The branching nerve of HDA and the Kan condition

We have already seen that in reasonable situations, i.e. when the path ω-category associated to a higher dimensional automata (HDA) is an ω-groupoid, only the globular nerve satisfies the Kan condition. Indeed for the branching and merging nerves, this condition generally does not hold. This drawback is overcome here by introducing two new nerves (the left and right globular nerves) satisfying the Kan condition in any reasonable situation and having conjecturally the same simplicial homology as the branching and merging nerves respectively for any ω-category freely generated by a cubical set.