5
$\begingroup$

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.

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ 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. $\endgroup$
    – David White
    Mar 19, 2013 at 23:18
  • $\begingroup$ 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. $\endgroup$
    – Eric Wofsey
    Mar 20, 2013 at 1:31
  • $\begingroup$ @Eric Wolfsey: I had in mind diagrams indexed by Δ^op, similarly to the first example. I edited the question accordingly. $\endgroup$
    – Dmitri Pavlov
    Mar 20, 2013 at 18:33

1 Answer 1

Reset to default
12
$\begingroup$

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.

Improve this answer
$\endgroup$
2
  • $\begingroup$ 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? $\endgroup$
    – Dmitri Pavlov
    Mar 20, 2013 at 3:06
  • 1
    $\begingroup$ @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 $X$. $\endgroup$
    – Akhil Mathew
    Mar 20, 2013 at 23:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.