?? so is it true for abelian varieties that a functor preserves both ordinary tensor product and "convolution" iff it corresponds to "inclusion of a zariski-open subgroup" ??? .... ??? ...

?? first of all, this might be a bad example ... ??? shortage of zariski-open subgroups here ?? .... ???? .... ?? maybe should try something like "algebraic monoid" instead .... ??? .... ??? ....

?? second ... ??? whether the idea makes any sense might depend on ... ??? whether the (?? ...) idea made any sense in the toric case .... ???? ...... ?? "combined doctrine" .... ???? ....

??? confusion about .... "toric map" .... ??? and how such relates to toric convolution .... ??? hmm, taking "toric mayhem" idea somewhat seriously ??? ..... ?? index-raising/-lowering games ... ??? .... ?? relationship between functor f preserving some operation and some adjoint of f preserving some "raising/lowering" of it .... ??? ....

??? "lax interchange law" as maybe hinting that one of the two operations involved has an adjoint (?? ...) that strongly interchanges with the other one .... ??? .....

?? cartesian product as adjoint to diagonal ..... ???? ..... level slip games here ??? .... ??? some sort of "microcosm principle" ??? .... ???? ..... and / or antithesis thereof ... ???? ..... ?? microcosm principle as maybe prototypical level slip ??? ... ??? ....

(reminds me of idea from paper notes that i still haven't copied here yet ... "s |-> _set_^s" as a sort of categorification (2-)functor .... ??? .....)

?? adjoint to toric convolution .... ????? ..... ????? ....

?? well, so what sort of adjoint string are we getting toric convolution and/or cartesian product to belong to, from theorem about accidental geometric morphisms ?? .... ???? ....

?? so ... partialness of toric multiplication in non-affine case as giving non-essential geometric morphism which is "geometric diagonal" of topos .... ???? .... so there's topos-wise f^* and f_* here .... ?? topos-wise f^* for geometric diagonal of a presheaf topos, for example, is ... ?? ... "diagonal presheaf of a double presheaf" ????? ..... ???? hmmm, so _is_ this what cartesian product of presheaves looks like when you "construe it as a single-variable functor" in the way that we've been practicing ?? ....(?? using universal property of tensor product of cocomplete categories ?? .... ??? ....) ... ??? bit hard to tell when everything seems so tautological .... ???? ... ??? more hints about "cartesian microcosm principle" ???? ... ???? ..... ?? let's try assuming so for a moment .... ???? .... ?? then .... ??? f_* as right adjoint to that .... ???? ..... ?? right kan extension ??? .... ?? taking a single presheaf x to the double presheaf x# with x#(y,z) = x(y) X x(z) ??? ..... that was just a stupid guess ... ?? .... ??? diagonal presheaf of representable (by object pair ... ?? ...) double presheaf as ..... ?? hmm, might take a bit of time to straighten this out ... ??? ....

??? "diagonal action of bi-action" .... ?? "diagonal module of bi-module" .... ??? "using comultiplication to tensor modules together" .... ???? ..... ???? ..... ?? then to define cocommutatie comonoids .... ????? .... ??? ... ????

??? "toric hom" ... ??? "toric deconvolution" ??? ..... ???? ...