Clarify definition of 'enough points'

Question

I have checked the stacks project, Sketches of an Elephant, and Sheaves in Geometry and Logic but cannot find a satisfactory answer. If a site or topos has enough points, does that mean, that there is a SET of points, such that isomorphisms can be checked on that set of points? Or do we run into set-theoretic troubles.

Answer 1

In the usual usage, for a topos to have enough points it is not required that there is a set of points such that isomorphisms can be detected by this set of points. It's enough if the full class of points does.

That said, it is a theorem that for Grothendieck toposes, the collection of all points detects isomorphisms if and only if there is a set of points which do so. This theorem is for instance stated on slide 3 of these slides by Olivia Caramello, and a proof is contained in Corollary C2.2.12 of the Elephant.

Finally, let me make a personal remark: As long as we don't precisely specify which foundations we work in, the question (and this answer) have a somewhat ill-defined flavor. Mike Shulman has a nice survey on available set-theoretic foundations, and then there are also foundations which are not set-theoretical in nature.

Addendum. Here is a concrete example. The ring classifier, that is the classifying topos of the theory of rings, that is the category $[\mathrm{Ring}_{\mathrm{fp}},\mathrm{Set}]$ of $\mathrm{Set}$-valued functors on the category of finitely presented rings, has a proper class of points, namely all rings. But for detecting isomorphisms it suffices to check at the finitely presented rings (which are incidentally precisely the "essential points" in the technical sense of essential geometric morphism), of which there is still a proper class, but we can restrict to some small, even countable skeleton of those.