Chern-Weil map for principal bundles over groupoids

The theory of principal G-bundles over a Lie groupoid is an important one unifying various types of principal G-bundles including those over manifolds those over orbifolds as well as equivariant principal G-bundles. In this paper we study differential geometry of these objects including connections and holonomy maps. We also introduce a Chern-Weil map for these principal bundles and prove that the characteristic classes obtained coincide with the universal characteristic classes. As an application we recover the equivariant Chern-Weil map of Bott-Tu. We also obtain an explicit chain map between the Weil model and the simplicial model of equivariant cohomology which reduces to the Bott-Shulman map S(g*)^G → H* (BG) when the manifold is a point.