Covering morphisms of crossed complexes and of cubical omega-groupoids with connection are closed under tensor product

TL;DR

This paper proves that covering morphisms of crossed complexes and cubical omega-groupoids with connections are preserved under tensor products, leading to new insights into free resolutions of groups and groupoids.

Contribution

It establishes that the tensor product of free crossed resolutions remains free, using the equivalence between crossed complexes and cubical omega-groupoids with connections.

Findings

Tensor product of free crossed resolutions is also free.

Covering morphisms are closed under tensor product.

The equivalence of categories facilitates the proof.

Abstract

The aim is the theorems of the title and the corollary that the tensor product of two free crossed resolutions of groups or groupoids is also a free crossed resolution of the product group or groupoid. The route to this corollary is through the equivalence of the category of crossed complexes with that of cubical omega-groupoids with connections where the initial definition of the tensor product lies. It is also in the latter category that we are able to apply techniques of dense subcategories to identify the tensor product of covering morphisms as a covering morphism.