半单张量范畴的扭曲德利涅积
Twisted Deligne products of semisimple tensor categories
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中文总结 AI 辅助
探讨半单张量范畴$\mathcal C$、$\mathcal D$的扭曲德利涅积分类，给出特定条件下唯一扭曲德利涅积是普通积的结论，借助相关工作实现群论分类及融合范畴精确分解，附录引入范畴$n$-上链概念并回答相关问题。
中文摘要 AI 辅助
我们讨论了两个半单张量范畴$\mathcal C$、$\mathcal D$的扭曲德利涅积的分类，即它们的格罗滕迪克环张量积的范畴化，其中因子由$\mathcal C$和$\mathcal D$范畴化。特别地，我们表明，如果两个因子都没有非平凡分次，或者如果一个因子既没有非平凡分次也没有恒等函子上的张量结构，那么唯一的扭曲德利涅积就是普通的那个。利用 Müller、Peña Pollastri 和 Plavnik 在 arXiv:2405.10207 上的工作，这原则上给出了扭曲德利涅积的群论分类，更一般地，给出了任意融合范畴的精确分解。在附录中，我们引入了$n = 2,3,4$时的范畴$n$-上链的概念，并表明它们都是基础环的通用分次群的群$n$-上链的拉回。在$4$-上链的情况下，这回答了 Johnson-Freyd、Ostrik 和 Yu 在 arXiv:2601.09060中的一个问题。
英文摘要
We discuss the classification of twisted Deligne products of two semisimple tensor categories $\mathcal C,\mathcal D$, i.e., categorifications of the tensor product of their Grothendieck rings in which the factors are categorified by $\mathcal C$ and $\mathcal D$. In particular, we show that if both factors have no non-trivial gradings, or if one factor has neither non-trivial gradings nor tensor structures on the identity functor, then the only twisted Deligne product is the ordinary one. Using the work arXiv:2405.10207 by Müller, Peña Pollastri and Plavnik, this gives, in principle, a group-theoretical classification of twisted Deligne products and, more generally, exact factorizations of arbitrary fusion categories. In the Appendix we introduce the notion of categorical $n$-cocycles for $n=2,3,4$ and show that they are all pullbacks of group $n$-cocycles from the universal grading group of the underlying based ring. In the case of $4$-cocycles, this answers a question of Johnson-Freyd, Ostrik and Yu from arXiv:2601.09060.
