The étale topos of a scheme is the classifying topos of which groupoid? [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.
 A: 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 $T$.


*

*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 $M$).

*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.

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

*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.
