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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.
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