# Operational total space theory of principal 2-bundles II: 2-connections and 1- and 2--gauge transformations

## TL;DR

This paper develops a novel operational framework for understanding 2-connections and gauge transformations in principal 2-bundles, extending classical bundle theory to higher categorical structures.

## Contribution

It introduces an original formulation of 2-connections and gauge transformations within the operational total space theory for principal 2-bundles, expanding the geometric understanding of higher gauge theories.

## Key findings

- Formulation of 2-connections using the operational framework.
- Definition of 1- and 2-gauge transformations in the new setting.
- Extension of classical bundle concepts to principal 2-bundles.

## Abstract

The geometry of the total space of a principal bundle with regard to the action of the bundle's structure group is elegantly described by the bundle's operation, a collection of derivations consisting of the de Rham differential and the contraction and Lie derivatives of all vertical vector fields and satisfying the six Cartan relations. Connections and gauge transformations are defined by the way they behave under the action of the operation's derivations. In the first paper of a series of two extending the ordinary theory, we constructed an operational total space theory of strict principal 2--bundles with reference to the action of the structure strict 2--group. Expressing this latter through a crossed module $(\mathsans{E},\mathsans{G})$, the operation is based on the derived Lie group $\mathfrak{e}[1]\rtimes\mathsans{G}$. In this paper, the second of the series, an original formulation of the theory of $2$--connections and $1$-- and $2$--gauge transformations of principal $2$--bundles based on the operational framework is provided.