomological functor which takes coproducts to products. Then H is representable. (That is, there is an object W of D such that for all X.) For another version, let D be a compactly generated triangulated category, T any triangulated category. If an exact functor sends coproducts to coproducts, then F has a right adjoint. The Brown representability theorem can be used to define various functors between triangulated categories. In particular, Neeman used it to simplify and generalize the construction of the exceptional inverse image functor for a morphism f of schemes, the central feature of coherent duality theory. t-structures For every abelian category A, the derived category D(A) is a triangulated category, containing