Spaces of directed paths on pre-cubical sets

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.