I know that there are already several discussions on this topic (and they already helped me a lot, so far), but I couldn't find one completely solving my problem.

Question 1. Fix a scheme $X$. I know that the small étale topos associated to $X$ (i.e. the topos of sheaves over the site $X_{\mathrm{ét}}$ whose objects are étale maps toward $X$) has enough points (maybe an hypothesis is needed, like local noetherianity of $X$). Does the same hold true for the big étale topos associated to $X$ (i.e. the topos of sheaves over the site $(\mathbf{Sch}/X, \tau_{ét})$, maybe with suitable assumptions on $X$)?

Explanation of the title. It seems to me that this question is closely related to the following other question (a bit ill-posed, though): given the knowledge of $(\mathbf{Sch}/X, \tau_{ét})$ (i.e. we can decide whether a sieve is a covering sieve), can we decide whether a map is an étale map?

This originated as follow: I know that geometric morphisms $x \colon \mathbf{Set} \to \mathrm{Sh}(\mathcal C,J)$ corresponds (functorially) to flat and $J$-continuous functors $A_x \colon \mathcal C \to \mathbf{Set}$. The correspondence is given by $$ x^* = - \otimes_{\mathcal C} A_x, \qquad A_x = x^* \circ \rho \circ \mathbf y $$ where $\mathbf y \colon \mathcal C \to \mathrm{PSh}(\mathcal C)$ is the Yoneda embedding and $\rho \colon \mathrm{PSh}(\mathcal C) \to \mathrm{Sh}(\mathcal C,J)$ is the sheafification functor.

Given a point $x \colon \mathbf{Set} \to \mathrm{Sh}(\mathcal C,J)$ we can form the category of elements of $A_x$, $\int_{\mathcal C} A_x$. It comes equipped with a forgetful functor $$ P \colon \int_{\mathcal C} A_x \to \mathcal C $$ and if I didn't any mistake I should have proven that for any sheaf $F$ we have $$ F \otimes_{\mathcal C} A_x = \mathrm{colim}_{\int_{\mathcal C} A_x} F \circ P $$

Philosophically (and by no means mathematically) we can rephrase saying that in a sense, if this is correct, we are reviewing $P$ as a "generalized neighbourhood of a (possibly imaginary) point of the site $(\mathcal C,J)$" (this intuition is strenghtened by the fact that flatness of $A_x$ is equivalent to say that $\int_{\mathcal C} A_x$ is filtered).

I tried to make me some example to see whether what I wrote makes any sense or not. If $X$ is a topological space and we work over the small (open) site of $X$, namely $O(X)$, this is really what I expect (the stalk functor precisely coincides with the usual stalk functor encountered in "naif sheaf theory"). In a similar fashion, the same construction gives back the stalk functor usually employed in étale cohomology (direct limit over étale neighbourhoods). However, if I work over the big (open) site of the topological space $X$ (or over the big étale site of a scheme X), my intution breaks down completely, so that I am lead to the point of doubting that in those cases there are enough points.

How did I find myself wondering about such questions? Well, I was originally trying to solve an exercise in the super classical article of Jardine, Simplicial presheaves:

Exercise. If a site $(\mathcal C,J)$ has "enough points" (I interpret with $\mathrm{Sh}(\mathcal C,J)$ has enough points), then a map of simplicial presheaves $f \colon F \to G$ is a local fibration if and only if $f_x \colon F_x \to G_x$ is a Kan fibration for every point of $(\mathcal C,J)$.

Using the machinery of flat and continuous functors, I am able to prove that if $f$ is a local fibration, then $f_x$ is a Kan fibration for every point of $(\mathcal C,J)$ (without assuming that there are enough points). For the converse, using the same machinery, I am able to prove the statements for all the "small sites" I know, but I am embarassed in dealing with "big sites". For example, let us take the big étale site of $X$. In the category of elements $\int_{\mathbf{Sch}/X} A_x$ of a point, the maps are not étale! Therefore, when I produce a lifting using smallness of $\Delta^n$, I cannot be sure of factorizing through an étale cover of $X$... Therefore I am lead to the following question:

Question 2. If a site $(\mathcal C, J)$ has enough points (same interpretation of above), can we decide whether a sieve on an object $U$ is a covering sieve using only the data of points?

