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Classification of adjustments on central crossed modules

Matthias Ludewig, Konrad Waldorf

TL;DR

The paper develops a comprehensive framework for adjustments on central crossed modules of Lie groups and their infinitesimal counterparts on Lie algebras. It establishes a precise criterion: infinitesimal adjustments exist if and only if the Kassel–Loday class $ ext{KL}( rak{G})$ lies in the image of the Lie-algebraic Chern–Weil map $ ext{cw}$, and it shows how these adjustments integrate under suitable connectedness assumptions. By exploiting a hierarchy of groupoids and a bicategorical Grothendieck construction, the authors show that adjusted crossed modules are classified by their adjusted Kassel–Loday class, and they provide explicit constructions (both algebraic and group-theoretic) realizing prescribed adjusted classes. The approach yields concrete examples, such as the string 2-group and categorical tori, and clarifies how adjustments transform under butterflies, enabling a robust, functorial understanding of higher gauge-theoretic refinements that circumvent fake-flatness constraints. This framework advances higher gauge theory by providing algebraic and geometric tools to manage nontrivial higher structures via explicit cohomological invariants.

Abstract

Adjustments are additional structures on crossed modules of Lie groups, serving as a tool in higher gauge theory to circumvent the fake flatness of connections on 2-bundles. In this article, we investigate the existence and classification of adjustments, as well as their covariance under weak equivalences. Our approach is based on a differentiation/integration correspondence with an infinitesimal version of adjustments on the associated crossed module of Lie algebras, which we then study using Lie algebra techniques. Our main result is that infinitesimal adjustments exist if and only if the Kassel-Loday classof the crossed module lies in the image of the (Lie algebraic) Chern-Weil homomorphism.

Classification of adjustments on central crossed modules

TL;DR

The paper develops a comprehensive framework for adjustments on central crossed modules of Lie groups and their infinitesimal counterparts on Lie algebras. It establishes a precise criterion: infinitesimal adjustments exist if and only if the Kassel–Loday class lies in the image of the Lie-algebraic Chern–Weil map , and it shows how these adjustments integrate under suitable connectedness assumptions. By exploiting a hierarchy of groupoids and a bicategorical Grothendieck construction, the authors show that adjusted crossed modules are classified by their adjusted Kassel–Loday class, and they provide explicit constructions (both algebraic and group-theoretic) realizing prescribed adjusted classes. The approach yields concrete examples, such as the string 2-group and categorical tori, and clarifies how adjustments transform under butterflies, enabling a robust, functorial understanding of higher gauge-theoretic refinements that circumvent fake-flatness constraints. This framework advances higher gauge theory by providing algebraic and geometric tools to manage nontrivial higher structures via explicit cohomological invariants.

Abstract

Adjustments are additional structures on crossed modules of Lie groups, serving as a tool in higher gauge theory to circumvent the fake flatness of connections on 2-bundles. In this article, we investigate the existence and classification of adjustments, as well as their covariance under weak equivalences. Our approach is based on a differentiation/integration correspondence with an infinitesimal version of adjustments on the associated crossed module of Lie algebras, which we then study using Lie algebra techniques. Our main result is that infinitesimal adjustments exist if and only if the Kassel-Loday classof the crossed module lies in the image of the (Lie algebraic) Chern-Weil homomorphism.
Paper Structure (27 sections, 44 theorems, 271 equations)

This paper contains 27 sections, 44 theorems, 271 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Adjustments
  4. Adjustments on central crossed modules of Lie groups
  5. Infinitesimal adjustments and their integration
  6. Classification of central crossed modules of Lie algebras
  7. The Kassel-Loday class
  8. Invariance under butterflies
  9. The classification of crossed modules
  10. Classification of adjustments
  11. The Chern-Weil homomorphism
  12. Existence of infinitesimal adjustments
  13. Covariance under butterflies
  14. Groupoids of adjustments
  15. The groupoid of adjustments
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $\Gamma =(H \stackrel{t}{\to} G \stackrel{\alpha}{\to} \mathrm{Aut}(H))$ be a central crossed module of Lie groups, let $\mathfrak{G}$ be the corresponding crossed module of Lie algebras, and let $s$ be any section of $\mathfrak{G}$. Differentiation constitutes maps which are injective when $G$ is connected, and bijections when $G$ is connected and simply-connected and $H$ is connected.

Theorems & Definitions (104)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • ...and 94 more