Chern–Weil map for principal bundles over groupoids

TLDR: The theory of principal G-bundles over a Lie groupoid is an important one unifying various types of principal bundles, including those over manifolds, those over orbifolds, as well as equivariant principal bundles.

Abstract: 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 $$S(\mathfrak{g}^*)^G \to H^*(BG)$$ when the manifold is a point.