?? just how annoyingly far would you have to relax the requirements in order to get toric varieties to qualify as monoidal instead of semi-monoidal wrt geometric morphisms ?? ....

?? hmmm, derived / homotopy level here ??? ... ?? "infinity-topos" ?? (?? maybe avoid being pinned down here ... ?? ...) .... ?? maybe mention in paper ... ?? ... ?? "secret motive" squared ???...

?? also mention ... ?? non-toric analog of "toric-geometric interpretation of accidental-topos-geometric morphism" ... ???? ... ?? arguably, left-exact left adjoint ... ?? ...

?? also ... arbitrary grothendieck topology being compatible with "toric structure" ... ?? non-vacuousness of that ??? ... ?? special role of double negation topology ... = "the torus" ... ??