The choice of cofibrations of higher dimensional transition systems

It is proved that there exists a left determined model structure of weak transition systems with respect to the class of monomorphisms and that it restricts to left determined model structures on cubical and regular transition systems. Then it is proved that, in these three model structures, for any higher dimensional transition system containing at least one transition, the fibrant replacement contains a transition between each pair of states. This means that the fibrant replacement functor does not preserve the causal structure. As a conclusion, we explain why working with star-shaped transition systems is a solution to this problem.