Hello,

A flat principal $G$-bundle over $X$ is determined by its holonomies, which are (after picking a trivialization) group homomorphisms $\pi_1(X)\rightarrow G$. The fiber of the bundle is not canonically identified with $G$, so these maps are only determined up conjugation by $G$. Equivalently these are gauge transformations at the basepoint where $\pi_1$ is evaluated.

Now let $H \overset{t}\rightarrow G$ present a 2-group. With respect to some trivialization, a flat 2-connection on a principal 2-bundle assigns an element of $G$ to a closed curve, and an element $h$ of $H$ to a surface bounding a curve $\gamma$ such that $\gamma$ gets $t(h)$. Flat means that $h$ only depends on the homotopy class of this surface (homotopies fixing $\gamma$).

Thus, I expect flat 2-connections are determined by a functor from the path 2-group of $X$ into $(H\rightarrow G)$. My question is : what is the degeneracy of this presentation? In other words, what is analogous to "up to conjugation in $G$" for flat 1-bundles?

Thanks.