A Chern-Weil theory for Lie groupoids with Cartan connections

Abstract

We extend Chern-Weil theory to Lie groupoids carrying a Cartan connection. For a principal $(\Sigma ,\omega )$-bundle $(P,\theta )$ over $X$ we construct a map of differential graded algebras from an $\omega$-basic subcomplex of the Weil algebra of the Lie algebroid of $\Sigma$ to ${\Omega }^{\bullet }(X)$, whose map in cohomology does not depend on $\theta$. It recovers the classical Chern-Weil map for Lie groups, the Cartan model for action groupoids, and the characteristic classes of Cartan geometries for gauge groupoids.