Oriented combinatorial topology and concurrency

Higher dimensional automata (HDA) provide valuable models of concurrent processes.Much current research related to HDA aims to further develop algebraic topological notionsrequired to analyse HDA in order to determine computer scientific properties includingdeadlock, safety, unreachable states, etc. It is well-known that classical algebraic topologywill not suffice since the sequences of actions represented by (1-dimensional) paths need tobe monotone with respect to a multi-dimensional coordinate system (the coordinate systemmight be thought of as time, or its coordinates can be thought of separately as progress withrespect to particular processes). The extent to which higher dimensional paths inherit anorientation as a result of either the coordinate system, or the definition of homotopy, variesaccording to the precise notions of directed algebraic topology that are utilised. This paperconsiders an extreme position in which all higher dimensional paths, like 1-dimensionalpaths, are oriented and can only be composed when orientations are compatible. This pointof view has arisen both from software engineering considerations and from considerationsof the history of classical combinatorial topology.