TITLE: What does an etale topos classify?
QUESTION [24 upvotes]: Any Grothendieck topos E is the "classifying topos" of some geometric theory, in the sense that geometric morphisms F→E can be identified with "models of that theory" internal to the topos F.  For the topos of sheaves on a site C, the corresonding theory may tautologically be taken to be "the theory of cover-preserving flat functors on C."  However, for some naturally arising toposes of interest, the classified theory has a different, more intuitive expression.  For instance, the topos of simplicial sets classifies linear orders with distinct endpoints, and the "Zariski topos" classifies local rings.
My question is: if X is a scheme—say affine for simplicity—then what theory does its (petit) etale topos $Sh(X_{et})$ classify?  Can it be expressed in a nice intuitive way, better than "cover-preserving flat functors on the etale site"?  I hope/suspect that it should have something to do with "geometric points of X" but I'm not sure how to formulate that as a geometric theory.

REPLY [14 votes]: For a general scheme $(X,\mathbb{O}_X)$, the étale topos over $X$ classify the strict henselisation (also called the separable closure) of the locale ring $\mathbb{O}_X$.
The result is clearly due to Monique Hakim in her thesis ("topos annelés et schémas relatifs") but as she did not use at all the concept of internal logic there is no description of the precise geometric theory involved. So if you look for a full and definite answer to this question, you should look in :
G. C. Wraith - Generic Galois theory of local rings
which indeed give a clear description of the theory involved.