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

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.