Categorical quantum symmetries and ribbon tensor 2-categories

Abstract

In a companion work on the combinatorial quantization of 4d 2-Chern-Simons theory, the author has constructed the Hopf category of quantum 2-gauge transformations $\tilde{C}=\mathbb{U}_q\mathfrak{G}$ acting on the discrete surface-holonomy configurations on a lattice. We prove in this article that the 2-$\mathsf{Hilb}$-enriched 2-representation 2-category $\operatorname{2Rep}(\tilde C)$ of finite semisimple $\mathbb{C}$-linear $\tilde C$-module categories is braided, planar-pivotal, and lax rigid, hence $\operatorname{2Rep}(\tilde C)$ provides an example of a ribbon tensor 2-category. We explicitly construct the ribbon balancing functors, and exhibit their coherence conditions against the rigid dagger structures. This allows one to refine the various notions of \textit{framing} in a 2-category with duals that have been previously studied in the literature. Following the 2-tangle hypothesis of Baez-Langford, framed invariants of 2-tangles can then be constructed from ribbon 2-functors into $\operatorname{2Rep}(\tilde C)$, analogous to the definition of decorated ribbon graphs in the Reshetikhin-Turaev construction. We will also prove that, in the classical limit $q\rightarrow 1$, the 2-category $\operatorname{2Rep}(\mathbb{U}_{q=1}\mathfrak{G})$ becomes strict pivotal in the sense of Douglas-Reutter.