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By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. Butz. and I. Moerdijk. Representing topoi by topological grupoids. Journal of Pure and Applied Algebra 130, 223-235, 1998) that topoi "with enough points" admit actually a representation as classifying topoi of topological groupoids.

Now my question is the following: take a well-known topos, as the étale topos for a scheme. This is the classifying topos of a localic groupoid, but which one? Do you know if someone has ever investigated that? Thank you in advance.

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    $\begingroup$ You might want to look at the work of Ingo Blechschmidt, whose is working on similar things (characterizing the big and small Zariski topoi for example), the answer might just be in his Thesis. github.com/iblech/internal-methods $\endgroup$ – user45878 Apr 8 '18 at 20:25
  • $\begingroup$ The answer isn't actually in that work, at least from a first look, but thank you very much for the reference! It really looks interesting. Any other suggestions are welcome. $\endgroup$ – W. Rether Apr 9 '18 at 17:50
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    $\begingroup$ This is an excellent question. While some parts of my thesis might be tangentially relevant, it doesn't contain an answer to this question. Could you migrate this question to MathOverflow? $\endgroup$ – Ingo Blechschmidt Jun 3 '18 at 10:30
  • $\begingroup$ I know very little algebraic geometry, but my understanding from what I have heard other people say in passing is the following: if $X=\mathrm{Spec}(R)$ for a local ring $R$, then the \'etale topos of $X$ is sheaves on the absolute Galois group of the residue field of $R$; in general, the topological groupoid you get is obtained by gluing these together. It looks like Pirashvili - The \'Etale Fundamental Groupoid as a Terminal Costack has a theorem along these lines, but I have not read it. (Aside: Wraith - Generic Galois Theory of Local Rings answers the question in your title, but not body). $\endgroup$ – ne- Jul 17 '18 at 18:03
  • $\begingroup$ This is too old to migrate but you may want to consider asking it there. $\endgroup$ – Alexander Gruber Oct 16 '18 at 14:57

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