Classifying theory for simplicial parametrized groups

Abstract

In this paper we describe a classifying theory for families of simplicial topological groups. If $B$ is a topological space and $G$ is a simplicial topological group, then we can consider the non-abelian cohomology $H(B,G)$ of $B$ with coefficients in $G$. If $G$ is a topological group, thought of as a constant simplicial group, then the set $H(B,G)$ is the set of isomorphism classes of principal $G$ bundles, or $G$ torsors, on $B$. For more general simplicial groups $G$, the set $H(B,G)$ parametrizes the set of equivalence classes of higher $G$ torsors on $B$. In this paper we consider a more general setting where $G$ is replaced by a simplicial group in the category of spaces over $B$. The main result of the paper is that under suitable conditions on $B$ and $G$ there is an isomorphism between $H(B,G)$ and the set of isomorphism classes of fiberwise principal bundles on $B$, with structure group $|G|$ given by the fiberwise geometric realization of $G$.