Spaces of directed paths on pre-cubical sets

The spaces of directed paths on the geometric realizations of pre-cubical sets, called also □-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)wv 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)wv are provided.