Flat principal 2-group bundles on $S^2$ and their holonomy

Abstract

Let $\mathcal{G}(G,A,\alpha)$ be a discrete 2-group, given as the {categorical} extension of an ordinary group $G$ by an abelian group $A$. We define principal $\mathcal{G}$-bundles over a manifold in terms of \v{C}ech data over a good open cover. In the case of the manifold $S^2$, we calculate this bicategory explicitly, and identify its 2-skeleton. We compare with the holonomy bicategory $\mathrm{Fun}(*/\!\!/ \pi_{\le 2}(S^2), * /\!\!/ \mathcal{G})$, and provide a concrete equivalence between the two.