TITLE: Cohomology theories as colimits
QUESTION [8 upvotes]: 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.

REPLY [3 votes]: Suppose you just look at colimits indexed by $\mathbb{N}$, and fix the cohomological degree $n$.  I claim that there must exist $j$ such that the map $h_j^n(X)\to h^n(X)$ is surjective for all $X$.  Indeed, if not, we can choose spaces $X_j$ and classes $a_j\in h^n(X_j)$ that do not come from $h^n_j(X_j)$.  Put $X=\coprod_jX_j$.  As $h$ is assumed to be a cohomology theory, there is a unique element $a\in h^n(X)$ with $a|_{X_j}=a_j$ for all $j$.  If this comes from some $a'\in h^n_j(X)$ then we see that $a'|_{X_j}$ maps to $a_j$, contrary to assumption.  (We do not need to assume that $h_j^n$ converts coproducts to products for this, we are just using functoriality.)  
I suspect that under mild assumptions one can prove something stronger: there is a cofinal sequence $j_1<j_2<\dotsb$ and functors $u_k$ such that $h_{j_k}^n=h^n\oplus u_k$ with the map $h_{j_k}\to h_{j_{k+1}}$ being zero on $u_k$.  This means that there are no really interesting examples.  All this depends on the colimit formula being valid for infinite complexes, of course.