发表机构
Tohoku University（东北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过引入一个4-范畴并证明其单对象版本（Gray 4-群）与3-交叉模范畴等价，精确建立了3-交叉模与高维范畴之间的联系，有望用于刻画曲面纽结和高维流形的拓扑性质。
AI 中文摘要
本文研究了3-交叉模范畴与Gray型4-群范畴之间的关系。3-交叉模的概念最早由Arvasi等人引入，其动机源于这样一个问题：何种代数结构能完全编码同伦4-型。另一方面，从高维群等价于高维范畴的代数实现这一观点出发——正如Sarikaya--Ulualan所建立的2-交叉模与Gray 3-群之间的关系所例证的那样——Arvasi等人的3-交叉模如何与任何高维范畴相关联尚不清楚。在我们之前的论文中，我们提出了3-交叉模的一个新定义，并观察到它在高维范畴方面具有自然的解释。在本文中，我们将这一解释精确化：我们引入一个4-范畴，当限制到单个对象和单个1-态射时，它退化为一个半严格辫子幺半2-范畴，并证明我们的3-交叉模范畴等价于Gray 4-群范畴，后者被定义为该4-范畴的单对象版本，其中所有态射都是可逆的。因此，我们期望这些结构能够正确捕捉曲面纽结和高维流形的拓扑性质。
英文摘要
In this paper, we investigate the relation between the category of 3-crossed modules and the category of Gray-type 4-groups. The notion of a 3-crossed module was first introduced by Arvasi \textit{et al.}, motivated by the question of what kind of algebraic structure completely encodes a homotopy 4-type. On the other hand, from the point of view that higher groups are equivalent to algebraic realizations of higher categories -- as exemplified by the relationship between 2-crossed modules and Gray 3-groups established by Sarikaya--Ulualan -- it had not been clear how the 3-crossed modules of Arvasi \textit{et al.} relate to any higher category. In our previous paper, we proposed a new definition of a 3-crossed module and observed that it admits a natural interpretation in terms of higher categories. In this paper, we make this interpretation precise: we introduce a 4-category, which reduces to a semistrict braided monoidal 2-category when restricted to a single object and a single 1-morphism, and prove that the category of our 3-crossed modules is equivalent to the category of Gray 4-groups, defined as single-object versions of this 4-category in which all morphisms are invertible. Furthermore, while Gray 4-groups provide a conceptual framework for 4-dimensional higher structures, they are often computationally intractable. Our equivalence establishes 3-crossed modules as a concrete, group-theoretic calculus for Gray 4-groups, providing a powerful computational tool for studying surface knots, state-sum invariants, and higher gauge theories.
Commentsv2: Title and abstract updated to emphasize computational applications to higher symmetries, TQFTs, and surface knots; cross-listed to math.QA, math.GT, and hep-th. 62 pages
