Spaces of directed paths on pre-cubical sets

TL;DR

This paper constructs a minimal CW-complex model homotopy equivalent to the space of directed paths in pre-cubical sets, providing explicit formulas for cell incidences, advancing the understanding of concurrent computation models.

Contribution

It introduces a functorial, minimal CW-complex construction for the space of directed paths on pre-cubical sets, with explicit incidence formulas.

Findings

Constructed a CW-complex $W(K)_v^w$ homotopy equivalent to directed path spaces.

Provided explicit formulas for incidence numbers of cells.

Established functoriality and minimality of the construction.

Abstract

The spaces of directed paths on the geometric realizations of pre-cubical sets, called also $\square$--sets, can be interpreted as the spaces of possible executions of Higher Dimensional Automata, which are models for concurrent computations. In this paper we construct, for a sufficiently good pre-cubical set $K$, a CW-complex $W(K)_v^w$ that is homotopy equivalent to the space of directed paths between given vertices $v$, $w$ of $K$. This construction is functorial with respect to $K$, and minimal among all functorial constructions. Furthermore, explicit formulas for incidence numbers of the cells of $W(K)_v^w$ are provided.