In what follows, all schemes are qcqs. Also, let $\operatorname{\acute{E}t}(X)$ denote the petit étale topos of a scheme $X$.
Let $Y\to X$ be an $X$-scheme. Say that $Y$ is a special $X$-scheme if for any $X$-scheme $Z\to X$, the universal geometric morphism
$$\operatorname{\acute{E}t}(Y\times_X Z)\to \operatorname{\acute{E}t}(Y)\times_{\operatorname{\acute{E}t}(X)} \operatorname{\acute{E}t}(Z)$$
is an equivalence of toposes.
Is there a known characterization of the special $X$-schemes for a scheme $X$?
I know of two classes of $X$-schemes that satisfy this property:
- If $Y\to X$ is a closed immersion, then $Y$ is a special $X$-scheme
- If $Y\to X$ is pro-étale and qcqs, then $Y$ is a special $X$-scheme (for qcqs étale morphisms, this is obvious, as it follows from slicing of sites, and to get pro-étale morphisms, apply some version of Noetherian approximation)
Are there any other kinds of maps that satisfy this property? What about in the case where $X$ is Noetherian?
Edit:
I have a guess. If anyone can think of an obvious counterexample, it would be much appreciated. If we restrict to the case where $X$ is Noetherian and try to classify only the special $X$-schemes $Y\to X$ of finite type, then it seems like from Will Sawin's idea, the condition should be something like quasi-finite.
At least when $X$ is the spectrum of a field, the quasi-finite (in fact finite) $X$-schemes will be (possibly empty) finite disjoint unions of finite field extensions. So reducing to the connected case, notice that a finite field extension will be the composite of a separable finite field extension and a purely inseparable finite field extension. But finite separable field extensions are étale, and purely inseparable field extensions are universal homeomorphisms.
I'm not sure if this is true, but I wonder if one can play a similar game in the case of a more general base scheme than a field, first using the fact that quasi-finite maps are (étale) locally finite maps, then factoring a finite map as some kind of étale map followed by a 'totally ramified' one that is a universal homeomorphism.
