Classification of adjustments on central crossed modules

TL;DR

This paper studies the existence and classification of adjustments on crossed modules of Lie groups, linking their infinitesimal versions to Lie algebra cohomology and Chern-Weil theory, with implications for higher gauge theory.

Contribution

It establishes a criterion for the existence of infinitesimal adjustments based on the Kassel-Loday class and Chern-Weil homomorphism, connecting geometric structures to algebraic invariants.

Findings

Infinitesimal adjustments exist iff the Kassel-Loday class is in the Chern-Weil image.

Provides a classification framework for adjustments via Lie algebra techniques.

Links adjustments to higher gauge theory and Lie algebra cohomology.

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.