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We construct and study general and integrable connections on Lie groupoids and differentiable stacks, as well as on principal bundles over them using an Atiyah exact sequence of vector bundles associated to transversal tangential distributions.

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In this paper, we introduce the notion of a topological groupoid extension and relate it to the already existing notion of a gerbe over a topological stack. We further study the properties of a gerbe over a Serre, Hurewicz stack.