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.