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From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories
 J. Pure Appl. Alg
, 2003
"... 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 f ..."
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Cited by 52 (6 self)
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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 = FVect, where F is a field. An object X ∈ A with twosided 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 ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A 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 3manifolds 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 ≈ ˆH − 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 math.CT/0111205. 1
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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YETTERDRINFELD MODULES FOR TURAEV CROSSED STRUCTURES
, 2002
"... Abstract. We provide an analog of the JoyalStreet center construction and of the KasselTuraev categorical quantum double in the context of the crossed categories introduced by Turaev. Then, we focus or attention to the case of categories of representation. In particular, we introduce the notion of ..."
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Abstract. We provide an analog of the JoyalStreet center construction and of the KasselTuraev categorical quantum double in the context of the crossed categories introduced by Turaev. Then, we focus or attention to the case of categories of representation. In particular, we introduce the notion of a YetterDrinfeld module over a crossed group coalgebra H and we prove that both the
Tannaka Reconstruction for Crossed Hopf Group Algebras
, 2008
"... We provide an analog of Tannaka Theory for Hopf algebras in the context of crossed Hopf group coalgebras introduced by Turaev. Following Street and our previous work on the quantum double of crossed structures, we give a construction, via Tannaka Theory, of the quantum double of crossed ..."
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We provide an analog of Tannaka Theory for Hopf algebras in the context of crossed Hopf group coalgebras introduced by Turaev. Following Street and our previous work on the quantum double of crossed structures, we give a construction, via Tannaka Theory, of the quantum double of crossed
3. The outer dual and the inner dual of a Tcoalgebra 6
, 2002
"... