Whenever I'm doing sheaf things and I have a construction that involves sheafifying, I find it convenient to think "the thing that has the same stalks but sections must be locally trivial sections to espace etale" - it's a nice way to think of/characterize sheafification, sections of sheaves, define pullbacks, etc.

Can we do the same thing on a general, reasonable site (e.g. etale, flat, etc), i.e. for say $\mathscr{F}$ a sheaf on a scheme $X$ on the etale site, can we form something like $\coprod \mathscr{F}_x$ as $x$ runs over geometric points, suitably "Grothendieck topologized" so that the Grothendieck continuous sections over $U \to X$ are precisely $\mathscr{F}(U \to X)$?

  • $\begingroup$ What is a "Grothendieck continuous" morphism? And what category is it supposed to be a morphism of? $\endgroup$ – Kevin Carlson Mar 14 at 16:21
  • $\begingroup$ I don't know, that's part of the question of what such a formulation would mean. $\endgroup$ – edgarlorp Mar 14 at 17:53
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    $\begingroup$ Ok. The problem is that Grothendieck topologies very specifically generalize the property of topological spaces that they produce a sheaf theory, and not that they produce a notion of continuous map. $\endgroup$ – Kevin Carlson Mar 14 at 19:51

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