?? so ... can we actually get nice "categorified bialgebra" from toric variety by taking either ordinary or toric quasicoherent sheaves and dualizing (?? ...) one of {ordinary tensor product, toric convolution} ?? ... ??? does one of these choices work better and if so then which one ?? ...

?? add addendum to thesis sketch about this ?? ...

?? so ... ??? consider accidental topos t of toric variety ... but just as cocomplete category ??? ..... with also "comultiplication" given by ... ?? diagonal functor t -> tXt ... ?? hmmm, maybe not quite make sense yet ??? ...... ?? want to go from t to t#t, not to tXt .... ????? .... hmmmmm ..... ??? ....

?? but now doesn't it seem like ... we're heading in a direction that would work just as well in the non-toric case ??? .... ??? which we're pretty sure that we don't want to be ..... ??? ..... ???? ....

?? or .... ???? ordinary tensor product and this "duplication" operation as having "frobenius" relationship to each other ??? .... ???? whereas .... maybe this "duplication" (??? ...) operation and toric convolution really do have a "bimonoid" relationship ... ??? ..... ???? ..... ?? and maybe it really is non-vacuous for that to happen ..... ????? ...... ????? .....

?? some peculiar confusing things going on here, but ..... ????? ...... duality flips ...... ???? .....

???? "toric duplication" ???? ..... ????? .....

?? stable comonoidal category ..... ???? ....modules of a commutative ring .... ???? more generally, quasicoherent sheaves over a nice (?? ...) scheme .... ???? ..... ??? ....

?? look concretely at alleged "duplication" operation in case of accidental topos of projective line ??? .... ??? ....

?? let's try "the other choice" here ... ?? for example in the affine case ... ?? tensor product of reps of comm monoid by just thinking of them as modules of comm monoid ring .... ?? "duplication" by .... ???????? .... ????? ....

??? adjoint vs converse here ??? .... ????? ....... ????

?? level/duality slip/flip .... ???? ......... ?????? ......

?? comm monoid m .... rep r of mXm .... ???? ..... ?? "first duplicate then tensor, vs vice versa" .... ??? .....

?? duplicate using multiplication, tensor using comultiplication .... ???? slip/flip .... ???? ...

?? accidental topos .... ???? cartesian product and "duplication" ???? ....

??? seem to be transporting bialgebra structure _contra_variantly along "module category" process ??? ..... ????? ..... ??? ....

?? "coefficients" idea here ?? ...

?? algebra / bialgebra duality .... ??? ..... "glueing" ... ?? ... ??? ...