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

AbstractWe 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∈A with two-sided dual X̄ gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE={U,B} such that EndE(U)≃⊗A and such that there are J,J̄:B⇌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≈B, and establish a correspondence between Frobenius algebras in A and tensor categories B≈A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A≈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≈Ĥ−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)