In this article we study the fundamental category (Goubault and Raussen, 2002; Goubault, 2000) of a partially ordered topological space (Nachbin, 1965; Johnstone, 1982), as arising in, e.g., concurrency theory (Fajstrup et al., 1999). The ldquoalgebrardquo 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 (Goubault, 2002; Grandis, 2001). We end up by giving some hints about how to carry out some computations in simple cases.