Characterizing flat 2-connections by their holonomy

Question

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.

Answer 1

Let $\mathcal{P}$ be a principal 2-bundle with structure 2-group a crossed module $t:H \to G$. Then, the holonomy of a connection on $\mathcal{P}$ around a surface $\Sigma$ is a well-defined element $$\mathrm{Hol}_{\mathcal{P}}(\Sigma) \in H/[G,H],$$ where $[G,H]$ is the normal subgroup of $H$ generated by all elements of the form $h^{-1}\alpha(g,h)$, where $\alpha$ denotes the action of $G$ on $H$ of the crossed module. Above quotient generalizes the concept of "conjugacy class".

This is explained in detail in my paper with Urs Schreiber

• Connections on non-abelian Gerbes and their Holonomy,

see Example 5.7.

To which extend the elements $\mathrm{Hol}_{\mathcal{P}}(\Sigma) \in H/[G,H]$ characterize the 2-bundle $\mathcal{P}$ I cannot say.

Maybe you are willing to drop "flat" upon replacing "holonomy" by "parallel transport": even in the classical world it is true that every principal $G$-bundle with connection is characterized by its parallel transport, may it be flat or not.

The same statement remains true in the context of connections on 2-bundles: every principal 2-bundle with connection is characterized by its 2-transport. This is one of the main statements of my above-mentioned paper with Urs.