A global perspective to connections on principal 2-bundles

TL;DR

This paper develops a global framework for connections on principal 2-bundles using Lie 2-algebra-valued forms on Lie groupoids, advancing higher gauge theory with a classification via non-abelian differential cohomology.

Contribution

It introduces a notion of Lie 2-algebra-valued differential forms on Lie groupoids and defines principal 2-bundle connections, providing a global perspective in higher gauge theory.

Findings

Defined Lie 2-algebra-valued differential forms on Lie groupoids

Established a classification of principal 2-bundle connections via non-abelian differential cohomology

Provided a consistent framework for higher gauge theory

Abstract

For a strict Lie 2-group, we develop a notion of Lie 2-algebra-valued differential forms on Lie groupoids, furnishing a differential graded-commutative Lie algebra equipped with an adjoint action of the Lie 2-group and a pullback operation along Morita equivalences between Lie groupoids. Using this notion, we define connections on principal 2-bundles as Lie 2-algebra-valued 1-forms on the total space Lie groupoid of the 2-bundle, satisfying a condition in complete analogy to connections on ordinary principal bundles. We carefully treat various notions of curvature, and prove a classification result by the non-abelian differential cohomology of Breen-Messing. This provides a consistent, global perspective to higher gauge theory.