I have recently been made aware of the fact that the category $\mathbf{Zar} = \mathbf{Sh}(\mathbf{Ring}_\mathrm{fp}, J)$ of sheaves on the category of finitely presented rings equipped with the Zariski topology is the classifying topos of local rings. (Mac Lane & Moerdijk, Sheaves in Geometry and Logic, $\S$VIII.6.) I have also heard of similar results regarding the étale (resp. Nisnevich) topology and strictly Henselian (resp. Henselian) rings.

I was wondering what consequences this has for the study and practice of algebraic geometry. In particular,

  • how could an algebraic geometer make use of these facts in their day-to-day work?
  • could (and if so, should) a student take them as a guiding principle in their study of the vast discipline of algebraic geometry?

and, perhaps going a bit too far afield,

  • does any of this connect to the story of $\mathbb{A}^1$-homotopy theory?

It is entirely possible that the only conclusion one should draw from these facts is that one should expect to encounter local (resp. Henselian, strictly Henselian) rings while practicing algebraic geometry, though I hope there is more to the story.

  • $\begingroup$ Also, perhaps this should be tagged as a soft question? $\endgroup$ – Brian Shin May 29 '17 at 22:28

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