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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.

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up vote 19 down vote accepted

It classifies what the Grothendieck school calls "strict local rings". The points of such a topos are strict Henselian rings (Henselian rings with separably closed residue field). See Monique Hakim's thesis (Topos annelés et schémas relatifs $\operatorname{III.2-4}$) for a proof and a more precise definition of what constitutes a "strict local ring" in a topos.

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I should also note that the corresponding question for the fppf-topos appears to be an open problem. – Harry Gindi Dec 8 '10 at 23:47
Where does X come in? – Mike Shulman Dec 9 '10 at 18:56
@Mike: It's just a change of base. If X is non-affine, then it may be a bit more complicated though. – Harry Gindi Dec 9 '10 at 19:00
For instance, changing base to $Spec R$ for a ring $R$ means that we get the strict henselian R-algebras as points. – Harry Gindi Dec 9 '10 at 19:01
Hm. This paper claims that the little etale topos of R classifies "strict Henselizations" of the ring R, whereas it is the big etale topos of R which classifies strict Henselian R-algebras. – Mike Shulman Dec 16 '10 at 3:19

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.

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