??so what about ... ?? toric projective line ... ??maybe "toric small zariski topos" here ends up being just presheaves on walking pushout data ??? ??or something ??... ??that is, something about "family of set-pairs" or something ... ??with structure sheaf of commutative monoids being ... ?? Z-indexed family of pairs ... first coordinate singleton if 0 <= n and nothington otherwise; second coordinate singleton if n <= 0 and nothington otherwise ... ??or something ?? ... ??which sounds vaguely suggestive in various ways ... ????something about "formal inverse" and / or "riemann-roch" ... ??or something, and so forth ... ???...

??then seems somewhat clear what sheaf of actions over structure sheaf amounts to here ... ??and seems like ... "sheaf : quasicoherent sheaf :: lax pullback : strong
pullback" here ....?? or something ?? .... ??so does that at all match our other intuitions (and so forth ...) about what non-quasicoherent sheaves are like ??? .... ??hmmm, perhaps it sort of does??? ???because it seems like you could have
extra local sections over the "overlap", for example, or over one of the big pieces, which don't contribute to any global sections ... ??or something ?? ... ??though on the other hand, might need a bit more work to match other vague idea about ... "strong pullback, but of the wrong diagram ..." ... that is, "at the affine level is where it goes wrong" ... or something ... ??is there maybe something here about ... ??systematically re-expressing lax colimit as strong colimit of different diagram ??? .... ???or something ?? ... and so forth ... ?? ???hmm, vaguely reminds me of something about ... ??"homotopy colimit as certain sort of weighted colimit" and so forth .... ???.... ??or is something backwards here, or something ?? ...

??what about that other (...) intuition here, something about "localness wrt base space vs wrt total space" ?? ... or something ...