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I would like to know if it is possible to characterize the property "quasi-compact" in the category of schemes by means of a pure categorical language. This would imply in particular that every equivalence of categories $\text{Sch} \to \text{Sch}$ preserves quasi-compact schemes. Together with Jonathan's answer here, this would answer affirmatively my question about the rigidity of the category of schemes, at least over a field $k$.

For example the property of being empty is categorical, because it just says that the scheme is initial. The terminal scheme is $\text{Spec}(\mathbb{Z})$, so this is also categorical. Further examples of categorical properties or schemes: Spectra of fields, the underlying set of a scheme (in particular surjective morphisms), connected schemes, $\text{Spec}(\mathbb{Z}_p)$, and much more, see here. The usual definition of quasi-compact involves open immersions, which are, a priori, not categorical.

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    $\begingroup$ Related: nlab.mathforge.org/nlab/show/compact+object $\endgroup$
    – Qiaochu Yuan
    Commented May 28, 2011 at 10:13
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    $\begingroup$ @Qiaochu: The same nlab article shows that even for topological spaces this does not give the correct notion. It is better suited for algebraic categories. $\endgroup$
    – Martin Brandenburg
    Commented May 28, 2011 at 11:08
  • $\begingroup$ I remember that the proper monomorphisms are the closed immersions and I thought that there was a similar story for open immersions but I've forgotten it. For properness you might be able to use some valuative criterion, but in this arguably pathological situation where maps aren't of finite type and schemes aren't noetherian perhaps anything can happen. $\endgroup$
    – Kevin Buzzard
    Commented May 28, 2011 at 11:32
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    $\begingroup$ How about this? If $A\to X$ is a closed immersion then the corresponding open immersion is the terminal example of a map $Y\to X$ such that $A\times_XY$ is empty. $\endgroup$
    – Tom Goodwillie
    Commented May 28, 2011 at 11:50
  • $\begingroup$ Open/closed immersions are the étale/proper monomorphisms, but every definition of étale/proper uses some finiteness condition which is - a priori - not categorical. $\endgroup$
    – Martin Brandenburg
    Commented May 28, 2011 at 12:37

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(This is a paraphrased version of something I learned from BCnrd. Consequently I'm posting it as a CW answer.)

There is currently no known useful criterion for quasi-compactness from the functor of points. Nonetheless "locally of finite presentation" does have one (cf. EGA IV-8.14).

This means in particular that to check for properness via the valuative criterion (which is purely functorial, of course), the extra steps of ensuring finite type need additional work---locally of finite type can be checked thanks to the criterion just listed, but one also wants quasi-compactness.

So in practice what one often does is to take something known to be quasi-compact (e.g. a concrete scheme) and surject this onto the moduli space in question. Surjectiveness can be checked functorially (using the spectra of large algebraically closed fields). This is the main way of establishing that an Artin stack or algebraic space is quasi-compact.

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    $\begingroup$ Okay, but I don't see any connection with my question. The functor of points only adresses affine-valued points, whereas in my question it is "available" on the whole category of schemes. $\endgroup$
    – Martin Brandenburg
    Commented May 30, 2011 at 9:30
  • $\begingroup$ @Martin: Ah, ok, I see I may have misunderstood the question. Sorry for the confusion -- said confusion was entirely mine, and not BCnrd's. $\endgroup$
    – Akhil Mathew
    Commented May 31, 2011 at 2:15

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