Correspondences of ribbon categories

Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally, we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.