  We provide a new perspective on parallel 2-transport and principal 2-group
bundles with 2-connection. We define parallel 2-transport as a 2-functor from
the thin fundamental 2-groupoid to the 2-category of 2-group torsors. The
definition of the 2-category of 2-group torsors is new, and we develop the
tools necessary for computations in this 2-category. We prove a version of the
non-Abelian Stokes Theorem and the Ambrose-Singer Theorem for 2-transport. This
definition motivated by the fact that principal $G$-bundles with connection are
equivalent to functors from the thin fundamental groupoid to the category of
$G$-torsors. In the same lines we deduce a notion of principal 2-bundle with
2-connection, and show it is equivalent to our notion 2-transport functors.
This gives a stricter notion than appears in the literature, which is more
concrete. It allows for computations of 2-holonomy which will be exploited in a
companion paper to define Wilson surface observables. Furthermore this notion
can be generalized to a concrete but strict notion of $n$-transport for
arbitrary $n$.
