Groupoids extension and Non-abelian gerbes

From CRCG-Wiki

We develop the theory of connections on Lie groupoid $G$-extensions, which we call "connections on $G$-gerbes", and study the induced connections on various associated bundles. In particular, we develop a cohomology theory which measures the existence of connections and curvings for $G$-gerbes over stacks. We also introduce $G$-central extensions of groupoids, generalizing the standard groupoid $S1$-central extensions. As an example, we apply our theory to study the differential geometry of $G$-gerbes over a manifold.