Ubiquity in Category for Metric Spaces and Transition Systems

We characterize the notion of ubiquity in category, introduced and developed by P. J. Cameron since 1985, in terms of prehomogeneity, a notion introduced by J. F. Pabion for relational structures in 1972. We apply this characterization first to metric spaces and particularly graphs equipped with the graphic distance, and then to transition systems. The transition system that we display on this occasion has the same homogeneity properties as Rado’s graph and could have similar randomness properties.