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Let $E_n$ be the little $n$-cubes operad which acts on $n$-fold loop spaces (up to group completion, an $E_n$-action is precisely the data needed to perform an $n$-fold delooping). I am looking for precise statements along these lines:

  • The $E_n$-operad can be modeled algebraically, over $\mathbb{Q}$ (even $\mathbb{Z}$?). These "spaces" are supposed to come from moduli stacks of curves via some sort of (Deligne-Mumford if $n = 2$?) compactification.
  • Let's take $n = 2$, so the spaces of the $E_2$-operad as the classifying spaces of braid groups (and in particular are not simply connected). Then applying the étale homotopy type gives an operad in pro-spaces; the homotopy inverse limit should give an operad in spaces, which at each level is the classifying space of profinite completion of the braid group.
  • This gives the profinite completion of the $E_2$-operad an action of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. This action is faithful.
  • Somehow, this is related to Drinfeld's associators, which are supposed to be related to deformations of braided monoidal (i.e., $E_2$) categories.

I have heard vaguely (from skimming various papers and YBL's answers here and here) that there is a story resembling the above four assertions, but I do not know what it is. Could anyone explain this relationship (or suggest references)?

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  • $\begingroup$ Do you know math.univ-lille1.fr/~fresse/OperadHomotopyBook? $\endgroup$ – archipelago Oct 5 '13 at 16:02
  • $\begingroup$ @archipelago: yes, I've started skimming Fresse's book, but I haven't quite been able to find these assertions. $\endgroup$ – Akhil Mathew Oct 5 '13 at 16:32
  • $\begingroup$ I don't have time to write anything long right now, but I want to comment that the usual explanation for the Galois action on the small disks is not the one you are describing but goes via the Grothendieck-Teichmuller group. GT can be defined as the automorphism group of a completion of the operad of fundamental groupoids of the small disks, you get different flavors of GT depending on what kind of completion you choose. Drinfeld-Ihara proved that the profinite GT contains the absolute Galois group. This is the way it's explained in Fresse's book. $\endgroup$ – Dan Petersen Oct 5 '13 at 19:59
  • $\begingroup$ At the risk of shameless self promotion you might find the brief summary in my paper on the little disk operad useful. It's on the arXiv. $\endgroup$ – Dan Petersen Oct 5 '13 at 20:00
  • $\begingroup$ @Dan: wait, I thought the inclusion of the absolute Galois group into the profinite Grothendieck-Teichmuller group exactly came from a version of this story. Are you claiming otherwise? $\endgroup$ – Qiaochu Yuan Oct 5 '13 at 21:12

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