I am looking for examples of cohomology theories that can be written as (filtered, or another nice class of) colimits of "simpler" functors, i.e. which $\{h^n : {\bf Top}^2 \to {\bf Ab}\}_n$ are such that $$ h^n(X) \cong \text{colim}_j\; h_j^n(X) $$ for a suitable diagram ${\cal J}\to [{\bf Top}^2,{\bf Ab}]$.

Of course this is a really vague question:

- A "cohomology theory" (and the category thereof) is what (for example) Rudyak I.3.8 defines as such.
- I'm not asking that the $h^n_j$ are cohomology theories themselves, but you can assume this additional requirement.
- You're quite free to interpret the word "simple" in the way to like more. I'm in fact explicitly asking for which meaning of "simple" this question has a good answer.

The question remains a bit vague: whatever $h^n(X)$ is, you can take a presentation for this abelian group and say that it is a colimit. Nevertheless I think that asking for a colimit *of functors* is a bit more restrictive and avoids trivial cases.

My feeling is that the answer is always "quasi-affirmative": a cohomology theory, i.e. a spectrum, belongs to a presentable quasicategory. But spectra and cohomology theories aren't really the same thing.

finitecomplexes, then this would present an arbitrary cohomology theory in terms of one built from cohomology theories associated to suspension spectra. It's up to you to decide whether that's any easier... $\endgroup$ – Dylan Wilson Aug 24 '17 at 19:22finite complexesare, though. $\endgroup$ – Dylan Wilson Aug 25 '17 at 9:302more comments