Commutative classifying space for simplicial groups

Abstract

In this paper, we introduce a simplicial analog of classifying spaces for commutativity which classify principal bundles with commutativity structure on their transition functions. Our construction $\overline{W}(\ddot{I}„,K)$, which takes as input a simplicial group $K$ and a cosimplicial group $\ddot{I}„$ that encodes the additional structure such as commutativity, is a variation of the $\overline{W}$-construction for simplicial groups. Our main result shows that the geometric realization of our $\overline{W}(\ddot{I}„,K)$ is homotopy equivalent to the topological classifying space $B(\ddot{I}„,|K|)$.

21 pages