[Paper Review] A global perspective to gerbes and their gauge stacks

Christoph Wockel|arXiv (Cornell University)|Mar 26, 2008

Homotopy and Cohomology in Algebraic Topology20 references5 citations

TL;DR

This paper introduces principal 2-bundles and demonstrates their classification via non-abelian Cech cohomology, establishing that their gauge 2-groups are described by 2-group-valued functors. Under mild conditions, these gauge 2-groups are shown to possess a natural smooth structure, extending classical bundle theory to higher gauge theory with global geometric insight.

ABSTRACT

In this paper we introduce principal 2-bundles and show how they are classified by non-abelian Cech cohomology. Moreover, we show that their gauge 2-groups can be described by 2-group-valued functors, much like in classical bundle theory. Using this, we show that, under some mild requirements, these gauge 2-groups possess a natural smooth structure. In the last section we provide some explicit examples.

Motivation & Objective

• To extend classical bundle theory to higher categories by introducing principal 2-bundles as a generalization of principal bundles.
• To classify principal 2-bundles using non-abelian Cech cohomology, providing a global classification framework.
• To describe the gauge 2-groups of these 2-bundles as 2-group-valued functors, mirroring classical gauge group constructions.
• To establish the existence of a natural smooth structure on these gauge 2-groups under mild conditions, enabling differential geometric treatment.

Proposed method

• Utilizes non-abelian Cech cohomology to classify principal 2-bundles over a space, generalizing the classifying theory of principal bundles.
• Models the gauge 2-groups as 2-group-valued functors on the category of open sets, extending the functorial description of gauge groups.
• Applies categorical and higher category theory to define and analyze the structure of 2-bundles and their automorphisms.
• Introduces a smooth structure on the gauge 2-groups by leveraging the smoothness of the underlying base space and transition data.
• Employs the framework of 2-groups and 2-categories to formalize higher gauge symmetries in a geometric and categorical setting.
• Provides explicit constructions and examples in the final section to illustrate the theory in concrete settings.

Experimental results

Research questions

• RQ1 How can principal 2-bundles be classified in a global, geometrically meaningful way?
• RQ2 In what sense do gauge 2-groups generalize classical gauge groups, and how can they be functorially described?
• RQ3 Under what conditions does a natural smooth structure exist on the gauge 2-group of a principal 2-bundle?
• RQ4 What role does non-abelian Cech cohomology play in the classification of higher bundles?
• RQ5 How do explicit examples of 2-bundles and their gauge 2-groups illustrate the theoretical framework?

Key findings

• Principal 2-bundles are classified by non-abelian Cech cohomology, generalizing the classical classification of principal bundles.
• The gauge 2-groups of principal 2-bundles are naturally described as 2-group-valued functors, extending the functorial description of gauge groups.
• Under mild assumptions, the gauge 2-groups admit a canonical smooth structure, enabling differential geometric analysis.
• The framework provides a consistent higher categorical generalization of classical gauge theory, with a global perspective on 2-connections and symmetries.
• Explicit examples in the final section demonstrate the applicability and consistency of the proposed classification and structure theory.