Higher dimensional transitions systems are thought of as modelling concurrent operation of interacting transition systems. They are related to applications of directed homotopy theory, and directed algebraic topology.
We will give the definition that Gaucher uses (see below). The version of Cattani and Sassone includes some extra conditions that yield a structure nearer to that of a process algebra.
A non-empty set, , of labels is fixed.
A weak higher dimensional transition system is a triple, , where is a set of states, a set of actions, is a labelling map and , where is a set of -transitions, such that two axioms hold, that ensure that the -transitions are closed under permutations of labels and also satisfy a coherence (patching) axiom.
The case corresponds to transition systems.