Exchange relations and Frobenius subalgebras

Bisch and Jones established a bijection between the intermediate subfactors of an irreducible subfactor and certain idempotents satisfying exchange relations. In this paper, we generalize this result to abelian monoidal categories through Frobenius subalgebras. As an application, we show that certain morphisms on a C*-correspondence arise from intermediate C*-subalgebras if and only if they are averaging operators.