  We study topological spaces with a distinguished set of paths, called
directed paths. Since these directed paths are generally not reversible, the
directed homotopy classes of directed paths do not assemble into a groupoid,
and there is no direct analog of the fundamental group. However, they do
assemble into a category, called the fundamental category. We define models of
the fundamental category, such as the fundamental bipartite graph, and minimal
extremal models which are shown to generalize the fundamental group. In
addition, we prove van Kampen theorems for subcategories, retracts, and models
of the fundamental category.
