2-Drinfel'd double symmetry of the 4d Kitaev model

Following the general theory of categorified quantum groups developed by the author previously (arxiv:2304.07398), we construct the 2-Drinfel'd double associated to a finite group $N=G_0$. For $N=\mathbb{Z}_2$, we explicitly compute the braided 2-categories of 2-representations of certain version of this 2-Drinfel'd double, and prove that they characterize precisely the 4d toric code and its spin-$\mathbb{Z}_2$ variant. This result relates the two descriptions (categorical vs. field theoretical) of 4d gapped topological phases in existing literature and, perhaps more strikingly, displays the first ever instances of higher Tannakian duality for braided 2-categories. In particular, we show that particular twists of the underlying 2-Drinfel'd double is responsible for much of the higher-structural properties that arise in 4d topological orders.