Tuesday, February 8, 2011

??so what about the possibility that... ???there's a nice doctrine interpretation from the ag doctrine to the "ringed g" doctrine, but/and that ... ??because the module category of a ring object in a topos (os... ???) is abelian (os...) ... ??the monad unit isn't invertible ??? or something???... ??hmm, do we know _any_ instances where it's invertible, and if so then what are they like???.... hmmm... ??relationship to various old confusions ... ???or something??? .... something about what we mean by "same theory ..." ... "... in different doctrine ..." ... ??? and so forth ... ???...

maybe ask/tell todd about some of the quasitopos ideas here ... ???....

??so what about whether the non-quasicoherent sheaves in the small zariski topos give (??in some maybe fairly obvious way??? or something??) similar examples in the big zariski topos?? ... ??and so forth ... ???...

???hmm, so ... ??what _about_ idea that we should expect monad unit here to be non-invertible???....

??so what _does_ it tend to mean in general when monad unit is invertible (or something...), and how does that relate to "same theory in different doctrine" stuff?? ... ???? ...

???and what about something about... ???getting algebra (2,1)-cat of monad involved here??? ???as yet another doctrine??? ???or something???.... ????..... hmmmm..... ??????.....

???any relationship between ... ???what's (???) going on here (??...), and ... ???something about bit about ... "full homotopy category

???something about (??loose ??...) analogy between "quasicoherent module in ringed topos" and "atom in commutative monoid" ??? ??or something?? ... ??something about "trying to recover the free generatros ... " ... ???or something ???....

???so what about something about trying to take seriously idea of ... doctrine interpretation from ag doctrine to "ringed g" doctrine as being expressed in terms of ... ??something about ... (2,1)-sketch for ag doctrine ??? .... or something... ??ask/tell todd about this ??? ....

???so what _about_ relationship between non-quasicoherent sheaves over (big and/or small ??? ...) zariski (??or something) ringed topos of commutative ring, and ... ??something about ... ??modules of category of quasicoherent sheaves ??? .... or something... and so forth ... ????....

??so what about something whether monad unit is ... ???? "(2,1) full and faithful" ?? ... ???or something??? .... and so forth ??? .... hmmmm.... ?????.....

...hmmm.... ????..... ???something about "good ag theory" here ??? ... ???or something????.... ... ????..... ??as one where the monad unit _is_ full and faithful ??? ???? or something??? ... vs ... ???? ???some vague analog here to something about "dense vs closed" ??? ??or something??? ... ?? ??or maybe closed vs dense" ?? ...??? ...

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