  A finite tensor category is called pointed if all its simple objects are
invertible. We find necessary and sufficient conditions for two pointed
semisimple categories to be dual to each other with respect to a module
category. Whenever the dual of a pointed semisimple category with respect to a
module category is pointed, we give explicit formulas for the Grothendieck ring
and for the associator of the dual. This leads to the definition of categorical
Morita equivalence on the set of all finite groups and on the set of all pairs
(G, w), where G is a finite group and w is a 3-cocycle on G. A
group-theoretical and cohomological interpretation of this relation is given. A
series of concrete examples of pairs of groups that are categorically Morita
equivalent but have non-isomorphic Grothendieck rings are given. In particular,
the representation categories of the Drinfeld doubles of the groups in each
example are equivalent as braided tensor categories and hence these groups
define the same modular data.

Quantum Algebra;Category Theory