Regular clock map and trace space

A regular clock map is a regular map of directed spaces from a saturated directed space to the directed circle. We prove that the category of regular clock maps is a small-orthogonality class of the category of clock maps. Hence it is locally presentable. Any geometric realization of precubical sets and of transverse sets gives rise to a regular clock map. Finally, we prove that for the underlying directed space of a regular clock map, the canonical quotient from directed paths to traces is always a homotopy equivalence.