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Hello !

If $X$ is a scheme, we can consider the etale topos of $X$ whose object are etale scheme above $X$ with the etale topology.

My question is : is there a know way to express this topos as the classifying topos of some geometric theory ? Of course it is possible, just because it's a grothendieck topos, but I'm looking for an explicit theory at least on some particular case (like when $X$ is affine, or when $X$ is the spectrum of the ring of integer of a number field, or when $X$ is a projective curve over a finite field... )

For example, if $A$ is a ring, then the Zariski topos of $Spec A$ (topos of finite presentation scheme above $Spec A$ with the Zariski topology) is the classifying topos of the theory of local $A$ algebra. (the universal local $A$ algebra being the structural sheaf).

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I cannot give the details, but my guess is that the etale topos should be the classifying topos of the theory of strictly local $A$ algebras. By a strictly local $A$ algebra I mean a henselian local algebra with separably closed residue field. I don't know if this is a honest algebraic theory.

Bonus: in this vein the Nisnevich topos should be the classifying topos of the theory of henselian local $A$ algebras. The proof should follow similar lines to the previous one.

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    $\begingroup$ The link by Achilleas K supports your guess. $\endgroup$
    – Martin Brandenburg
    Commented Mar 23, 2012 at 17:12

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