PB-groupoids vs VB-groupoids

Abstract

In this paper, we establish the principal bundle counterpart of the well-known and widely applied notion of vector bundle groupoid (VB-groupoid). In particular, we provide a general notion of principal bundle groupoid (PB-groupoid) as a diagram of Lie groupoids and principal bundles, together with the action of a (strict) Lie $2$-groupoid. This recovers as particular cases various constructions involving structural Lie $2$-groups, Lie groupoids and Lie groups.
Moreover, given any VB-groupoid, we detect which frames are compatible with the underlying structure and show that the sets of such frames has a natural PB-groupoid structure, with the action of the general linear $2$-groupoid of the appropriate ranks. We use this construction, as well as that of the associated bundle induced by a $2$-representation, to extend the classical correspondence between vector bundles and principal bundles to a new correspondence between VB-groupoids and PB-groupoids. We conclude with several examples and applications.