?? pathological for one cone in fan to be contained within another without being a face of it ?? ....

?? maybe prohibiting this kind of pathology still allows the fans on a given lattice to nicely form a lattice (!! different meaning !! ...) ... ??? ... with the dominance relations realized as nice (?? but generally not geometrically monic ??? ...) morphisms .... ???? .....

?? how this (...??...) might relate to "borrowing geometric (?? ... homotopy-)colimits from TAG theories" ?? .... ???? ...

?? "torus" definition of "toric variety" ... ?? as telling us anything about whether to exclude this sort of alleged pathology ?? .... ???? ....