Question

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.

Answer 1

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.