?? so ... for an affine toric variety, the tensor product of toric quasicoherent sheaves is clearly just day convolution wrt the symmetric monoidal structure on the corresponding commutative monoid (as one object category) .... being somewhat careless about opposite categories here ... and this day convolution can be thought of as the extra left adjoint of an essential geometric morphism ..... ???? ..... ?? meaning essentially just a functor, namely that tensor product functor on the one-object category ... ??? ....

?? then non-affine case .... ???? ....

?? also .... ?? geometric morphism here as _flat toric geometric morphism_ .... ????? ..... ??? .... ?? fan picture ?? ... flatness ... ??? contrast between binary and nullary cases ... ???? .....

?? snake eating own tail here ... ??? ....

?? hmmm .... ?? isn't x -> 1 just as flat (?? ...) as x -> x^2 ?? .... ??? no wait, it's flatter .... ?????? ...... ???? .... ???? confusion .... ????? ..... ??? affine vs non-affine here ??? .....

?? linear / bilinear confusion here ??? .... ??? .....

?? so ... ?? possibility what we've got here is the multiplication operations as "toric geometric morphisms" ??? .... ???? 1 -> x as pretty non-flat ... ?? x^2 -> as pretty flat .... ???? .... ??? hmmmm .... ???? torus case vs more general case ???? ...... ???? .....

?? "non-toric analog" .... ???? ..... ??? of ... ??? ..... ??? ... ?? problematic ?? .... ???? ....

?? toric geometric interpretation of essentialness here ?? ..... ??? ...... hmmmm ....

?? weird interrelationship between ordinary tensor product and "toric convolution" here ??? .... ???? ...... ????? .......