The standard uses of toposes in algebraic geometry come from sites that look roughly like the syntactic sites of theories of local rings that they classify. This isn't particularly surprising, since this corresponds to allowing localizations as coverings in the site. Are there any interesting uses of classifying toposes of theories of rings that are not necessarily local?

  • $\begingroup$ Are you asking for example of interesting Grothendieck topology on the category of affine scheme which do not contains the Zariski topology ? $\endgroup$ – Simon Henry Apr 15 '20 at 21:23
  • $\begingroup$ @SimonHenry That, plus some description of the resulting topos as the classifying topos of a theory of rings. $\endgroup$ – Cameron Zwarich Apr 15 '20 at 21:36
  • $\begingroup$ Well the theory of all rings is classified by the category of functors from finitely-presented rings to Sets. I do not know any other example of such topos used in algebraic geometry, but maybe a geometer can comment on this. $\endgroup$ – Simon Henry Apr 15 '20 at 22:44
  • $\begingroup$ There are examples where the rings are not just local, but this is probably not what you want. $\endgroup$ – David Roberts Apr 16 '20 at 2:07
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    $\begingroup$ Sorry for storm of replies, but here is Hutzler's thesis, Internal language and classified theories of toposes in algebraic geometry $\endgroup$ – David Roberts Apr 16 '20 at 5:42

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