The classical Brown representability theorem is for set valued functors. Is there a version for abelian group valued functors, and ring valued functors?
In other words say we have an abelian group valued functor F on the category of CW top. spaces, satisfying the necessery condition that F maps colimits to limits. What extra conditions on F do we need to ensure that the classifying object is an H-space. Actually Brown doesn't state this, but at a brief glance his paper seems to prove that F just needs to satisfy excision that is we have exact sequences $$0 \to F (V \cap W) \to F(V) \oplus F (W) \to F (V \cup W) \to 0,$$ for V,W open sets in X. Is this right? What about the case of ring valued functors, when are they representable by (E_\infty? whatever that is)-ring space.