# Cohomology theories for spaces vs cohomology theories for spectra

It is a standard consequence of the Brown Representability Theorem for $$\operatorname{Ho}(\operatorname{Top}_*)$$ that the category of generalized cohomology theories for spaces (pointed CW complexes, more specifically) is equivalent to the stable homotopy category $$\operatorname{SHC}$$ (defined as the homotopy category of the stable model structure on sequential spectra) via representability. On the other hand, we can define a cohomology theory for spectra (as in Barnes and Roitzheim's Foundations of Stable Homotopy Theory) to be a contravariant functor $$E^*:\operatorname{SHC}^{op}\to\operatorname{Ab}_*$$ such that

1. Each exact triangle in $$\operatorname{SHC}$$ gives rise to a long exact sequence in $$\operatorname{Ab}$$
2. $$E^n$$ preserves products for each $$n$$ (that is, it sends wedge sums to products of abelian groups)
3. $$E^*$$ preserves suspension up to a specified natural isomorphism $$E^{n+1}(\Sigma X)\cong E^n(X)$$.

Then it is easy to check that $$[,E]$$ is always a cohomology theory for spectra. But is every cohomology theory for spectra thus represented? Given such a cohomology theory $$E^*$$, we obtain an associated cohomology theory for spaces by restricting to suspension spectra. Thus, by the result for spaces mentioned above, the question becomes whether a cohomology theory for spectra is determined by its restriction to spaces.

This is certainly true if we require $$E^*$$ to preserve sequential homotopy colimits, because any spectrum is weakly equivalent to a CW spectrum with basepoint at its unique $$0$$-cell, which is a sequential homotopy colimit of homotopy cofibers of coproducts of shifted sphere spectra. But without this requirement, does the result still hold? If so, why? If not, is there a standard counterexample?

• Regarding the non-equivalence of cohomology theories and the stable homotopy category see also this mathoverflow question: mathoverflow.net/questions/117684/… -- For representability results for cohomology theories see also the very nice book 'Axiomatic Stable Homotopy Theory' by Hovey, Palmieri and Strickland. – Lennart Meier Jul 9 at 11:24

• @DoronGrossman-Naples : it's possible that you get sequential colimits back from asking $E^n$ to respect all products, and not just finite ones – Maxime Ramzi Jul 9 at 8:20