??so what about something about ... taking any "de-glueing" of a "scheme" (or something...) into affine pieces and interpreting it as giving a ringed (or whatever ... monoided in the toric case ...) topos ... ???and then applying to that ringed topos hakim/tierney "spectrum" idea ... ???or something ... and so forth ...

??also... take such de-glueing... in toric case ... say for projective line ... and get commutatively monoided pre-sheaf topos ... ??or something ??? ... and then take the topos of actions of that commutative monoid object ... ???and then find in there "toric quasicoherent sheaves" as alleged full subcategory... (?? not sure what sort of formal properties the inclusion functor should have here... ?? or something ... and so forth ...) ... ??and consider whether this leads to way of interpreting the toric quasicoherent sheaves as forming a pre-sheaf topos ... ??or perhaps at least gives some hints as to whether or not this is possible ... ???.... or something ... and so forth ... ????....

??hmm, so what about something about ... ???(almost ... something about karoubi envelope ...) recovering the site category from a presheaf topos as the connected projectives, or something... ??also something about algebroid version of this ... ??relationship to something about "enough projectives" and "cohomological approach to non-/affineness" and so forth ...

(??did i mention recently something about ... ??"enough projectives" (or something) as something measured by "cohomology" (or something...) vs as some sort of precondition for getting it (...) to work well ... ?? ... or something...)

??something about ... "best presheaf topos approximation to given topos" ... ??or even to given category or something ??...

though there's approximation from above vs below, and maybe the one that's sort of working nice here isn't the "useful" one ??? ... ??or something ??? ....