This question is closely related to my previous question about modules over truncated sphere spectra, in particular, it has the same motivation.
Recall that every space (or ∞-groupoid) can be represented as the homotopy colimit of some simplicial diagram of 0-truncated spaces, i.e., sets considered as homotopy 0-types. For example, if X is a Kan simplicial set, then the homotopy colimit of X considered as a functor X: Δop → Set → Space is equivalent to X itself.
For connective spectra (i.e., group-like E∞-spaces) a more complicated picture emerges: homotopy colimits of simplicial diagrams of 0-truncated connective spectra (i.e., abelian groups) are precisely Eilenberg-MacLane spectra (of connective chain complexes of abelian groups). More generally, homotopy colimits of simplicial diagrams of k-truncated connective spectra for some k>0 are modules over the k-truncation π≤kS of the sphere spectrum S. In particular, not every connective spectrum can be represented in this way. (Presumably, being a module over π≤kS is also a sufficient condition, but I haven't checked any details to claim this.)
I wonder what happens in the intermediate case of (not necessarily group-like) E∞-spaces.
Which E∞-spaces can be represented as homotopy colimits of k-truncated E∞-spaces for some k>0? What if we require the diagrams to be simplicial, i.e., indexed by Δop?
In particular, if some E∞-spaces cannot be represented in this way, what tools do we have to detect this?
Specifically, I'm interested in the answer for the case of E∞-spaces coming from connective spectra like MU, MSO, KU, or KO.