Parallel 2-transport and 2-group torsors

TL;DR

This paper introduces a new, concrete framework for parallel 2-transport and principal 2-group bundles with 2-connection, enabling explicit computations and generalizations to higher dimensions.

Contribution

It defines a new 2-category of 2-group torsors, proves non-Abelian Stokes and Ambrose-Singer theorems for 2-transport, and establishes an equivalence with principal 2-bundles with 2-connection.

Findings

New 2-category of 2-group torsors developed

Non-Abelian Stokes and Ambrose-Singer theorems proved for 2-transport

Concrete framework enables computation of 2-holonomy and Wilson surface observables

Abstract

We provide a new perspective on parallel 2-transport and principal 2-group bundles with 2-connection. We define parallel 2-transport as a 2-functor from the thin fundamental 2-groupoid to the 2-category of 2-group torsors. The definition of the 2-category of 2-group torsors is new, and we develop the tools necessary for computations in this 2-category. We prove a version of the non-Abelian Stokes Theorem and the Ambrose-Singer Theorem for 2-transport. This definition motivated by the fact that principal $G$-bundles with connection are equivalent to functors from the thin fundamental groupoid to the category of $G$-torsors. In the same lines we deduce a notion of principal 2-bundle with 2-connection, and show it is equivalent to our notion 2-transport functors. This gives a stricter notion than appears in the literature, which is more concrete. It allows for computations of 2-holonomy which will be exploited in a companion paper to define Wilson surface observables. Furthermore this notion can be generalized to a concrete but strict notion of $n$-transport for arbitrary $n$.