Regular directed path and Moore flow
Philippe Gaucher

Abstract. Abstract. Using the notion of tame regular d-path of the topological n-cube, we introduce 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 associated Moore flow gives rise to a functor from precubical sets to Moore flows which is weakly equivalent in the h-model structure to a colimit-preserving functor. The two functors coincide when the precubical set is spatial, and in particular proper. As a consequence, it is given a model category interpretation of the known fact that the space of tame regular d-paths of a precubical set is homotopy equivalent to a CW-complex. We conclude by introducing the regular realization of a precubical set as a multipointed d-space and with some observations about the homotopical properties of tameness

Rend. Mat. Appl. (7) 45 (2024) 111-151; pdf