From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence of tensor categories

Abstract

We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A=F-Vect, where F is a field. An object X in A with two-sided dual X^ gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with Obj(E)={X,Y} such that End_E(X,X) is equivalent to A and such that there are J: Y->X, J^: X->Y producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to E. We define weak monoidal Morita equivalence (wMe) of tensor categories and establish a correspondence between Frobenius algebras in A and tensor categories B wMe A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence of A and B implies (for A,B semisimple and spherical or *-categories) that A and B have the same dimension, braided equivalent `center' (quantum double) and define the same state sum invariants of closed oriented 3-manifolds as defined by Barrett and Westbury. If H is a finite dimensional semisimple and cosemisimple Hopf algebra then H-mod and H^-mod are wMe. The present formalism permits a fairly complete analysis of the quantum double of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205.