Regular directed path and Moore flow
Résumé
Using the notion of regular d-path of the topological n-cube, we construct the regular realization of a precubical set as a multipointed d-space. Its execution paths correspond to the tame regular d-paths of the precubical set between two vertices. By considering the associated Moore flow, we obtain a colimit-preserving functor from precubical sets to Moore flows. As a consequence, we obtain that the associated Moore flow is always m-cofibrant. This provides a model category interpretation of the known fact that the space of tame regular d-paths of a precubical set between two vertices is homotopy equivalent to a CW-complex. Finally, we prove that the flow associated with the regular realization of a precubical set coincides with the natural realization of a precubical set as a flow.
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