The theory of directed complexes is a higher-dimensional generalisation of the theory of directed graphs. In a directed graph, the simple directed paths form a subset of the free category which they generate; if the graph has no directed cycles, then the simple directed paths constitute the entire category. Generalising this, in a directed complex there is a class of split subsets which is contained in and generates a free ω-category; when a simple loop-freeness condition is satisfied, the split sets constitute the entire ω-category. The class of directed complexes is closed under the natural product and join constructions. The free ω-categories generated by directed complexes include the important examples associated to cubes and simplexes.