Discrete torsion in gauging non-invertible symmetries

In this paper we explain that there exist two complementary generalizations of discrete torsion for non-invertible symmetries in 2d QFT's. Both characterizations are counted by H² (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. We also explain how this same generalization of discrete torsion gives rise to physically-sensible twists on gaugeable algebras and fiber functors.