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Consider the étale site $X_{ét}$ of a scheme $X$. As a category, this is the collection of all étale schemes over $X$.

Now, is this a set (i.e., is the étale site a small category)? If $X=Spec\ k$, one could suspect that every set has a scheme structure which is étale over $X$. Namely, $\coprod Spec \ k$, where the coproduct is taken over the cardinality of the set. If so, the class of étale schemes over $X$ would be a proper class.

Is this true?

Thank you in advance.

P.S.: My question is motivated by the following: if $X_{ét}$ is a proper class, I'm afraid $Sets^{X_{ét}^{op}}$ shouldn't be a category. But the theory of étale cohomology and of the étale topos is based on the fact that its "subcategory" of sheaves on the étale site $Sh(X_{ét})$ is indeed a category (which we call the étale topos). So, either $Sh(X_{ét})$ while $Sets^{X_{ét}^{op}}$ is not, which would be strange to me, or we have a problem.

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    $\begingroup$ The étale site is small, because of the "locally of finite presentation" condition. An étale map is analogous to a finite covering space in topology. $\endgroup$
    – Kevin Carlson
    Commented Nov 2, 2018 at 23:14
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    $\begingroup$ @KevinCarlson Do you mean "locally small?" Unless I'm missing something, every isomorphism type of a scheme is represented by proper-class-many specific set-theoretic objects (analogously to how the category of finite groups is not small), so it shouldn't be small. $\endgroup$
    – Noah Schweber
    Commented Nov 2, 2018 at 23:41
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    $\begingroup$ @NoahSchweber I mean “essentially small”, i.e. equivalent to a small category. Sorry to be imprecise, it’s just that literal smallness isn’t the significant concept. $\endgroup$
    – Kevin Carlson
    Commented Nov 3, 2018 at 2:03
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    $\begingroup$ @KevinCarlson Oh I agree, but the OP is asking about smallness per se. $\endgroup$
    – Noah Schweber
    Commented Nov 3, 2018 at 2:06
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    $\begingroup$ @KevinCarlson: It's not essentially small, though, because its objects are only locally of finite presentation. As mentioned in the question, you can take an arbitrarily large disjoint union of etale maps to get another one. $\endgroup$
    – Eric Wofsey
    Commented Nov 3, 2018 at 3:55

1 Answer 1

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You are right: $X_{ét}$ is large (in fact, essentially large). This means that there is no category of presheaves on $X_{ét}$ in ZFC, or if you are using universes you would need to go to a higher universe to talk about the category of presheaves.

However, the category of sheaves $Sh(X_{ét})$ is indeed a genuine category that can be defined without enlarging your universe. This is because there is a small set of objects of $X_{ét}$ which can be used to cover all other objects, and so a sheaf is uniquely determined by the values it takes on a small subcategory of $X_{ét}$. Indeed, since an étale map is locally of finite presentation, it suffices to consider affine schemes which are finitely presented étale covers of affine open subsets of $X$, and there is an (essentially) small set of these.

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    $\begingroup$ What about the big étale site? Is there a similar trick that we can do? For examples, do affine f.p. schemes suffice? $\endgroup$
    – W. Rether
    Commented Jan 7, 2019 at 19:49
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    $\begingroup$ My understanding is that there is no similar trick for the big étale site. To work rigorously with big étale sheaves in ZFC you must restrict the big étale site to a small subcategory which is large enough for whatever particular application you are doing at the moment. See for instance Definition 33.4.6 and the following material on the Stacks project. $\endgroup$
    – Eric Wofsey
    Commented Jan 7, 2019 at 20:10
  • $\begingroup$ Then I have a question. In her work Topos annelés et schemas relatifs, Hakim uses a site,say $S_1$, which has as objects finitely presented affine schemes over $\operatorname{Spec}\mathbb Z$, as morphisms étale morphisms, and as topology the étale topology. Then she proves things about the topos of sheaves of this site. It is common to say that "Hakim proved the classifying property for the étale topos". But why? If $S_1$ does not present the big étale topos, something is missing. (And of course it does not present the little étale topos, since we never require schemes to be étale over $X$.) $\endgroup$
    – W. Rether
    Commented Jan 7, 2019 at 20:28
  • $\begingroup$ I'm afraid that's beyond my expertise. $\endgroup$
    – Eric Wofsey
    Commented Jan 7, 2019 at 20:55
  • $\begingroup$ Could you give a reference for your answer that gives complete proves? Thank you. $\endgroup$
    – Andrés Ibáñez Núñez
    Commented Jun 21, 2020 at 11:49

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