  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
$\vP(K)_\bO^\bI$.
