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Using the notion of tame regular $d$-path of the topological $n$-cube, weintroduce the tame regular realization of a precubical set as a multipointed$d$-space. Its execution paths correspond to the nonconstant tame regular$d$-paths in the geometric realization of the precubical set. The associatedMoore flow gives rise to a functor from precubical sets to Moore flows which isweakly equivalent in the h-model structure to a colimit-preserving functor. Thetwo functors coincide when the precubical set is spatial, and in particularproper. As a consequence, it is given a model category interpretation of theknown fact that the space of tame regular $d$-paths of a precubical set ishomotopy equivalent to a CW-complex. We conclude by introducing the regularrealization of a precubical set as a multipointed $d$-space and with someobservations about the homotopical properties of tameness.

Regular directed path and Moore flow