Dihomotopy Classes of Dipaths in the Geometric Realization of a Cubical Set: from Discrete to Continuous and back again

This extended abstract should be considered as an overview of some results, and the more precise statements and proofs are found in the references. Hence the approach here is less rigorous than in the references. The new results presented at Dagstuhl are in [1] and [3]. The subject is directed topology in spaces which have been subdivided into cubes. Such spaces are used as the geometric model of Higher Dimensional Automata [6] and for Dijkstras PV-models [2], and going back and forth between the algebraic and the geometric representation is a very concrete example of a geometrization/discretization process or a discrete continuous correspondence.