Twisted Deligne products of semisimple tensor categories

Abstract

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.

Disclosure

“-cocycles on 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. Acknowledgements. In this paper, especially in the Appendix, we collaborated with ChatGPT 5.5 Pro. The work of P.E. was supported by the NSF grants DMS-2001318 and DMS-2502467. The work of D.N. was supported by the National Science Foundation under Grant No. DMS-2302267. 3”

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2026-08-23T19:01:21.693938Z

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