Parallel transport on a Lie 2-group bundle over a Lie groupoid along Haefliger paths

TL;DR

This paper develops a theory of parallel transport for principal 2-bundles over Lie groupoids, extending classical concepts to higher categorical structures and differentiable stacks, with applications to VB-groupoids.

Contribution

It introduces a Lie 2-group torsor version of the fibered category correspondence, enabling the definition of parallel transport along Haefliger paths in a higher geometric setting.

Findings

Established a smooth parallel transport functor for principal 2-bundles over Lie groupoids.

Extended the notion of principal bundles to differentiable stacks.

Analyzed parallel transport on associated VB-groupoids.

Abstract

We prove a Lie 2-group torsor version of the well-known one-one correspondence between fibered categories and pseudofunctors. Consequently, we obtain a weak version of the principal Lie group bundle over a Lie groupoid. The correspondence also enables us to extend a particular class of principal 2-bundles to be defined over differentiable stacks. We show that the differential geometric connection structures introduced in the authors' previous work, combine nicely with the underlying fibration structure of a principal 2-bundle over a Lie groupoid. This interrelation allows us to derive a notion of parallel transport in the framework of principal 2-bundles over Lie groupoids along a particular class of Haefliger paths. The corresponding parallel transport functor is shown to be smooth. We apply our results to examine the parallel transport on an associated VB-groupoid.