  We proved in a previous work that Cattani-Sassone's higher dimensional
transition systems can be interpreted as a small-orthogonality class of a
topological locally finitely presentable category of weak higher dimensional
transition systems. In this paper, we turn our attention to the full
subcategory of weak higher dimensional transition systems which are unions of
cubes. It is proved that there exists a left proper combinatorial model
structure such that two objects are weakly equivalent if and only if they have
the same cubes after simplification of the labelling. This model structure is
obtained by Bousfield localizing a model structure which is left determined
with respect to a class of maps which is not the class of monomorphisms. We
prove that the higher dimensional transition systems corresponding to two
process algebras are weakly equivalent if and only if they are isomorphic. We
also construct a second Bousfield localization in which two bisimilar cubical
transition systems are weakly equivalent. The appendix contains a technical
lemma about smallness of weak factorization systems in coreflective
subcategories which can be of independent interest. This paper is a first step
towards a homotopical interpretation of bisimulation for higher dimensional
transition systems.
