  Symmetric transverse sets were introduced to make the construction of the
parallel product with synchronization for process algebras functorial. It is
proved that one can do directed homotopy on symmetric transverse sets in the
following sense. A q-realization functor from symmetric transverse sets to
flows is introduced using a q-cofibrant replacement functor of flows. By
topologizing the cotransverse maps, the cotransverse topological cube is
constructed. It can be regarded both as a cotransverse topological space and as
a cotransverse Lawvere metric space. A natural realization functor from
symmetric transverse sets to flows is introduced using Raussen's notion of
natural $d$-path extended to symmetric transverse sets thanks to their
structure of Lawvere metric space. It is proved that these two realization
functors are homotopy equivalent on cofibrant symmetric transverse sets by
using the fact that the small category defining symmetric transverse sets is
c-Reedy in Shulman's sense. This generalizes to symmetric transverse sets
results previously obtained for precubical sets.
