Geometric Realisations of Cubical Sets with Connections, and Classifying Spaces of Categories

We investigate the category of cubical sets with some additional degeneracies called connections. We prove that the realisation of a cubical set with connections is independent, up to homotopy, of whether we collapse those extra degeneracies or not and that any cubical set which is Kan admits connections. Using this type of cubical sets we define the cubical classifying space of a category and prove that this is equivalent to the simplicial one.