# Universal coverings of simplicial groups

Tuesday, July 03rd, 2012 | Author: Konrad Voelkel

Every simplicial set admits a universal covering space. If the simplicial set is a simplicial group, the covering can be given a group structure, such that the covering map is a group homomorphism. This can be done "at once" without using geometric realizations, just by close analysis of the usual construction of the universal covering.

Let be a simplicial group, its path space, its loop space and the kernel of the projection , which consists of all contractible loops. The composition of with the inclusion yields a monomorphism whose cokernel we denote by .

I claim that this simplicial group already does the job.

Continue reading «Universal coverings of simplicial groups»

Category: English, Mathematics | 3 Comments