Connections on Lie groupoids and Chern–Weil theory

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Let [Formula: see text] be a Lie groupoid equipped with a connection, given by a smooth distribution [Formula: see text] transversal to the fibers of the source map. Under the assumption that the distribution [Formula: see text] is integrable, we define a version of de Rham cohomology for the pair [Formula: see text], and we study connections on principal [Formula: see text]-bundles over [Formula: see text] 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 [Formula: see text]-bundles over a pair [Formula: see text].