More precisely, I would like a criterion similar to the following:

a sieve $S$ on $U$ is a covering sieve if and only if for every point $A \colon \mathcal C \to \mathbf{Set}$ such that $A(U) \ne \emptyset$ there is $V \to U$ in $S$ such that $A(V) \ne \emptyset$.

However, this is false, and a counterexample can be found in the big Zariski site. Indeed, the points of this site are local rings. If $A$ is a local ring and $F$ is a sheaf on this big site, the stalk of $F$ at $A$ is given by $\hom(\mathrm{Spec}(A),F) = F(A)$ (this is obtained unravelling the correspondence given in [Mac Lane, Moerdijk, Theorem VIII.3]). In particular, my earlier formula holds as well, but since we are considering a corepresentable functor, the category of elements has an initial objects, making this statement not interesting. Anyway, fix a scheme $X$ corresponding to a variety (let's say over $\mathbb C$), consider the family of maps $\mathrm{Spec}(A) \to X$, where $A$ is a local ring. The image of such a map can contain at most one closed point of $X$, hence it cannot be open (because $X$ is a variety, hence every open contains infinitely many closed points). On the other side, the sieve generated by such maps satisfies tautologically the above condition.

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    $\begingroup$ The big étale site has enough points, because it can be replaced by an equivalent site "of finite type". More precisely, instead of considering $\mathbf{Sch}_{/ X}$, consider $\mathbf{Aff}_{/ X}$ (each with a suitable size restriction, of course). Since affine schemes are quasicompact, all covering sieves in this site are finitely generated. Thus the big étale topos is a coherent topos, and Deligne showed that coherent toposes have enough points. $\endgroup$ – Zhen Lin Oct 25 '13 at 10:18
  • $\begingroup$ That said, your question about the relationship between petit and gros toposes in general seems to be much harder, and not much is known. I do not think it is possible to determine whether a morphism is étale with only the data of the big site. Perhaps [Johnstone, Calibrated toposes] is relevant here. $\endgroup$ – Zhen Lin Oct 25 '13 at 10:21
  • $\begingroup$ Ok, thank you. I was wondering about the Deligne theorem, so your answer is completely satisfactory with respect to Question 1. But what are those points? They corresponds to $\mathrm{Spec}(k)$ where $k$ is separably closed? Or do they corresponds to strict henselian local rings? And this said, my way of computing stalks is correct? (Thank you also for the reference!) $\endgroup$ – Mauro Porta Oct 25 '13 at 10:25
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    $\begingroup$ For $X = \operatorname{Spec} A$, the big étale site of affine schemes finitely presented over $X$ classifies strictly henselian $A$-algebras. (Similarly, if we restrict to Zariski coverings, we get the classifying topos for local $A$-algebras.) This is explained in Hakim's thesis. $\endgroup$ – Zhen Lin Oct 25 '13 at 10:37
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    $\begingroup$ The thickenings can be much larger than infinitesimal. For instance, the points of the petit Zariski topos of a scheme $S$ are just the set-theoretical points of $S$. The points of the gros Zariski topos of $S$ are local rings $A$ together with a morphism $f : \operatorname{Spec} A \to S$. Any such point determines a set-theoretical point of $S$, by considering the image $f(\mathfrak{m})$ of the unique closed point of $\operatorname{Spec} A$ under $f$. But I find it hard to pretend that $\operatorname{Spec} A$ is an infinitesimal thickening of $f(\mathfrak{m})$. $\endgroup$ – Ingo Blechschmidt May 30 '17 at 18:55

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