# Operational total space theory of principal 2--bundles I: operational geometric framework

## TL;DR

This paper develops a geometric framework for principal 2-bundles using an operational approach, extending classical bundle theory to higher structures with potential applications in gauge theories.

## Contribution

It introduces an operational total space theory for strict principal 2-bundles using crossed modules, extending classical geometric concepts to higher gauge structures.

## Key findings

- Constructed an operational total space theory for principal 2-bundles.
- Expressed the theory via a derived Lie group from a crossed module.
- Laid groundwork for a subsequent formulation of 2-connections and gauge transformations.

## Abstract

It is a classic result that the geometry of the total space of a principal bundle with reference to the action of the bundle's structure group is codified in the bundle's operation, a collection of derivations comprising the de Rham differential and the contraction and Lie derivatives of all vertical vector fields and obeying the six Cartan relations. In particular, connections and gauge transformations can be defined through the way they are acted upon by the operation's derivations. In this paper, the first of a series of two extending the ordinary theory, we construct an operational total space theory of strict principal 2--bundles with regard to the action of the structure strict 2--group. Expressing this latter via a crossed module $(\mathsans{E},\mathsans{G})$, the operation is based on the derived Lie group $\mathfrak{e}[1]\rtimes\mathsans{G}$. In the second paper, an original formulation of the theory of $2$--connections and $1$-- and $2$--gauge transformations based on the operational framework worked out here will be provided.