From subfactors to categories and topology I:: Frobenius algebras in and Morita equivalence of tensor categories

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 Lambda = F-Vect, where F is a field. An object X is an element of A with two-sided dual (X) over bar 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 = {U,B} such that End(E)(U)(circle times)similar or equal toA and such that there are J,(J) over bar :B reversible arrow U producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A approximate to B, and establish a correspondence between Frobenius algebras in A and tensor categories B approximate to A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A approximate to B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles ('centers') and (if A,B are semisimple spherical or (*)-categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3-manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite-dimensional semisimple and cosemisimple Hopf algebras, for which we prove H - mod approximate to (H) over cap - mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper (J. Pure Appl. Algebra 180 (2003) 159-219). (C) 2003 Elsevier Science B.V. All rights reserved.