My impression is that there are at least two distinct notions of cubical set which have entered this discussion. One version describes cubical sets as presheaves on the Lawvere theory generated by two constants or 0-ary operations; this is close to what Vaughan described. More precisely, instead of taking the category whose objects are finite sets equipped with two distinct points (which is opposite to the Lawvere theory), he adds in a terminal object (where the two constants are forced to coincide), giving a category C. Anyway, whether one takes the Lawvere theory or C^{op}, the result is a category with finite cartesian products and an interval object, and one notion of cubical set is that of presheaf on this category. Whereas cubical sets in the sense described by Ross are different: they are presheaves on the free *monoidal* category with an interval object. This category does not include diagonal maps. I expect this is the notion of cubical set that Dmitry and Urs were actually concerned with, but in any event, both the cartesian version and the monoidal version of the cubical site appear in the literature, and it is important to clarify which notion is meant. Todd Trimble [For admin and other information see: http://www.mta.ca/~cat-dist/ ]