Representability theorems for presheaves of spectra

Abstract

Suppose that M is a simplicial model category and that F is a contravariant simplicial functor defined on M which takes values in pointed simplicial sets. This note displays conditions on the simplicial model category M and the functor F such that F is representable up to weak equivalence. The conditions on F are homotopy coherent versions of the classical conditions for Brown representability, while M should have the fundamental properties of the stable model structure for presheaves of spectra on a Grothendieck site.