My question is motivated by the discussion in this question.
Quick summary: It may be tempting to think that sheaves are determined by their stalks, but this is not the case (far from it). Sheaves are more than a bunch of stalks - the stalks are glued by the etale topology. This gluing together is a (perhaps the most) central aspect of its structure.
From this I understand that different sheaves could have the same stalks. I managed to think of two reasons for this:
- Given a presheaf, one may glue its stalks in different ways. Perhaps it's possible to construct sheaves in different, non-universal ways: instead of going to the etale bundle and then taking the sheaf of sections, to use some other topology or whatever.
- The "origins" of different sheaves with the same stalks are really just different presheaves with the same stalks. This happens because the etale space topology depends on the presheaf itself: the basis is the set of images of the maps $\dot s:x\mapsto \mathrm{germ}_x(s)$ where $s\in PU$ is a section of the presheaf $P$ (I think this is a final topology).
Since the category of etale bundles is equivalent to the category of sheaves, it seems that option 1 is "wrong" in the sense that even though it may be possible to realize sheaves by some strange construction, they're always realizable as sheaves of sections of etale bundles.
This temps me to conclude that option 2 is the "explanation" for different sheaves with the same stalks: it happens simply because presheaves are not determined by their stalks. This seems too simple to be true... What am I missing?
As has been pointed in the comments, my question is vague. I'm trying to understand where the differences in the etale space topology (that make different sheaves with the same stalks possible) come from.