# Spaces of directed paths on pre-cubical sets II

## TL;DR

This paper proves that the space of directed paths on a pre-cubical set is homotopy equivalent to a subspace of tame paths, and relates this to automata step executions via the cube chain category.

## Contribution

It establishes a homotopy equivalence between all directed paths and tame paths on pre-cubical sets, and links the path space to the cube chain category of the set.

## Key findings

- Homotopy equivalence between path space and tame path subspace.
- Weak homotopy equivalence of the cube chain category's nerve to the path space.
- Application to higher dimensional automata and step executions.

## Abstract

For a given pre-cubical set ($\square$--set) $K$ with two distinguished vertices $\bO$, $\bI$, we prove that the space $\vP(K)_\bO^\bI$ of d-paths on the geometric realization of $K$ with source $\bO$ and target $\bI$ is homotopy equivalent to its subspace $\vP^t(K)_\bO^\bI$ of tame d-paths. When $K$ is the underlying $\square$--set of a Higher Dimensional Automaton $A$, tame d-paths on $K$ represent step executions of $A$. Then, we define the cube chain category of $K$ and prove that its nerve is weakly homotopy equivalent to $\vP(K)_\bO^\bI$.