范畴量子对称性与带状张量 2-范畴
Categorical quantum symmetries and ribbon tensor 2-categories
AI总结：
本文证明量子 2-规范变换 Hopf 范畴的 2-表示 2-范畴构成带状张量 2-范畴，并构造平衡函子及 2-缠绕标架不变量。
摘要：
在一篇关于四维 2-Chern-Simons 理论组合量子化的配套工作中，作者构造了作用于格点上离散曲面和乐构型的量子 2-规范变换 Hopf 范畴 $\tilde{C}=\mathbb{U}_q\mathfrak{G}$。本文证明，由有限半单 $\mathbb{C}$-线性 $\tilde C$-模范畴构成的 2-$\mathsf{Hilb}$-充实 2-表示 2-范畴 $\operatorname{2Rep}(\tilde C)$ 是辫化的、平面枢纽的且松刚性的，因此 $\operatorname{2Rep}(\tilde C)$ 给出了带状张量 2-范畴的一个例子。我们显式构造了带状平衡函子，并展示了它们相对于刚性 dagger 结构的相容条件。这使得人们可以细化以往文献中研究过的、带对偶的 2-范畴中各种“标架”概念。遵循 Baez-Langford 的 2-缠绕假设，随后可由进入 $\operatorname{2Rep}(\tilde C)$ 的带状 2-函子构造 2-缠绕的标架不变量，这类似于 Reshetikhin-Turaev 构造中装饰带状图的定义。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 的意义下变为严格枢纽的。
英文摘要：
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.
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