?? "toric convolution" (functor, correspondence, ... ???) on abelian variety, for example ?? ... ??? .... cocommutative comonoids therefor ... ?? topos of "nice" (??? ....) such ???? .... ???? .... ????? ....

?? defining (??? ...) "nice" in terms of .... (?? maybe ... ?? forming a topos and ... ??? ....) ???? .... ??? recovering the quasicoherent sheaves as the k-module objects ?? .... ??? .... ??? ..... ??? hmmm, what's k here ??? .... ??? maybe somewhat problematic ?? ... ?? and / or related to ring over which the abelian variety lives ??? ..... ???? .... ??? .... ?? ... ??? complications ?? ... ???? .....

?? hard to imagine getting continuous spectrum of accidental toposes here .... ?? but then, spectrum of abelian varieties as interesting from discrete (?? ... rational / algebraic ... ??? ....) viewpoint .... ??? speaking of which, did we try thinking about that discrete aspect in connection with possibility of getting concept of "quasicoherent sheaf over stack of torsors of elliptic curve x" to "work" ?? .... ??? .... ?? also any relationship of that to this ?? ... ?? ....

?? general idea of .... ???? model of prop in monoidal bicategory where morphisms are correspondences .... ??? ..... ???? .....

?? try playing around with endo-correspondences of projective line ... ?? ...

?? seems like ... ??? we're mostly in some sort of fortunate circumstance where we don't have to worry much about stuff like "sesquicoherence" ?? ... ??? .....

????? idle .... ???????? .... :

"??? so ... affine toric variety : bistable bialgebra :: toric variety : ?? some sort of "bistable bialgebroid" ????? ..... ?? any relationship to "hopf algebroid" concept that we've heard about ?? ... would seem funny if so, since the "bistable bialgebroids" that we have in mind conspicuously pretty much seem to reduce to bistable bialgebras in the "hopf" case ... ?? nevertheless i've been idly trying to dream up connections involving the contexts in which hopf algebroids appear, according to hazy recollections ... ?? ..."

"?? idea that existence of non-affine toric variety can be interpreted as failure of certain naive generalization of certain "tannaka-krein" theorem from case of bicommutative hopf algebras to case of bicommutative bialgebras, and that such failure might be expected to persist beyond bicommutative case, giving examples of some sort of "non-affine generalized toric variety" ... ??? ....

?? was idly wondering whether this might relate to some concept of "generalized toric variety" (?? ...) that i think alex mentioned ben webster as having worked on ... maybe still wondering that, but may have changed my mind about relaxation of which half of bicommutativity might be involved ... ???"

?? somewhat connected idle wonderings ??? .... ???? ....

?? i forget to what extent (and in which notebook ... ??? ...) i mentioned ... ?? idea that, based on googling "hopf algebroid", trying to connect it with the idea of non-affine toric variety (?? as involving "glueing together bistable bialgebras" .... ???? ....) seems probably a stretch ... ??? ..... ??? .....