Betweenness, monotonicity and road systems: a categorical interpretation

We apply a categorical lens to the study of R-relations (those betweenness relations generated via road systems) and closures thereof. As an application, we construct the antisymmetric closure of an R-relation as an inverse limit of length omega and expose it as a reflector between complete lattices and their distributive counterparts. We also study the Dedekind-MacNeille completion from a betweenness perspective. As an aside, and by exploiting the Grothendieck construction, we show that R-relations enjoy a succinct description as complete lattices with non-empty (join)-meet and (meet)-join preserving functions with these fi?brations being isomorphic.