  Let G be a general (not necessarily finite dimensional compact) Lie group,
let g be its Lie algebra, let Cg be the cone on g in the category of
differential graded Lie algebras, and consider the functor which assigns to a
chain complex V the V-valued total de Rham complex of G. We describe the
G-equivariant de Rham cohomology in terms of a suitable relative differential
graded Ext, defined on the appropriate category of (G,Cg)-modules. The meaning
of "relative" is made precise via the dual standard construction associated
with the monad involving the aforementioned functor and the associated
forgetful functor. The corresponding infinitesimal equivariant cohomology is
the relative differential Ext over Cg relative to g. The functor under
discussion decomposes into two functors, the functor which determines
differentiable cohomology in the sense of Hochschild-Mostow and the functor
which determines the infinitesimal equivariant theory, suitably interpreted.
This functor decomposition, in turn, entails an extension of a Decomposition
Lemma due to Bott. Appropriate models for the differential graded Ext involving
a comparison between a suitably defined simplicial Weil coalgebra and the Weil
coalgebra dual to the familiar ordinary Weil algebra yield small models for
equivariant de Rham cohomology including the standard Weil and Cartan models
for the special case where the group G is compact and connected. Koszul duality
in de Rham theory results from these considerations in a straightforward
manner.
