发表机构
Institut Denis Poisson, CNRS, Université de Tours; Fachbereich Mathematik, Universität Hamburg（图尔大学; 汉堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为交叉模引入交叉辫子（及缎带）范畴，统一推广了多种已知结构，并利用上同调分类了分级向量空间上的此类结构，还通过扭曲局部模构造了实例。
AI 中文摘要
对于交叉模 $\chi: G \to H$，我们引入了 $\chi$-交叉辫子（相应地，$\chi$-交叉缎带）范畴的概念，其中这些范畴由群 $G$ 分级并带有 $H$-作用。我们的定义统一并推广了几个熟知的概念：取 $\chi = id: G \to G$ 并带有共轭作用，则恢复出 $G$-交叉辫子范畴；取 $\chi: G \to \{*\}$（其中 $G$ 为阿贝尔群）则得到 $G$-分级辫子范畴；取 $\chi: \{*\} \to G$ 则得到带有 $G$-作用的辫子范畴。$\chi$-交叉辫子范畴之间的等价关系通常比 $G$-交叉辫子范畴之间的等价关系更精细。我们利用上同调数据对 $G$-分级向量空间范畴上的 $\chi$-交叉辫子结构进行了分类，并给出了循环群的显式例子。给定一个在辫子幺半范畴中带有 $G$-和 $H$-作用的双中心代数，我们定义了扭曲局部模的概念，并展示了它们如何产生 $\chi$-交叉辫子范畴。此外，我们给出了充分条件，使得这些范畴额外成为 $\chi$-交叉缎带范畴或允许正交 $G$-分解。
英文摘要
For a crossed module $χ: G \to H$, we introduce the notion of $χ$-crossed braided (resp. ribbon) categories, where the categories are graded by group $G$ and carry an $H$-action. Our definition unifies and generalises several familiar notions: taking $χ= id: G \to G$ with the conjugation action recovers $G$-crossed braided categories; taking $χ: G \to \{*\}$ for abelian $G$ yields $G$-graded braided categories; taking $χ: \{*\} \to G$ leads to braided categories equipped with a $G$-action. The equivalence relation between $χ$-crossed braided categories is typically finer than that between $G$-crossed braided ones. We classify $χ$-crossed braided structures on the category of $G$-graded vector spaces in terms of cohomological data, and give explicit examples for cyclic groups. Given a doubly central algebra with $G$- and $H$-actions in a braided monoidal category, we define a notion of twisted-local modules and show how they give rise to $χ$-crossed braided categories. We furthermore give sufficient conditions so that these categories are additionally $χ$-crossed ribbon or admit an orthogonal $G$-decomposition.
Comments60 pages
