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 W (, K) W ¯ (τ, K), which takes as input a simplicial group K and a cosimplicial group τ that encodes the additional structure such as commutativity, is a variation of the W W ¯ -construction for simplicial groups. Our main result shows that the geometric realization of our W (, K) W ¯ (τ, K) is homotopy equivalent to the topological classifying space B (, |K|) B (τ, | K |).
Okay et al. (Thu,) studied this question.