Summary: In this article we study the fundamental category of a partially ordered topological space, as arising in, e.g., concurrency theory. The “algebra” of dipaths modulo dihomotopy (the fundamental category) of such a po-space is essentially finite in a number of situations: We define a component category of a category of fractions with respect to a suitable system, which contains all relevant information. Furthermore, some of these simpler invariants are conjectured to also satisfy some form of a van Kampen theorem, as the fundamental category does. We end up by giving some hints about how to carry out some computations in simple cases.