# Is there a fiber bundle/espace-etale interpretation of sheaves on a Grothendieck site

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 $$\mathscr{F}(U \to X)$$?

• What is a "Grothendieck continuous" morphism? And what category is it supposed to be a morphism of? – Kevin Carlson Mar 14 at 16:21
• I don't know, that's part of the question of what such a formulation would mean. – edgarlorp Mar 14 at 17:53
• 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. – Kevin Carlson Mar 14 at 19:51