On the Eilenberg–Zilber theorem for crossed complexes

We give a natural strong deformation retraction from the fundamental homotopy crossed complex of a product of simplicial sets onto the tensor product of the corresponding crossed complexes. This generalises the classical theorem of Eilenberg and Zilber to a non-abelian setting. Explicit crossed complex homomorphisms analogous to the shuffle and Alexander–Whitney chain maps are given. We proceed to give a simplicially–enriched structure to the category of crossed complexes and to the simplical nerve N:Crs→S, but note that the fundamental crossed complex functoradjoint to N has only a lax or simplicially coherent enrichment.