I asked a part of this in an earlier question, but that part of my question didn't receive precedence.

Etale site is useful - examples of using the small fppf site?

Let $X$ be a scheme (assume it is as nice as you like). There is a description of "points" in the (small) etale site $X_{et}$, and these are the geometric points of $X$. More generally, I've heard that the notion of "points" makes sense in any site (maybe "any" is a little too strong?).

1.) Can you give me a reference defining "points" in other sites. Specifically, I am interested in the small fppf site over a scheme and the big etale site. Is the notion of "points" a useless notion in sites other Zariski and small etale?

2.) What are "points" in other sites "supposed" to do? Is there an analogy that we keep in mind (as to why they are called points)? In the case of the Zariski site, the "points" have a natural structure of a locally ringed space - (the local rings being the stalks in the Zariski site) and this gives a canonically associated locally ringed space to a given site. An analogy similar to this doesn't seem to hold in the small etale site over a scheme.

3.) To whatever a "point" is, I expect one would have a naturally associated local ring. Is this the case in the small fppf site over a (nice?) scheme? This is of course the case in the etale and Zariski site. The small fppf site over a scheme seems a little strange, since limits tend not to be directed.

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    $\begingroup$ There's an example of Deligne of a site with no points. $\endgroup$
    – arsmath
    Dec 30 '12 at 8:35
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    $\begingroup$ Regarding 1) : in SGA4 you can find the more general definition of a point of a topos, it is simply a morphism from the topos Sets to the given topos. $\endgroup$
    – Niels
    Dec 30 '12 at 9:22
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    $\begingroup$ Articles by Stefan Schröer and by Ofer Gabber and Shane Kelly discuss the points of the fppf site. $\endgroup$ May 13 '17 at 22:18

See SGA 4, Exposé VIII, 7.8, which defines an abstract point of a site as a functor from the topos of that site (i.e. the category of sheaves of sets on that site) to the category of sets that commutes with arbitrary inductive (direct) limits and finite projective (inverse) limits.*

One should think of a functor as being the functor that takes a sheaf to its stalk at that point (the "fiber functor" of the point). Since every fiber functor preserves such limits, these are reasonable axioms to postulate.

Grothendieck then proves in this exposé that in the étale site, every functor satisfying the axiom above comes from a (unique) geometric point (up to isomorphism in a suitably-defined sense).

Furthermore, and I can only find this referenced in Brian Conrad's unavailable draft book on the Ramanujan conjecture, in a (classical) topological space where every irreducible subset has a unique generic point (also known as a sober space), such as a normal Hausdorff space or the underlying space of a scheme, every abstract point of the category of sheaves on the topological space corresponds to a unique classical point. This is Proposition 2, Section 3, Chapter IX of Sheaves in Geometry and Logic.

Why characterize points in terms of their functors? This fits in with the general Tannakian formalism, which emphasizes fiber functors, and in particular, defines the fundamental group to be the group of automorphisms of the fiber functor. I.e., we should think of a point as the functor that assigns to a sheaf its stalk at that point.

This also fits into a more general philosophy that we should ignore the site and focus on the topos entirely. One can in fact put a Grothendieck topology on the topos so that it is equal to the category of sheaves on itself.

*Edit: One can also find this in Sheaves in Geometry and Logic, by Maclane and Moerdijk, Chapter VII, Section 5.

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    $\begingroup$ Spaces of the form you mention in your fourth paragraph are called "sober", and the correspondence between points of the topos and points of the space is a straightforward corollary of the fact that sober spaces can be recovered up to homeomorphism from their frame of open subsets, which is manifestly frame of subterminal objects of the corresponding sheaf topos. This can be found in the usual topos theory references such as Sheaves in geometry and logic or Sketches of an elephant. $\endgroup$
    – Zhen Lin
    Dec 30 '12 at 11:03

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