Regular directed path and Moore flow

TL;DR

This paper introduces a new topological model for precubical sets using tame regular paths and Moore flows, providing a homotopical interpretation and a functorial framework for their realization.

Contribution

It develops the tame regular realization of precubical sets as multipointed d-spaces and establishes a functor to Moore flows that aligns with known homotopy equivalences.

Findings

The tame regular realization models nonconstant tame regular d-paths.

The Moore flow functor is weakly equivalent to a colimit-preserving functor.

The space of tame regular d-paths is homotopy equivalent to a CW-complex.

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.