标题： 范畴量子对称性和辫子张量2-范畴 显示英文标题

标题： Categorical quantum symmetries and ribbon tensor 2-categories

摘要： 在关于4d 2-Chern-Simons理论的组合量化的一篇相关工作中，作者构造了作用于格点上的离散表面-holonomy配置上的量子2-规范变换的Hopf范畴$\tilde{C}=\mathbb{U}_q\mathfrak{G}$。 我们在本文中证明了有限半单$\mathbb{C}$-线性$\tilde C$-模范畴的 2-$\mathsf{Hilb}$-丰富 2-表示 2-范畴$\operatorname{2Rep}(\tilde C)$是辫子的、平面-可逆的，并且是松紧的，因此$\operatorname{2Rep}(\tilde C)$提供了一个辫子张量 2-范畴的例子。 我们显式构造了辫子平衡函子，并展示了它们与紧致自伴结构的一致性条件。 这使得人们能够在具有对偶的2-范畴中精炼之前文献中研究过的各种\textit{框架}概念。 根据Baez-Langford的2扭结假说，可以构造出2扭结的框架不变量，这些不变量来自进入$\operatorname{2Rep}(\tilde C)$的丝带2-函子，类似于Reshetikhin-Turaev构造中装饰丝带图的定义。我们还将证明，在经典极限$q\rightarrow 1$下，2-范畴$\operatorname{2Rep}(\mathbb{U}_{q=1}\mathfrak{G})$在Douglas-Reutter的意义下变为严格前 pivotal。 显示英文摘要

摘要： 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.