In recent years, several geometric models have been introduced and investigated as tools in the analysis of concurrent processes in Computer Science. A common feature of these models is, that an execution corresponds to an oriented path (dipath), and that homotopies preserving the orientations have equivalent computations as a result. As an algebraic topologist, one is tempted to apply the methods of the subject. What are the counterparts of the connected components, the fundamental group, homology, induced maps etc.? Answers are not that obvious. Because of orientations, the fundamental group has to be replaced by the (more complicated) fundamental category. To arrive at a component category, we propose to form a quotient category of a category of fractions of the fundamental category with respect to a certain system of weakly invertible morphisms. The talk will present some first results and several open questions, e.g. concerning the definition of a dihomology category, connections to directed coverings and some difficulties with naturality and homotopy invariance. (joint with Lisbeth Fajstrup)