Representability theorems for presheaves of spectra

The Brown representability theorem gives a list of conditions for the representablity of a set-valued contravariant functor which is defined on the classical stable homotopy category. It has had many uses through the years, and has long been part of the canon of Algebraic Topology. It is entirely reasonable to ask for a more general version of the theorem, which would give conditions on a closed model category N and a contravariant set-valued functor G defined on the homotopy category Ho(N ) so that the functor G is representable. One could call this a Brown representability theorem for N , although some might say that it is a “cohomological” Brown representability result [1], [9]. A result of this type is proved in this paper, and appears as Theorem 19. The conditions for the result are essentially classical: the model category N must have a set of compact generators, suitably defined, while the functor G should take coproducts to products and should satisfy a Mayer-Vietoris property. Theorem 19 asserts that G is representable under these circumstances. The proof displayed for this result is the standard argument (see also the proof of Theorem 3.1 in [8], or Heller’s purely categorical formulation in [2]), albeit translated into the language of model categories. The ideas behind Theorem 19 and its proof are not new. Multiple settings in which Theorem 19 applies are displayed in the third section, following the proof. The basic message is that there are classical Brown representability results for various stable model structures arising from simplicial presheaves — these include presheaves of spectra, diagrams of spectra, and motivic T -spectra, as well as presheaves of chain complexes — so long as the underlying local model structure is defined on a rather forgiving Grothendieck topology, for which a set of compact generators can be defined in a traditional way.