On the Eilenberg–Zilber theorem for crossed complexes

✍ Scribed by A.P. Tonks

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 192 KB
• Volume: 179
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(02)00160-3

✦ Synopsis

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 shu e 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 functor adjoint to N has only a lax or simplicially coherent enrichment.