?? bistable "categorified bialgebra" where multiplication is cartesian ??? ....

?? or anything in between "distributive" (?? ...) and "grothendieck topos" ?? ... ????

?? "cocomm comonoid right adjoint to inclusion" game here ?? ... ??? ...

?? "toric category" .... ??? "torus category" if topos is boolean ... ??? ....

?? universal property of particular otric variety ... nice simple / semi-famous examples ... ??? ....