Short Journey to the Land of Gerbes on Differentiable Stacks

We present an introduction to gerbes on differentiable stacks from the viewpoint of Lie groupoids, sheaf-theoretic descent, and differential cohomology fore the interested students and researchers alike. After recalling the basic notions of differentiable stacks and gerbes, we describe how an S¹-gerbe on a differentiable stack can be represented by an S¹-central extension of a Lie groupoid. We discuss connections, curvings, Dixmier–Douady classes, and the corresponding Chern–Weil picture. We also explain the relation with ordinary gerbes on manifolds and with gerbes on orbifolds. Several examples are given, including quotient stacks and classifying stacks. The final section formulates a classification principle in terms of degree-three cohomology and explains the role of Morita invariance.