  A nice differential-geometric framework for (non-abelian) higher gauge theory
is provided by principal 2-bundles, i.e. categorified principal bundles. Their
total spaces are Lie groupoids, local trivializations are kinds of Morita
equivalences, and connections are Lie-2-algebra-valued 1-forms. In this
article, we construct explicitly the parallel transport of a connection on a
principal 2-bundle. Parallel transport along a path is a Morita equivalence
between the fibres over the end points, and parallel transport along a surface
is an intertwiner between Morita equivalences. We prove that our constructions
fit into the general axiomatic framework for categorified parallel transport
and surface holonomy.
