Comparing cubical and globular directed paths

TL;DR

This paper introduces a new, canonical realization functor from precubical sets to flows, ensuring homotopy equivalence without reliance on non-canonical cofibrant replacements, and relates execution path spaces to geometric paths.

Contribution

It constructs a homotopy-invariant realization functor from precubical sets to flows that avoids non-canonical choices, improving the understanding of directed path spaces.

Findings

The new realization functor is homotopy equivalent to previous ones.

The flow obtained is m-cofibrant, not q-cofibrant.

The space of execution paths is homotopy equivalent to nonconstant d-paths.

Abstract

A flow is a directed space structure on a homotopy type. It is already known that the underlying homotopy type of the realization of a precubical set as a flow is homotopy equivalent to the realization of the precubical set as a topological space. This realization depends on the non-canonical choice of a q-cofibrant replacement. We construct a new realization functor from precubical sets to flows which is homotopy equivalent to the previous one and which does not depend on the choice of any cofibrant replacement functor. The main tool is the notion of natural $d$-path introduced by Raussen. The flow we obtain for a given precubical set is not anymore q-cofibrant but is still m-cofibrant. As an application, we prove that the space of execution paths of the realization of a precubical set as a flow is homotopy equivalent to the space of nonconstant $d$-paths between vertices in the geometric realization of the precubical set.