Dear Categorists, Any comment is welcome. pg. Title: This paper is the third paper of a series devoted to higher dimensional transition systems. It is proved that there exists a model category of labelled symmetric precubical sets which is Quillen equivalent to the Bousfield localization of the left determined model category of cubical transition systems by the cubification functor. The realization functor from labelled symmetric precubical sets to cubical transition systems is not a left Quillen functor: it is only a left adjoint. It is proved that the two model categories are related to each other by a zig-zag of Quillen equivalences of length two. The class of cofibrations of this new model category structure is strictly larger than the class of monomorphisms. The weak equivalences are closely related to bisimulation. Similar results are obtained by restricting the constructions to the labelled symmetric precubical sets satisfying the HDA paradigm. Url: http://www.pps.univ-paris-diderot.fr/~gaucher/HomotopyPrecubicalSet.ps http://www.pps.univ-paris-diderot.fr/~gaucher/HomotopyPrecubicalSet.pdf [For admin and other information see: http://www.mta.ca/~cat-dist/ ]