  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.
