As it is described for example in [Mac Lane-Moerdijk, Sheaves in Geometry and Logic, II.6.], one can construct the sheafification functor very lucidly by associating to a presheaf a certain bundle (cf. espace etale) and then taking its sheaf of sections.

This construction is outlined in the reference for presheaves $\mathcal{O}(X)^{op}\to Sets$ where $\mathcal{O}(X)$ is the category of opens for a topological space $X$.

Does this method work for presheaves $C^{op}\to Sets$ on a general small Grothendieck site, too, and is this written down somewhere?

(Let's assume also that the associated topos has enough points.)

  • $\begingroup$ Martin Brandenburg [mentioned this][1] as a "common false belief"—the problem is that we might have to take a colimit over an indexing category that is not essentially small. But it should be possible to sheafify presheaves on most reasonable sites. For example, the category of étale maps to a scheme is essentially small (I think), so this problem does not arise. [1]: mathoverflow.net/questions/23478/… $\endgroup$ – Andrew Dudzik Oct 20 '11 at 17:02
  • $\begingroup$ Thank you Andrew. Of course, $C$ has to be (essentially) small. $\endgroup$ – K Shao Oct 20 '11 at 17:21
  • $\begingroup$ See the first proof on p61 of Milne's book on etale cohomology for the etale site. Something similar may work in general provided there are enough points. $\endgroup$ – anon Oct 20 '11 at 17:56

It depends on what you mean by an étalé "space". As long as $C$ has a small set of topological generators (i.e. as long as $Sh(C,J)$ isn't too large to be a topos), there always exists a certain version of an étalé space: If $F$ is a sheaf on $(C,J),$ the slice topos $Sh(C,J)/F$ has a canonical étale projection $$\pi_F:Sh(C,J)/F \to Sh(C,J).$$ This map is a local homeomorphism of topoi. This topos with this local homeomorphism is the étalé space of $F.$ Indeed, we may make this construction for each object $c \in C,$ call it $U(c):=Sh(C,J)/y(c),$ where $y(c)$ is the (possibly sheafified) Yoneda embedded object. Then, "sections of $\pi_F$ over $U(c) \to Sh(C,J)$" are in bijection with elements of $F(c).$ If the Grothendieck site $(C,J)$ happened to be the canonical site of a topological space, then each slice $Sh(C,J)/F$ is equivalent to sheaves on the étalé space of that sheaf, and the projection corresponds to the usual one. In particular, $U(c) \to Sh(C,J)$ corresponds to the inclusion of an open subset. So, this is reduces to the usual construction for spaces. Another example is, if $(C,J)$ were the small étale site of some scheme $S$, then each $Sh(C,J)/F$ is the small étale site of some algebraic space (with no seperation conditions) étale over $S,$ with $\pi_F$ corresponding to the étale map from this algebraic space to $S.$


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