Representability theorems for presheaves of spectra
The Brown representability theorem gives conditions for the representability of a set-valued contravariant functor defined on the classical stable homotopy category. Let \({\mathrm {s{\mathbf Set}}_*}\) be the category of pointed simplicial sets. Suppose \(\mathcal{M}\) is a closed simplicial model category and \(F:\mathcal{M}^{op}\to {\mathrm {s{\mathbf Set}}_*}\) is a contravariant functor. The main result of this paper gives conditions on \(\mathcal{M}\) and \(F\) which imply that \(F\) is sectionwise weakly equivalent to a representable functor \(hom( \;,Y)\), where \(Y\) is some fibrant object of \(\mathcal{M}\). The conditions on the simplicial model category \(\mathcal{M}\) are abstractions of the general behaviour of the stable model structure of presheaves of spectra. The functor \(F\) should preserve weak equivalences between cofibrant objects and should take homotopy colimits to homotopy inverse limits.
scientific article; zbMATH DE number 5802621