Adjusted parallel transport for higher gauge theories

Many physical theories, including notably string theory, require non-abelian higher gauge fields defining higher holonomy. Previous approaches to such higher connections on categorified principal bundles require these to be fake flat. This condition, however, renders them locally gauge equivalent to connections on abelian gerbes. For particular higher gauge groups, for example 2-group models of the string group, this limitation can be overcome by generalizing the notion of higher connection. Starting from this observation, we define a corresponding generalized higher holonomy functor which is free from the fake flatness condition, leading to a truly non-abelian parallel transport.