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[Sent here from Math.StackExchange by suggestion of an user.]

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, acrtually a topological groupoid since it is "coherent" and thus has enough points by Deligne's theorem; but which groupoid is this in concrete? Do you know if someone has ever investigated that? Thank you in advance.

P.S.: I have been suggested to look at the proof and try to reconstruct the particular case, which I am going to try.

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    $\begingroup$ The first step in Joyal and Tierney's construction of the localic cover is to choose a small generating family in the topos, so this is unlikely to ever lead to something concrete. The localic cover is the classifying topos for enumerations (= surjections from $\mathbb N$) of $\coprod_i X_i$ where $\{X_i\}$ is small generating family (e.g. etale $X$-schemes that are affine over an affine open in $X$). $\endgroup$ – Marc Hoyois Jun 18 '18 at 1:18
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I'm not familiar with Joyal and Tierney's construction, but Butz and Moerdijk's construction seems fairly straightforward to me. Also, doing the etale topos is a bit beyond my reach, so here's a description for the case of the classifying topos $\mathcal E_T$ of some first-order theory $T$.

  1. Select a set $\mathcal M$ of enough points of the topos. A point of the topos $\mathcal E_T$ is a model $M$ of the theory $T$. Here we can take $\mathcal M$ to be the set of all models of cardinality at most the size of the language of $T$ (cardinality of a model $M$ can be determined by adding up the sizes of all definable sets in $M$).

  2. An object of your topological groupoid is a model $M$ from the set $\mathcal M$, equipped with an enumeration by a fixed set $I$ which is "infinitely redundant": every element of the model gets counted infinitely many times. The enumeration is a technicality in order to get the topology to work out correctly.

  3. A morphism of your topological groupoid is an isomorphism, which need not respect the enumerations.

  4. The topology is such that for each $i \in I$ and each definable set $D$, the set of all enumerated models $f: I \to M$ such that $f(i) \in D$ is an open set.

So basically, Butz and Moerdijk express a topos $\mathcal E$ as equivariant sheaves on the topological groupoid of points of $\mathcal E$, cut down in size to make it small, and fattened up a bit (passing to an equivalent groupoid) with enumerations in order to be able to define the topology.

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  • $\begingroup$ I should point out that I think Butz and Moerdijk explain this pretty clearly and in a bit more detail in section 2 of their paper. $\endgroup$ – Tim Campion Jun 19 '18 at 4:56
  • $\begingroup$ Thanks. Yes, the construction is pretty clear in abstract. The point of view that you enlighten could perhaps suggest that, knowing the theory that the étale topos classifies (those "strict local rings"), one could move from that and construct the groupoid...? $\endgroup$ – W. Rether Jun 20 '18 at 15:07
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    $\begingroup$ Tentatively, that seems right -- this should be the topological groupoid of ($I$-enumerated) strictly henselian local rings (with a cardinality bound) and isomorphisms, with topology such that for every definable set $D$ in the coherent theory of strictly henselian rings (in particular, this is an intuitionistic theory), the set of enumerated models $f: I\to R$ such that $f(i) \in D$ is an open set. Working over a base scheme $S$, this should all be internal to sheaves on $S$, I suppose. $\endgroup$ – Tim Campion Jun 21 '18 at 0:12
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    $\begingroup$ I'm actually not clear on what theory exactly is classified by the etale topos $\mathcal E_S$ over a base scheme $S$. I think it might be "the theory of strictly henselian algebras over $\mathcal O_S$", whatever that means. But whatever it is, the topological groupoid one gets will still be the groupoid of points of $\mathcal E_S$, with some appropriate topology. $\endgroup$ – Tim Campion Jul 1 '18 at 16:08
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    $\begingroup$ I can confirm your guesses regarding what is classified by the étale topos. A reference in the case of $X = \mathrm{Spec}(R)$ is a very nice paper by Mathieu Anél. What Tim is saying doesn't have an external meaning, since $\mathcal{O}_S$ is not a ring, but a ring object in $\mathrm{Sh}(S)$. However, it has from the internal point of view of $\mathrm{Sh}(S)$; it correctly characterizes the étale topos in the world of toposes over $\mathrm{Sh}(S)$. This doesn't seem to be explicitly written down anywhere. It might be the case that my thesis comes closest. $\endgroup$ – Ingo Blechschmidt Jul 2 '18 at 9:05

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