Regular directed path and Moore flow

TL;DR

This work develops canonical, tameness-aware realizations of precubical sets within a Moore-flow framework, defining tame regular realizations $|K|^t_{reg}$ and regular realizations $|K|_{reg}$ as multipointed $d$-spaces whose execution paths capture true concurrent behavior. It proves that the associated Moore-flow functor is weakly equivalent, up to the $h$-model structure, to a colimit-preserving replacement $[-]_{reg}$, with $[K]_{reg}$ being $m$-cofibrant and coinciding with ${oldsymbol M}^{oldsymbol G}(|K|^t_{reg})$ for spatial precubical sets. The paper builds a cube-chain–based realization, relates tameness to $L_1$-arc length, and shows that spaces of tame regular $d$-paths are homotopy equivalent to CW-complexes in a purely model-category framework. It further compares tame and regular realizations, establishing natural weak equivalences and highlighting when they align, thereby providing a robust, canonical, and cofibrant-friendly perspective on path spaces in concurrent computation models.

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.

Figures (3)

• Figure 1: $|X|=[0,1]\times [0,1]$, $X^0=\{0\}\times [0,1] \cup \{(x,x)\mid x\in [0,1]\}$, $\mathbb{P}^{top}_{(0,t),(t,t)}X= \mathcal{G}(1,1)$ for all $t\in ]0,1]$, $\mathbb{P}^{top}_{(0,0),(0,0)}X= \{(0,0)\}$ and $\mathbb{P}^{top}_{\alpha,\beta}X=\varnothing$ otherwise, there is no composable execution paths.
• Figure 2: The left $d$-path is tame, the right $d$-path is not tame.
• Figure 3: Non-tame $d$-path from $0_3$ to $1_3$ in the boundary of the $3$-cube

Theorems & Definitions (96)

• Theorem
• Theorem
• Theorem
• Theorem
• Proposition 1.1
• proof
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Example 1.6