Twisted tensor products of n-groupoids and crossed complexes

For any 1-reduced simplicial setX, we define a crossed complex of groups PCrsX, which we define as a twisted tensor product of the crossed cobar construction ΩCrsX and thef undamental crossed complexπX. In fact, we prove that PCrsXis contractible. Therefore PCrsXis a crossed complex model for the path space ofX. It is also an example of a crossed complex model of the total space of a fibration, ΩX−→PX−→X.This generalises from chain complexes to crossed complexes the theorem proved by J.F. Adams, and P. J. Hilton in their paper [3]. Our definition of twisted tensor products of crossed complexes also defines a twisted tensor product of n-groupoids, for all n. This comes from the fact that there is an equivalence of categories (∞-groupoids←→crossed complexes) which was proved by R. Brown and P. J. Higgins in their paper [12].We recall the classical Eilenberg-Zilber theorem for chain complexes, and its generalisation for crossed complexes, which show that the tensor product provides an algebraic model forthe Cartesian product of the fibration X−→X×Y−→Y.We also extend our theorems to 0-reduced simplicial sets X. In this case we generalise the crossed cobar construction ΩCrsX from 1-reduced simplicial sets to the group-completed crossed cobar constructionˆΩCrsXfor 0-reduced simplicial sets and define a crossed complex of groupoids PCrsX, a twisted tensor product with the twisted boundary maps∂Pn:PCrsnX= (ˆΩCrsX⊗φπX)n−→PCrsn−1= (ˆΩCrsX⊗φπX)n−1, ∂2= 0.We end by defining a contracting homotopy {ηn:PCrsnX→PCrsn+1X}which shows that this crossed complex of groupoids is still a model for the path space onX.