i think that i've mentioned that the question of how interesting the paper that we're trying to write turns out to be probably depends somewhat (perhaps obviously) on how well-studied the concept of "toric quasicoherent sheaf" already is; and that as usual for me, i'm not familiar enough with the literature to have much idea how well-studied it is...
but i've reached a point where i think i understand the topic well enough that probably even i should be able to find where it's studied in the literature...
this is mostly a result of what i mentioned in my e-mail the other day, about understanding the geometric interpretation of the underlying ordinary quasicoherent sheaf of a toric quasicoherent sheaf...
having thought about a bit, i'd say that the key idea about this geometric interpretation is that the underlying ordinary quasicoherent sheaf is (at least locally) cocommutative comonoidal wrt the "convolution" tensor product of quasicoherent sheaves over an affine toric variety....
??? blecchh, am i still / again horribly confused here ??? .... convolution wrt lattice vs wrt torus .... ???? .... ?? commutative monoid ring as bialgebra .... mult from mult, comult from diagonal ... ??? but in affine toric variety picture it's somewhat reversed ... ??? algebra/geometry duality for affine toric varieties as sort of extension of fourier duality (??? ....) from abelian group case to commutative monoid case ??? .... ???anyway, somehow it sort of makes sense to call the tensor product coming from the comultiplication here "the ocnvolution product" .... convolution on the affine toric variety ... comultiplication on the commutative monoid ring ... diagonal on the commutative monoid .... ??? .....
??? skycraper sheaves closed under convolution tensor product here ???? ..... giving actual monoid structure on affine toric variety ???? ....
?? for affine toric variety mutliplication is already totally defined, so no need to adjoin "undefined" element ... ????? ..... ?? confusion about adjoining "undefined" (??? =?= "zero" ???? ....) vs adjoining "1" ...... ???? .....
?? freyd trick using "undefined" to construe category as semigroup / monoid ... ??? ..... ???? .... cocone category .... ????? .....
??? _non_-cocommutative comonoid for convolution of quasicoherent sheaves over affine toric variety ??? .... ???? .....
?? "sheaf of convolution cocommutative coalgebras over structure sheaf of "monoidally" ringed topos" ..... ???? .....
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