2026/07/21 by Pavel Etingof, Dmitri Nikshych, Victor Ostrik
#math.CT #math.QA #math.RT
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.