old discussion with urs (where at the time it was pretty difficult to understadn what he was saying, though by now it seems like it should be somewhat les difficult...) ...


me:

[my interest in trying to understand these ideas has been motivated in large part by my interest in trying to understand algebraic geometry. i know that a nice algebraic “stack” or “scheme” can be construed as the moduli stack of models of a “theory” expressed in the “doctrine” of symmetric monoidal finitely cocomplete algebroids; the objects of the theory are known as “coherent sheaves”.]

urs:

[I am wondering how this is related to the following general abstract conception of coherent sheaves of modules:

For C⊂Alg op the ∞-site of formal duals of ∞-algebras in question, the ∞-stack of quasicoherent ∞-stacks is the almost-tautological one that assigns SpecA↦AMod.]


there's more to try to decipher, but maybe i should start with trying to decipher the above... which now seems like it might be more or less clear, though i should probably check to see if it actually hangs together the way that i suspect that it might ...

?? hmm, i guess that urs's point here, or at least one perhaps minor one, is supposed to be a sort of "site-independence" idea (??allegedly same such idea promoted by johnstone in baby elephant preface ?? ...) that he alluded to somewhere nearby ... ?? that there's some sort of topos whose syntactic category can be thought of as sheaves over site s for (of course) various different s, for one of which the underlying category is _affine scheme_, while for another of which it's c = category of some sort of more general schemes, and there's a certain stack over this topos which can thus be construed as a pre-stack with extra property over either _affine scheme_ (in which case it assigns to each affine scheme the category (???....) of modules of its affine coordinate algebra) or c (in which case it assigns to each c-object the category of quasicoherent sheaves over it) ... which you could try to philosophize as saying something like "from a sufficiently site-independent viewpoint there's no essential distinction between the concept of module and the concept of quasicoherent sheaf" ...

(??i should worry here for at least a moment about whether it's really true that for a pre-stack to be a stack is really just a mere property (rather than structure ... ??), just as in case of pre-sheaf being a sheaf ... ??? ...)

?? but i'm not at all sure to what extent i would agree or disagree with such a philosophy ... even though part of what he's getting at is evidently related to things that i sometimes say... along the lines of "defining quasicoherent sheaf concept as globalization of concept of module" ... not clear to me how similar / different these philosophies are ... could be subtle or not-so-subtle differences, migth be worth thinking about ... ?? ...

?? "no essential distinction ..." vs "one concept arising by conscious act of globalization of other concept" ... ??? .....

?? urs's formulation as perhaps involving / depending on grothendieck topology ... ?? vs mine as not so much... ??maybe except ... ???more like trying to conjecturally extract grothendieck topology from it ... ???? also ... ???syntax vs semantics ... ??????? ...... ????? ......