It is well known that the collection of linear frames of a smooth n-manifold M defines a principal GL(n, R)-bundle over M (called the frame bundle); more generally, this construction makes sense for any vector bundle over M. Conversely, any principal bundle together with a representation induces an associated vector bundle; these processes establish therefore a correspondence between vector bundles on one side, and principal bundles with representations on the other side. In differential geometry there are several natural instances where diagrams of Lie groupoids and vector bundles, together with suitable compatibilities, appear. They are known as vector bundle groupoids (VB-groupoids) and their theory has been fairly developed in the past decades, with applications e.g. to representations up to homotopy and deformation theory. On the other hand, little is known about the principal bundle counterpart of these objects.
In this talk, I will recall the notions mentioned above and then introduce a special class of frames of VB-groupoids which interact nicely with the groupoid structure. I will then use them to associate to any given VB-groupoid a diagram of Lie groupoids and principal bundles, together with the action of a (strict) Lie 2-groupoid GL(l, k); this will lead to the general notion of a principal bundle groupoid (PB-groupoid). Moreover, I will sketch how to generalise the standard correspondence between vector bundles and principal bundles to a correspondence between VB-groupoids and PB-groupoids. I will conclude discussing a few examples and future applications.
This is joint work with Alfonso Garmendia.