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For nice topological spaces (say Haudorff spaces) $X$ and $Y$, there is a bijection between continuous maps $X\to Y$ and isomorphism classes of geometric morphisms $\mathrm{Sh}(X)\to \mathrm{Sh}(Y)$.

Question: Is there a similar statement for "nice" schemes, i.e., that morphisms of schemes $X\to Y$ correspond bijectively to isomorphism classes of geometric morphisms from the étale topos of $X$ to the étale topos of $Y$?

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    $\begingroup$ "Correspond bijectively" is a rather weak relationship. I think for silly cardinality reasons it will often be true. Perhaps this is the question you mean to ask (as in your title): is the (pseudo)functor taking schemes to their étale toposes fully faithful? Then Jens's answer shows the answer is no. $\endgroup$
    – Zhen Lin
    Commented Jul 5, 2022 at 9:10
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    $\begingroup$ I suspect the answer become yes if we work with the gros étale topos and take the slice of the category of toposes at the étale topos of the base scheme. At least it works for the gros Zariski topos $\endgroup$
    – Simon Henry
    Commented Jul 5, 2022 at 9:54
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    $\begingroup$ @SimonHenry Yes, good point. This is because for any topos $\mathcal{E}$ there is a natural bijection between morphisms $X \to Y$ in $\mathcal{E}$ and isomorphism classes of geometric morphisms $\mathcal{E}/X \to \mathcal{E}/Y$ over $\mathcal{E}$. If $\mathcal{E}$ is the gros étale topos of the base scheme $S$, then $\mathcal{E}$ contains as full subcategory the schemes that are finitely presented over $S$, and for each such scheme $X$, its gros étale topos is $\mathcal{E}/X$. $\endgroup$
    – Jens Hemelaer
    Commented Jul 5, 2022 at 10:25
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    $\begingroup$ Maybe as a helpful intution: a scheme is not the analog of a topological space, a scheme is an analog of a complex manifold. The étale topos is the analog of a topological space. And you don't expect the functor obtained by taking the underlying space of a complex manifold to be fully faithful... $\endgroup$
    – Denis Nardin
    Commented Jul 5, 2022 at 17:23
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    $\begingroup$ The Frobenius map induces the identity on the etale topos. So the answer is negative. $\endgroup$
    – Piotr Achinger
    Commented Jul 6, 2022 at 9:14

3 Answers 3

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Below is a proof that the (pseudo)functor that sends a scheme to its petit étale topos is not fully faithful, for any category of schemes over an algebraically closed base field $k$, assuming that this category contains both $\mathrm{Spec}(k)$ and $\mathrm{Spec}(k[t])$.

Proof: The morphisms of schemes $\mathrm{Spec}(k) \to \mathrm{Spec}(k[t])$ over $k$ correspond to maximal ideals of $k[t]$. At the level of toposes, these give geometric morphisms $\mathbf{Sets} \to \mathcal{E}$, with $\mathcal{E}$ the étale topos of $\mathrm{Spec}(k[t])$. However, not every geometric morphism $\mathbf{Sets} \to \mathcal{E}$ is of this form. There is another one induced by the morphism of schemes $\mathbf{Spec}(k(t)^\mathrm{sep}) \to \mathrm{Spec}(k[t])$, with $k(t)^\mathrm{sep}$ the separable closure of $k(t)$. $\square$

For other reasonable categories of schemes, I expect that the answer to your question will still be "no". However, in practice it might be enough that the functor is "almost fully faithful", see the positive results described in the answer by D.-C. Cisinki.

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    $\begingroup$ Your answer reads like a proof of some statement. What are you proving? $\endgroup$
    – user485276
    Commented Jul 5, 2022 at 8:51
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    $\begingroup$ I'm showing that the (pseudo)functor from "nice" schemes to toposes is not fully faithful. I'm assuming that the category of nice schemes you have in mind is a category of schemes over an algebraically closed base field k, and that both Spec(k) and Spec(k[t]) are nice. $\endgroup$
    – Jens Hemelaer
    Commented Jul 5, 2022 at 9:30
  • $\begingroup$ I edited the question to make this more clear, and mentioned how my answer relates to the one by D.-C. Cisinski. $\endgroup$
    – Jens Hemelaer
    Commented Jul 7, 2022 at 7:44
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There is no way it is true in general, but there are results in this direction nevertheless: Theorem 3.1 in this paper of Voevodsky establishes (a kind of fully) faithfulness for normal schemes of finite type over a number field, in the spirit of a conjecture of Grothendieck (in his letter to Falting, an English translation of which is published in Geometric Galois actions, 1, ser. London Math. Soc. Lecture Note Ser. Vol. 242, 1997, and in his Esquisse d'un programme).

In particular, if $X$ and $Y$ are normal schemes of finite type over a number field $K$ (in fact any field of finite type of characteristic zero) and if their small étale topoi are equivalent over the small étale topos of $K$, then $X\cong Y$ as $K$-schemes.

There are also Torelli-like theorems saying that one can sometimes determine the isomorphism class of a scheme from its topological space only (i.e. from the 0-localic topos associated to its étale topos). This paper of János Kollár, Max Lieblich, Martin Olsson, Will Sawin shows that, for $K$ an uncountable algebraically closed field of characteristic zero $K$, one can reconstruct a normal geometrically integral proper scheme of dimension at least 2 from its underlying topological space. Their results also prove, for $K$ an uncountable algebraically closedfield of characteristic zero, one can reconstruct a proper normal $K$-scheme of dimension at least 2 from its category of constructible abelian étale sheaves.

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Take a look at the paper of Barwick, Glasman and Haine (https://arxiv.org/pdf/1807.03281.pdf).

In particular the section "Exodromy for schemes & the Reconstruction Theorem".

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    $\begingroup$ You are right to point to the work of Barwick, Glasman and Haine (not Barwick alone). It explains the dependency of the small étale topos (in particular, the fact that two schemes related through a universal homeomorphism have the same étale topoi) and state a nice conjecture. However, the only positive results in this direction are all derived from the ones I mention in my answer. $\endgroup$
    – D.-C. Cisinski
    Commented Jul 5, 2022 at 14:43
  • $\begingroup$ Thanks for the clarification. $\endgroup$
    – user234212323
    Commented Jul 5, 2022 at 17:52

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