Abstract
Let
Y
be a CW-complex with a single 0-cell, let
K
be its Kan group, a free simplicial group whose realization is a model for the space
ΩY
of based loops on
Y
, and let
G
be a Lie group, not necessarily connected. By means of simplicial techniques involving fundamental results of {\smc Kan's} and the standard
W
- and bar constructions, we obtain a weak
G
-equivariant homotopy equivalence from the geometric realization $|\roman{Hom}(K,G)|$ of the cosimplicial manifold $\roman{Hom}(K,G)$ of homomorphisms from
K
to
G
to the space $\roman{Map}^o(Y,BG)$ of based maps from
Y
to the classifying space
BG
of
G
where
G
acts on
BG
by conjugation. Thus when
Y
is a smooth manifold, the universal bundle on
BG
being endowed with a universal connection, the space $|\roman{Hom}(K,G)|$ may be viewed as a model for the space of based gauge equivalence classes of connections on
Y
for all topological types of
G
-bundles on
Y
thereby yielding a rigorous approach to lattice gauge theory; this is illustrated in low dimensions.