AI 中文总结
该研究构建了基于$n$-Morita范畴的高阶Morita理论框架，证明了相关函子的等价性，阐明了双双模等概念在拓扑序缺陷研究中的作用。
AI 中文摘要
基于我们之前利用拓扑图像开展的$E_2$-代数的2-Morita等价性研究，我们在$n$-Morita范畴$\text{Mrt}_{E_n}(\text{C})$的框架下，构建了适用于不同维度拓扑序的Morita等价性的系统理论。该框架将各类$n$-Morita等价性概念统一为$\text{Mrt}_{E_n}(\text{C})$中对象的等价关系。我们还对比了Haugseng以及Gwilliam–Scheimbauer提出的高阶Morita范畴的构造。对于$n=1,2$，我们证明函子$\text{Mod}_n:\text{Mrt}_{E_n}(\text{C})\to \text{Mrt}_{E_{n-1}}(\text{LMod}^{\text{rep}}(\text{C}))$是一个等价，建立了高阶Morita理论的代数描述与其在模范畴中实现之间的关联。在此表述中，双双模的概念自然产生，为局域模和受限模提供了统一框架，其明确的定向数据及对应的融合规则，阐明了拓扑序中凝聚产生的缺陷之间的关系。
英文摘要
Building on our previous work on 2-Morita equivalence for $E_2$-algebras using topological pictures, we develop a systematic framework for Morita equivalence of topological orders in different dimensions in terms of $n$-Morita categories $\mathrm{Mrt}_{E_n}(\mathcal{C})$. In this framework, various notions of $n$-Morita equivalence are unified as equivalences of objects in $\mathrm{Mrt}_{E_n}(\mathcal{C})$. We also compare the constructions of higher Morita categories due to Haugseng and to Gwilliam--Scheimbauer. For $n=1,2$, we prove that the functor $\mathrm{Mod}_n:\mathrm{Mrt}_{E_n}(\mathcal{C})\to \mathrm{Mrt}_{E_{n-1}}(\mathrm{LMod}^{\mathrm{rep}}(\mathcal{C}))$ is an equivalence, relating the algebraic description of higher Morita theory to its realization in terms of module categories. In this formulation, the notion of a bi-bimodule arises naturally and provides a common framework for local and confined modules. Its explicit orientation data, together with the corresponding fusion rules, clarifies the relations among defects arising from condensation in topological orders.
Comments51 pages, 11 figures
