Discrete torsion in gauging non-invertible symmetries

In this paper we discuss generalizations of discrete torsion to noninvertible symmetries in 2d QFTs. One point of this paper is to explain that there are two complementary generalizations. Both generalizations are counted by $H^2(G,U(1))$ when one specializes to ordinary finite groups $G$. However, the counting is different for more general fusion categories. Furthermore, only one generalizes the picture of discrete torsion as differences in choices of gauge actions on B fields. Explaining this in detail, how one of the generalizations of discrete torsion to noninvertible cases encodes actions on B fields, is the other point of this paper. In particular, this generalizes old results in ordinary orbifolds that discrete torsion is a choice of group action on the B field. We also explain how this same generalization of discrete torsion gives rise to physically-sensible twists on gaugeable algebras and fiber functors.