(for todd)

just some afterthoughts from the discussion this morning ...

so one of the things that we've been talking about is this general phenomenon of a right-adjoint "spectrum" process that yields a pre-stack from an object of some sort, and a kan extension process left-adjoint to this that tries to re-assemble an object of that same sort from a pre-stack ...

and sometimes i've been calling this left-adjoint "reassembly" process a "globalization" process, for hopefully evident reasons ...

but it occurs to me now that maybe we should similarly call the right-adjoint spectrum process a "localization" process ... that is, maybe it really does make good conceptual sense to think of "localization" and "globalization" as adjoints to each other in this context ... not that the word "localization" is completely untaken and free to be given a new meaning, but... not even clear that some of its existing meanings aren't special cases of this proposed meaning ...

but in any case, we've also been talking about considering the category (or higher category) of "fixed points" of the globalization/localization adjunction here ...

and i just wanted to mention the idea that perhaps one of the ways in which the concept of "stack" (as opposed to the concept of mere "pre-stack") emerges here is precisely from the condition on a pre-stack of being a fixed point of this adjunction ...

that is, we can ask what is the least restrictive "sheaf condition" (or in this context really a "stack condition" ...) aka "grothendieck topology" for which all those pre-stacks which qualify as fixed-points under the adjunction are in fact stacks ...

and if the site category that we started with is "topos-like" (mainly in the sense of having "distributivity" aka "exactness" properties similar to those holding in toposes) then it may be that the sheaves for this topology (as a special case of the stacks) are not so far different from the original site objects, and also not so far different from the pre-sheaves that are fixed-points under the adjunction ...

whereas if, as seems to happen in the "toric" case, the original site category is rather un-topos-like, then it may be that the sheaves for this topology are considerably more general than the original site objects, and than those pre-sheaves that are fixed-points under the adjunction ...

i guess that one sort of "consistency check" that i should probably apply to my attempted reasoning here is as follows: is it true that the "sheafification" and/or "stackification" functors here are of the correct "handedness" (that is, left- vs right-adjoint) as would be expected for a process forcing a pre-stack to become a fixed-point of the globalization/localization adjunction ???

not sure yet just how much sense all of that makes ... ??...

??hmmm, i guess tha there's some possibility (that maybe we even already alluded to) that maybe processes of _both_ handednesses might exist here ... ???....