"?? using zero quasicoherent sheaf as "undefined" succeeds in making global toric convolution tensor product well-defined ??? ...."

?? well, bit more subtle (but also straightforward ... in some sense ...) than that ... global toric convolution as colimit wrt fan of local such ... ?? ...

?? superman-flavor equivalence theorem ... ?? role in pushing / establishing boundary of definition ... ?? cases at hand ... nice accidental topos ... and so forth ... ?? ...

?? "toric convolution" ... "generalized day convolution" ..... ????? ...... ??? bialgebra duality .... ??? .... ??? "generalized toric convolution" .... ????? ....

?? transport bialgebra duality across tannakian bridge ?? ...