@HarryGindi Riehl and Verity also have a treatment of straightening / unstraightening

@TimCampion It's not finished yet. It's not proven to be functorial

@DenisNardin I learned from a masterpiece numdam.org/item/BSMF_2010__138_2_181_0 that the Zariski+constructible+"S-topology" (defined in terms of generizations and specializations) makes the Zariski topology truly geometric :)

@EldenElmanto This seems interesting. Can you elaborate? I thought that Zariski=constructible+S-topology

In fact this is true more generally in spectral spaces, or stably compact spaces

@DenisNardin Right. I should say thinking of Zar=cbl + S is really an aid to think about how the zariski topology looks like on a scheme.

In fact it's even better: the category of spectral spaces and spectral map is equivalent to the category of posets in Stone spaces

For example Cor 2.9 rephrases universal epis in terms of these valuation rings and the idea uses Zar = cbl + S

Yeah, I like these things a lot :).

I know. It's lovely :)