  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 $G$.
