On the relative dual of an S1S1-gerbe over an orbifold

We construct a new effective orbifold Yˆ with an S1S1-gerbe c   to study an S1S1-gerbe tt on a G  -gerbe YY over an orbifold BB. We view the former as the relative dual, relative to BB, of the latter. We show that the two pairs (Y,t)(Y,t) and (Yˆ,c) have isomorphic categories of sheaves, and also the associated twisted groupoid algebras are Morita equivalent. As a corollary, the K-theory and cohomology groups of (Y,t)(Y,t) and (Yˆ,c) are isomorphic.