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.

Definition:

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 $T = \bigcup_{n\geq 1}T_n$, 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.

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