  Let G be a compact, connected Lie group, acting smoothly on a manifold M.
Goresky-Kottwitz-MacPherson described a small Cartan model for the equivariant
cohomology of M, quasi-isomorphic to the standard Cartan complex of equivariant
differential forms. In this paper, we construct an explicit cochain map from
the small Cartan model into the standard Cartan model, inducing an isomorphism
in cohomology. The construction involves the solution of a remarkable
inhomogeneous Maurer-Cartan equation. This solution has further applications to
the theory of transgression in the Weil algebra, and to the Chevalley-Koszul
theory of the cohomology of principal bundles.
