Absolute Lax 2-categories

We have introduced, in a previous paper, the fundamental lax 2-category of a ‘directed space’ X. Here we show that, when X has a T1-topology, this structure can be embedded into a larger one, with the same objects (the points of X), the same arrows (the directed paths) and the same cells (based on directed homotopies of paths), but a larger system of comparison cells. The new comparison cells are absolute, in the sense that they only depend on the arrows themselves rather than on their syntactic expression, as in the usual settings of lax or weak structures. It follows that, in the original structure, all the diagrams of comparison cells commute, even if not constructed in a natural way and even if the composed cells need not stay within the old system.