Chern-Weil theory for principal bundles over Lie groupoids and differentiable stacks

Let $mathbb{X}=[X_1rightrightarrows X_0]$ be a Lie groupoid equipped with a connection, given by a smooth distribution $mathcal{H} subset T X_1$ transversal to the fibers of the source map. Under the assumption that the distribution $mathcal{H}$ is integrable, we define a version of de Rham cohomology for the pair $(mathbb{X}, mathcal{H})$, and we study connections on principal $G$-bundles over $(mathbb{X}, mathcal{H})$ in terms of the associated Atiyah sequence of vector bundles. We also discuss associated constructions for differentiable stacks. Finally, we develop the corresponding Chern-Weil theory and describe characteristic classes of principal $G$-bundles over a pair $(mathbb{X}, mathcal{H})$.