Commutative classifying space for simplicial groups

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 $\overline W(\tau,K)$, which takes as input a simplicial group $K$ and a cosimplicial group $\tau$ 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 $\overline W(\tau,K)$ is homotopy equivalent to the topological classifying space $B(\tau,|K|)$.