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

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.