  We construct a model structure on the category of cubical sets with
connections whose cofibrations are the monomorphisms and whose fibrant objects
are defined by the right lifting property with respect to inner open boxes, the
cubical analogue of inner horns. We show that this model structure is Quillen
equivalent to the Joyal model structure on simplicial sets via the
triangulation functor. As an application, we show that cubical quasicategories
admit a convenient notion of a mapping space, which we use to characterize the
weak equivalences between fibrant objects in our model structure as
DK-equivalences.
