Etale topos as a classifyng topos ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:55:03Z http://mathoverflow.net/feeds/question/92012 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92012/etale-topos-as-a-classifyng-topos Etale topos as a classifyng topos ? Simon Henry 2012-03-23T14:37:29Z 2012-03-23T15:09:02Z <p>Hello !</p> <p>If \$X\$ is a scheme, we can consider the etale topos of \$X\$ whose object are etale scheme above \$X\$ with the etale topology.</p> <p>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... )</p> <p>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).</p> http://mathoverflow.net/questions/92012/etale-topos-as-a-classifyng-topos/92016#92016 Answer by Leo Alonso for Etale topos as a classifyng topos ? Leo Alonso 2012-03-23T15:09:02Z 2012-03-23T15:09:02Z <p>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 <em>strictly local</em> \$A\$ algebra I mean a henselian local algebra with separably closed residue field. I don't know if this is a honest algebraic theory.</p> <p>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.</p>