"toric ag as fourier duality generalized from abelian groups to commutative monoids" ....

(?? but ... ??? "global" aspect ??? .... ?? ... ?? categorification ?? ... ... ??? ...)

??the / a here ??? ..... ??? dualizing object/s ??? ....

incorporating semilattice duality ....

doctrines .... AG + extra (?? secretly "convolution" ?? ...) tensor product ....

vs ... "flat AG" ... "abelian + extra tensor product" .... ???? .... ?? .... ??? .....

??? "keeping biproduct as coproduct but taking new product" .... ???? .....

??? relationship between toric quasicoherent sheaf and corresponding ordinary quasicoherent sheaf as something like "fourier tranform" ?????? ...... ????? .......

??? "convolution tensor product of Z-reps" .... ??? tensor product of underlying vector spaces ... that's the "usual" tensor product of group reps .... becomes addition/convolution on fourier dual (??algebraic vs analytic ???? ....); hence name "convolution product" ..... ?????..... ????? other ("pointwise") tensor product as seeming culturally weirder here .... "intersection with multiplicity" ..... ???? distributivity here ??????

(a pointwise b) convolve c =?= (a convolve c) pointwise (b convolve c) ??????

wasn't expecting anything like this because of ..... "fourier _duality_" .... ?? and apparent lack of decategorification motivation .... ??? ....

??? semilattice duality ????? .....


??? "newton polytope" ... ????? .....

??? bi-distributivity =?=> bi-idempotence and bi-absorption ????? ..... ????....

x*x = (x*x)+0 = (x+0)*(x+0) .... ?? not too promising ?? ...

??? thing c st (a pointwise b) convolve c = (a convolve c) pointwise (b convolve c) for all a,b ... ???? ....

?? case b = "constantly 1" ... ???... ??? so c convolve constantly 1 must be constantly 1 ?? ..... using all a ??? .....

?? which maybe has flavor of ... requiring c to be "pointlike" ???? ..... ... ??? depending ... ???? ....

(a pointwise b) convolve (c + c') = (a pointwise b) convolve c + (a pointwise b) convolve c' = (if c and c' are "good") (a convolve c) pointwise (b convolve c) + (a convolve c') pointwise (b convolve c') =?= (a convolve (c+c')) pointwise (b convolve (c+c')) =
((a convolve c) + (a convolve c')) pointwise ((b convolve c) + (b convolve c')) = (a convolve c) pointwise (b convolve c) + (a convolve c') pointwise (b convolve c) + (a convolve c) pointwise (b convolve c') + (a convolve c') pointwise (b convolve c') .... ?? ...

?? so looks like sum of good tends to be bad ???? ..... matrix vs its main diagonal ... ??? ....

??? distributivity vs modularity ... ?????? .....

??? two kinds of tensor product of ab group reps, vs two kinds of hom .... ????? ..... ?? don't seem that related, since the tensors adjoint to the homs live in somewhat different places ... ?????? .....



?? toric quasicoherent sheaf vs generalized toric quasicoherent sheaf ... ???? "dirac measure" .... derivatives thereof .... ???? level slip ??? ..... ??? contrasting ideas of "measure" ???? .....


?? given theory of certain combined doctrine, taking just those objects obeying certain distributivity condition ... ???? ....

?? distributivity comparison morphism here ???? ..... ??? inclusion of main diagonal of matrix ???? ...

??? modularity ???? ..... ???? ..... _is_ there really a commonality here ???? ..... are modular lattices "laxly distributive" in particular ??? ..... (a sup b) inf c ... (a inf c) sup (b inf c) ... ???? .... ??? in the classic non-distributive lattice there's a comparison morphism from latter to former ??? is that correct ??? ..... ... ?? in classic non-modular lattice .... ???? .... ?? maybe still a comparison morphism ?? ... ????? .... obvious by universal property of latter as sup ??? .... ... ??? ..... but still .... ??? ......

?? "toric tannakian program/philosophy" .... ?? 2 versions ... toric quasicoherent sheaf vs [generalized toric quasicoherent sheaf ..... and/or ordinary quasicoherent sheaf with toric convolution tensor product .... at least locallly .... ???? .....] .... extra "toric convolution" tensor product" (??? at least locally ????? ....) vs"thorough-going set-based approach" .... ???? ....

analogy of cartesian product of toric quasicoherent sheaves to cartesian product of ordinary quasicoherent sheaves, vs to toric convolution tensor product of (at least locally ..... ????? .....) ordinary quasicoherent sheaves ... ??? relationship to "toric convolution tensor product as usurping biproduct's role as cartesian product though not as coproduct" ... ??? ....


?? relationship / vague similarity feeling of problematicness of tensor product on cocone category to problematicness of global toric convolution tensor product .... ???? .... ?? but problematicness dealt with by interesting new (??) "zero" (=?= "undefined") vs that dealt with by interesting new "one" .... ??? non-totalness of binary product vs of nullary .... ????? .....


??? locally a toric convolution comonoid .... ????? ....

??? "fourier analysis" ..... ??? analyzing vector space into comonoids vs vice versa .... ???? .... ??? chain complex vs filtration ??? ........ ????? .....

???? "tail tree structure" and toric convolution comonoidalness ....

???? toric convolution tensor product ..... ???? vague feeling of relationship to .... ???? ..... generalized day convolution as involving geometric morphism ... ??? ..... "adjoint to tensor product" .... ????? ......

??? toric orbit stack of toric variety, and "quantum double" .... ???? ....

?? more generally (?? ...), interaction between toric quasicoherent (and/or generalized such ...) structure and toric equivariance structure .... ???? ...... .... ?? ...

?? underlying real variety of elliptic curve .... ?? toric aspect ?? .... or higher abelian variety ??? ... ??? also higher genus curve .... ??? and relationship between last two .... ?? "jacobian" ... ??? ....