Categorical Morita equivalence of pointed fusion categories, and subgroups
Résumé
Work of Naidu answers completely the question as to when two pointed fusion categories are categorically Morita equivalent. The constructions that relate the groups and three-cocycles involved can be rather complicated to handle in concrete examples, however. We discuss simple necessary criteria for Morita equivalence based on the subgroups of the two groups and the restricted cocycles. While neither the groups, nor the subgroup lattices, nor the cocycles need to coincide, we find that for two morita equivalent fusion categories C, C', every fusion subcategory of C has a Morita equivalent copy related to C' , albeit not as a subcategory but a subquotient. If the subcategory corresponds to a Sylow subgroup, it is in fact Morita equivalent to a subcategory in C'.