TL;DR
This paper develops an explicit framework for parallel transport in principal 2-bundles, a key concept in higher gauge theory, by constructing Morita equivalences and intertwiners for paths and surfaces.
Contribution
It provides the first explicit construction of parallel transport in principal 2-bundles, integrating Morita equivalences and intertwiners within a categorified gauge theory framework.
Findings
Parallel transport along paths is a Morita equivalence.
Parallel transport along surfaces is an intertwiner.
The constructions align with axiomatic frameworks for categorified parallel transport.
Abstract
A nice differential-geometric framework for (non-abelian) higher gauge theory is provided by principal 2-bundles, i.e. categorified principal bundles. Their total spaces are Lie groupoids, local trivializations are kinds of Morita equivalences, and connections are Lie-2-algebra-valued 1-forms. In this article, we construct explicitly the parallel transport of a connection on a principal 2-bundle. Parallel transport along a path is a Morita equivalence between the fibres over the end points, and parallel transport along a surface is an intertwiner between Morita equivalences. We prove that our constructions fit into the general axiomatic framework for categorified parallel transport and surface holonomy.
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