??in light of recent partial straightening out of relationship between toric variety accidental topos and filteredly-cocomplete category associated to its fan, it looks to me at the moment like maybe "frankenstein doctrine" idea isn't really working ... ??or something?? ...

??but what about maybe something about ... relatively free cocompletion of only finitely cocomplete ag theory... ??as maybe thought of as theory of doctrine including something about filtered colimits ... ??or do i mean filtered limits here ???... hmm, probably lots of confusion here, but ... ???...

??something about "compactness" issues here, and so forth ??? ... ??quasiprojective vs projective and so forth ??? ....


???hmmm, so am i maybe now catching another big mistake (or another part of one same big mistake...) ... ??something about ... i wrote to todd ... :

"given a filteredly cocomplete category x and the corresponding topos x# of filtered-colimit-preserving set-valued functors on it, we have the yoneda embedding from x^op into x#, and i'm trying to tell you (in a particular case where we have a reasonable concrete description of x#, and would like to obtain a similarly reasonable concrete decription of x) how to reverse-engineer x from x# by finding x^op as a certain 3-object full subcategory of x#."

???so ... _is_ this screwed up ??? .... because of something about ... ??yoneda embedding here as maybe not actually landing in x# ???? .... ???or what???...

??well, so we should really test this example of "N-torsors", i think ... ??though there could be danger of extra-special coincidences of some kind here ... ??...

???so, a functor _N-torsor_ -> _set_ consists of ... ???an N-set, and a Z-set, and an N-equivariant map from the N-set to the Z-set ?? ... ???is that correct ????.... ??and the functor is filtered-colimit-preserving precisely in case the N-equivariant map is the comparison map from the N-set to its tensor product over N with Z ?? .... ???is that correct ??? ....

(??hmmm... ??so what _about_ relationship to "quasicoherent vs non-quasicoherent" and so forth ???? ...)

??so anyway ... ??we want to test whether each value of the yoneda embedding _N-torsor_^op -> [_N-torsor_,_set_] (contravariantly assigning to an object x the covariant functor "homming from x") is filtered-colimit-preserving ...

??so two cases to check ... homming from the N-torsor N, and homming from the N-torsor Z ...

so let's try homming from the N-torsor N .... ??seems like N represents concept of "element" ...

??something about ... ??in case of free filteredly-cocomplete category, of course the "generating objects" should get taken to connected projectives by the contravariant yoneda embedding ... ???what we're seeing here being part of that... ??or something ???....

and just as of course, the _non_-generating objects should get taken to _non_-[connected projective]s, right ??...

in any case, let's check "homming from Z" here to make sure about what's going on ... seems like it's _not_ going to preserve filtered colimits ...

so... "homming from Z" takes N to the empty set, but takes Z to Z ... and Z is _not_ the tensor product of the empty set over N with Z ...

so yeah, it seems clear that that message that i sent to todd was screwed up ...

so then what _about_ how to try to straighten out the situation?? well for one thing... instead of trying to recover a filteredly cocomplete category as a certain subcategory of the topos of filtered-colimit-preserving set-valued functors on it, why not simply recover it as the model category of that topos ??? .... ??to what extent does that "fix" various problems / confusion ???... and so forth ....???? ....

??maybe something about "isbell duality" (??or something ???) here ?? ... ?? ....

?????some further (?????....) confusion here ???? ..... ?????something about .... ????hadn't we pretty much convinced ourselves that the difference between the quasicoherent and the non-quasicoherent sheaves (in the toric case ...) was ... something about ... ??the quasicoherent ones as being sheaves wrt some (??further?? ... ??or something?? ...???) grothendieck topology?? (and then i was going to say: whereas it wasn't until after that that we caught the mistake about preserving filtered colimits as not being a sheaf condition; thus contaminiation by that mistake ... and so forth ... ) or maybe no, that's not quite what we'd convinced ourselves of ... ???rather maybe just the bit about "lax glueing vs strong glueing" or something ... ???also various other ways of thinking about it; would probably be good to go back and try to synthesize them all together, or something ... ???... ???so maybe now we're more or less claiming that the strong glueing arises from the lax one by imposing the further filtered-colimit-preservation property ... ???or something??? .... though hmmm, then why don't i rememebr anyone trying to express quasicoherence as something like a filtered-colimit-preservation property (or something ... from a certain point of view ... toric vs non-toric case here ...) .. ??? ... and so forth ... ???

??so ... might it be that the topos of toric quasicoherent sheaves arises from the topos of toric non-quasicoherent sheaves as the coalgebras for a nice comonad?? ... ???or something ?? ... ???if so then what about various possible nice conceptual interpretations here??? ... and so forth ... ???and again maybe something about "isbell conjugation" (and so forth ...) ... ??? .... ??...