Principal bundle groupoids, their gauge group and their nerve

Abstract

We consider groupoids in the category of principal bundles, which we call principal bundles (PB) groupoids. Inspired by work by Th. Nikolaus and K. Waldorf, we generalise bundle gerbes over manifolds to bundle gerbes over groupoids and discuss a functorial correspondence between PB groupoids and bundle gerbes over groupoids. From a PB groupoid over a fibre product groupoid, we build a bundle gerbe over another fibre product groupoid. Conversely, from a bundle gerbe over a Lie groupoid, we build a PB groupoid. It has a trivial base and from any PB groupoid with trivial base, we build a bundle gerbe over a Lie groupoid. In that case, the resulting bundle gerbe is isomorphic as a groupoid to a partial quotient of the PB groupoid. We describe the nerves of PB groupoids and their partial quotients, which are simplicial objects in the category of principal bundles. Applying this construction enables us to define the inner transformation group of the nerve of a partial quotient groupoid and to describe the transformations of the corresponding bundle gerbe.