?? toric line bundle ... ??? trying to formalize in nice "conventional algebraic-geometry" way, and then prove equivalent to ... even nicer way ... ??? .....

?? tendency for toric line bundle concept to be somewhat orthogonal to "loops and tails" idea .... ????? .....

?? module of "quantum double" ???? ..... ?????..... no, not correct yet .... but .... vague idea of combining being a module of the monoid ring with some extra structure associated with .... ????? comultiplication on monoid ring ... seems maybe more correct ... perhaps eventually obviously so ...

?? how coalgebra structure gets along with comultiplication here .... ????

?? hmmm .... i've been missing something a bit obvious here ??? .... modules of commutative monoid ring .... ?? as having two tensor products ???? ......

??but ... ??? special case of abelian group ring ... ????....

hmmm ... some confusion here .... probably not too bad ...

?? _invertible_ rep of abelian group ... ???? ....

?? line bundle over torus .... ?? ......

?? 1-dim fibers vs 1-dim total vsp and/or sections ... ??? ....

?? invertible line bundle over torus =?= invertible Z-rep ... ?? ...

?? hmmm, maybe there really are two different tensor products here ... ??? again seems amazing for me not to have really noticed (?? ...) it, if so .... ???? ....

?? must (?? ...) have noticed there are two _reasons_ for tensor products to exist here ... ?? mult and comult, or 2 mults (equal by eckmann-hilton ...) ...

?? p-tuples of vector spaces can be tensored pointwise, or convolved .... ??? ... and this is a pretty familiar idea .... with perhaps more familiar decategorification .... hmmmmm .....

?? wait a minute ... even on a non-abelian group (or monoid) you can both convolve and multiply pointwise .... hmmm, wait another minute ... that's using measure-function duality ..... ????? .....

????? haar measure ???? ..... ??? and what happens to it in trying to extend from group to monoid ??? ... ??? compactness and/or "moral compactness" issues ???? .... torus ..... ??? ....

"categorified haar measure" .... ??? ....

?? i was going to throw in something about "serre duality" here ...

but ... ?? actually some of what's going on here turns out to be very simple ... (not that it mightn't be interesting to try to straighten out some of the other stuff sometime ...) ...

?? commutative monoid acting on cocommutative coalgebra ... as equivalent to cocommutative monoid wrt convolution tensor product (that is, arising from comultiplication ...) of quasicoherent sheaves over affine toric variety ... ?? ....

?? what happens to convolution tensor product in passage from affine to more general toric variety ??? ....

?? well for one thing it becomes the cartesian product of toric quasicoherent sheaves, right ?? ...

??terry bisson ??? ..... ?? categorified fourier transform ... ??? ....

?? but still, "global funny stuff" here ??? .... relationship to "generalized day convolution" and so forth ??? .... toric variety as semi-group object, or almost so .... ???? .....