Tannaka-Krein duality for finite 2-groups

Abstract

Let $\mathcal{G}$ be a finite 2-group. We show that the 2-category $2\mathrm{Rep}(\mathcal{G})$ of finite semisimple 2-representations is a symmetric fusion 2-category. We also relate the auto-equivalence 2-group of the symmetric monoidal forgetful 2-functor $\omega : 2\mathrm{Rep}(\mathcal{G}) \to 2\mathrm{Vec}$ to the auto-equivalence 2-group of the regular algebra and show that they are equivalent to $\mathcal{G}$. This result categorifies the usual Tannaka-Krein duality for finite groups.