If I understand correctly, in 2.15 in notes on condensed mathematics, Scholze speaks of a notion of quasicompactness for a map of condensed sets. What's the definition?

@PiotrPstrągowski I think quasicompact objects make sense in a topos (e.g. stacks.math.columbia.edu/tag/090G) so maybe this is a quasicompact object of the slice topos? but maybe the easier thing is to figure out what happens in the proof before Scholze concludes something is quasicompact...

@S.carmeli actually, since asking this here i've determined that this is not true. (thanks to jay shah for helping me sort through this.) the $G/H$-action on the value $(Fin_G)^H \simeq Fin^H$ will be such that its fixedpoints is $Fin^G$, and so it can't be trivial unless $H$ is in fact a summand of $G$.

here's a basic group theory question that i'm stumped on -- perhaps someone in here will know how to answer it: math.stackexchange.com/questions/3843932/…

@AaronMazel-Gee hmmm so you claim that the action is trivial is wrong, or the entire tescription as unstreightening?

oh, i'm claiming that the action is nontrivial. certainly it can be described straightenedly.

OK, I'm reliefed:) that's what I referred to sorry for not being very precise

@PiotrPstrągowski I think Scholze means "relatively 0-coherent" in the sense of Lurie math.ias.edu/~lurie/papers/SAG-rootfile.pdf#subsection.A.2.1

I.e., a morphism f:X → Y in a topos is quasicompact if for every quasicompact object U and morphism U → Y, the pullback X ×_Y U is a quasicompact object.

@DylanWilson @PeterHaine These look very much related, so it must be some variant of this! Thanks. :)

Maybe one other thing to say related to what Dylan said: if T is a topos and Y ∈ T, then an object X ∈ T/Y of the slice topos over Y is quasicompact as an object of T/Y if and only if X is quasicompact when regarded as an object of T. So qusaicompactness of morphisms ≠ quasicompact as an object of the slice topos.