Classifying theory for simplicial parametrized groups

TL;DR

This paper develops a classifying theory for families of simplicial topological groups, linking non-abelian cohomology to fiberwise principal bundles, extending classical bundle classification to a more general simplicial setting.

Contribution

It introduces a new framework connecting non-abelian cohomology with fiberwise principal bundles for simplicial groups over a base space.

Findings

Establishes an isomorphism between cohomology sets and classes of fiberwise principal bundles.

Generalizes classical bundle classification to simplicial topological groups.

Provides conditions under which the classification holds.

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 $G$.