Directed path and Moore flow

TL;DR

This paper introduces the tame realization of precubical sets as multipointed d-spaces, extending previous work to non-regular cases and relating Moore flows to precubical sets.

Contribution

It extends the tame realization concept to non-regular precubical sets and establishes a functorial relationship with Moore flows.

Findings

The tame realization captures nonconstant tame d-paths in geometric realizations.

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

For spatial precubical sets, the functors coincide.

Abstract

This addendum extends prior work to the non-regular setting by introducing the tame realization of a precubical set as a multipointed $d$-space. Its execution paths are precisely the nonconstant tame $d$-paths in the geometric realization of the precubical set. The associated Moore flow induces a functor from precubical sets to Moore flows, which is naturally weakly equivalent, within the $h$-model structure, to a colimit-preserving functor whose image is included in the class of m-cofibrant Moore flows. For spatial (and thus proper) precubical sets, these functors coincide.