The path category P(K) of a simplicial complex K is a category which is built from vertices (objects) and 1-simplices (morphisms), subject to commutativity conditions associated to the 2-simplices of K. This construction extends to a functor from simplicial sets to categories which is left adjoint to the nerve. Here is why one cares: path category morphisms specialize to execution paths in higher dimensional automata. These objects are geometric models for behaviour of parallel processing systems, and techniques are required to distinguish execution paths between states in such a system. This calculational problem is non-trivial, since the path category functor is not a standard homotopy invariant and produces categories with little extra structure. The known viable lines of attack arise from higher category theory and homotopy coherence theory.