Cubical sets with symmetries and the homotopy coherent nerve functor

I will report on joint work with Vladimir Voevodsky on the homotopy theory of cubical sets with symmetries. These cubical sets are presheaves on the appropriately chosen Box category and we show that this Box category can be equipped with a topology for which the sheaves are precisely the simplicial sets. We then generalize several constructions and results of simplicial homotopy theory to the cubical setting, often simplifying the existing classical proofs.