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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