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, $\Sigma $, of labels is fixed.
A weak higher dimensional transition system is a triple, $(S,\mu :L\to \Sigma ,T)$, where $S$ is a set of states, $L$ a set of actions, $\mu $ is a labelling map and $T={\bigcup}_{n\ge 1}{T}_{n}$, where ${T}_{n}\subset S\times {L}^{n}\times S$ is a set of $n$-transitions, such that two axioms hold, that ensure that the $n$-transitions are closed under permutations of labels and also satisfy a coherence (patching) axiom.
The case $n=1$ corresponds to transition systems.
Gian Luca Cattani, Vladimiro Sassone Higher Dimensional Transition Systems In, 11th Symposium of Logics in Computer Science, LICS ‘96. IEEE Press, 55-62,(1996).
Philippe Gaucher, Directed algebraic topology and higher dimensional transition system, New-York Journal of Mathematics 16 (2010), 409 - 461, available as arxiv:0903.4276
Philippe Gaucher, Towards a homotopy theory of higher dimensional transition systems, Theory and Applications of Categories, Vol. 25, No. 12, 2011, pp. 295–341, available as arxiv:1011.0918