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Higher gauge theory I: 2-Bundles

Toby Bartels

TL;DR

The paper systematically promotes fibre bundles to a higher-categorical setting by defining $2$-bundles carried by a coherent $2$-group, and then develops a comprehensive 2-categorical framework of $2$-spaces, $2$-bundles, and their morphisms. It proves that, over a given $2$-space and with a chosen $2$-group, the $2$-category of $2$-bundles is equivalent to the $2$-category of $2$-transitions and, from the gerbe perspective, to the $2$-category of (nonabelian) gerbes, via a cohomological $2$-transition data. A semistrictification program is introduced to simplify coherence data, showing every $2$-transition is equivalent to a semistrict one, while crossed-modules provide a bridge to classical gerbe formalisms. The framework unifies bundles, vector bundles, and gerbes under higher gauge theory, and clarifies how ordinary bundle data arise as degenerate $2$-bundle cases, with potential applications to string-theoretic and higher-gauge physical models.

Abstract

I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2-groups, and bundles with a suitable notion of 2-bundle. To link this with previous work, I show that certain 2-categories of principal 2-bundles are equivalent to certain 2-categories of (nonabelian) gerbes. This relationship can be (and has been) extended to connections on 2-bundles and gerbes. The main theorem, from a perspective internal to this paper, is that the 2-category of 2-bundles over a given 2-space under a given 2-group is (up to equivalence) independent of the fibre and can be expressed in terms of cohomological data (called 2-transitions). From the perspective of linking to previous work on gerbes, the main theorem is that when the 2-space is the 2-space corresponding to a given space and the 2-group is the automorphism 2-group of a given group, then this 2-category is equivalent to the 2-category of gerbes over that space under that group (being described by the same cohomological data).

Higher gauge theory I: 2-Bundles

TL;DR

The paper systematically promotes fibre bundles to a higher-categorical setting by defining -bundles carried by a coherent -group, and then develops a comprehensive 2-categorical framework of -spaces, -bundles, and their morphisms. It proves that, over a given -space and with a chosen -group, the -category of -bundles is equivalent to the -category of -transitions and, from the gerbe perspective, to the -category of (nonabelian) gerbes, via a cohomological -transition data. A semistrictification program is introduced to simplify coherence data, showing every -transition is equivalent to a semistrict one, while crossed-modules provide a bridge to classical gerbe formalisms. The framework unifies bundles, vector bundles, and gerbes under higher gauge theory, and clarifies how ordinary bundle data arise as degenerate -bundle cases, with potential applications to string-theoretic and higher-gauge physical models.

Abstract

I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2-groups, and bundles with a suitable notion of 2-bundle. To link this with previous work, I show that certain 2-categories of principal 2-bundles are equivalent to certain 2-categories of (nonabelian) gerbes. This relationship can be (and has been) extended to connections on 2-bundles and gerbes. The main theorem, from a perspective internal to this paper, is that the 2-category of 2-bundles over a given 2-space under a given 2-group is (up to equivalence) independent of the fibre and can be expressed in terms of cohomological data (called 2-transitions). From the perspective of linking to previous work on gerbes, the main theorem is that when the 2-space is the 2-space corresponding to a given space and the 2-group is the automorphism 2-group of a given group, then this 2-category is equivalent to the 2-category of gerbes over that space under that group (being described by the same cohomological data).

Paper Structure

This paper contains 61 sections, 29 theorems, 172 equations.

Table of Contents

  1. Uncategorified bundles
  2. Categorical preliminaries
  3. Notation and terminology
  4. Products
  5. Pullbacks
  6. Equivalence relations
  7. Covers
  8. The category of spaces
  9. Products and pullbacks
  10. Open covers
  11. Bundles
  12. The category of bundles
  13. Trivial bundles
  14. Restrictions of bundles
  15. Locally trivial bundles
  16. ...and 46 more sections

Key Result

Proposition 1

Given the pullback diagram (pullback diagram) and pullbacks $P$ and $P '$, the spaces $P$ and $P '$ are isomorphic in ${\mathbf C}$.

Theorems & Definitions (29)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • Proposition 9
  • Proposition 10
  • ...and 19 more