carchedi

on somewhat short notice i may have a chance to ask dave carchedi some questions tomorrow, if i manage to avoid sleeping through the opportunity. i should try to figure out what questions to ask...

??something about the relationship between toposes and stacks on site _locale_ with probably some obvious (and/or perhaps slightly less obvious) grothendieck topology... ??maybe something about how this relates to "syntax/semantics adjointness for geometric doctrine", and so forth...

??something about this bit about "moduli stack of foliations" or something... and so forth or something...

??something about "higher-dimensional toposes" ... groupoid-based vs category-based... and so forth...

??something about philosophy that "focusing on stacks (as opposed to objects of groupoid-enriched categories...) is a bit silly, in a categorified version of the same way that focusing on sheaves (instead of objects of categories) is" .... or something like that... ??in connection with this, something about the idea of "obtaining grothendieck topology from knowledge of homotopy colimits in a groupoid-enriched category into which the putative site (ordinary) category embeds" ... or something...

hmm, one semi-obvious (in retrospect at least) pointthat carchedi made is that what they call the "topological stacks" is pretty much ess just the grothendieck toposes, and more special than the arbitrary stacks on the hopefully obvious site here.... ???or something like that... actually, sorry, from talking to carchedi again just now i see where i screwed up here again... might try to straighten this out at some point...

??maybe i should try asking them more specifically about what happens if we try to define a grothendieck topology on the category of ("finitary"?) affine schemes (or something like that) by using homotopy-colimits in the groupoid-enriched category of symmetric monoidal finitely cocomplete algebroids... or something like that...

??this idea that there's a tendency for stacks which are well-described by some kind of nice sheaves over them to coincide with those arising from groupoid objects (or something like that) seems somewhat unexpected (to me, but it might be just because i haven't been paying attention). at least we seem to be seeing something like that in the "topological" case; do we also see it in the "algebraico-geometric" case?