# Categorical characterization of quasi-compact schemes

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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–  Qiaochu Yuan May 28 '11 at 10:13
@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. –  Martin Brandenburg May 28 '11 at 11:08
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. –  Kevin Buzzard May 28 '11 at 11:32
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. –  Tom Goodwillie May 28 '11 at 11:50
Open/closed immersions are the étale/proper monomorphisms, but every definition of étale/proper uses some finiteness condition which is - a priori - not categorical. –  Martin Brandenburg May 28 '11 at 12:37