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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.

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I don't know the answer to your question, and didn't know the answer to the previous one either. But both times I thought of Goodwillie Calculus. Have you learned much about that subject? It seems to be valuable when trying to realize things as (ho)colimits of towers of truncated versions, and I feel like I've seen that machinery applied to $E_\infty$ things before. Just some idle speculation. – David White Mar 19 '13 at 23:18
Taking homotopy colimits of 0-truncated connective spectra gives you much more than just Eilenberg-MacLane spectra. For instance, it includes all 1-connective spectra with finitely many homotopy groups, because you can build them by repeatedly taking cofibers of maps from connective Eilenberg-MacLane spectra. – Eric Wofsey Mar 20 '13 at 1:31
@Eric Wolfsey: I had in mind diagrams indexed by Δ^op, similarly to the first example. I edited the question accordingly. – Dmitri Pavlov Mar 20 '13 at 18:33
up vote 10 down vote accepted

If you don't group complete, then free $E_{\infty}$-spaces are $1$-truncated. Consequently, for $k > 0$, the answer is "all $E_{\infty}$-spaces". When $k=0$, you'll get those which are homotopy equivalent to simplicial commutative monoids.

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Thanks a lot for the answer! Is this result still true if we insist that all diagrams for homotopy colimits are simplicial, i.e., indexed by Δ^op? – Dmitri Pavlov Mar 20 '13 at 3:06
@Dmitri: I believe that one way to see this is to use hypercovers. Namely, given an $E_\infty$-space $X$, choose a free one $F$ (on a discrete set) with a map $F \to X$ which is a surjection on $\pi_0$. Now form the fiber product $F \times_X F$ and choose an free $E_\infty$-space (on a discrete set) $F'$ with a map $F' \to F \times_X F$ inducing a surjection on $\pi_0$. This is a small piece of a simplicial resolution for $X$. Repeating this process, you get a simplicial $E_\infty$-space whose geometric realization is $X$. – Akhil Mathew Mar 20 '13 at 23:49

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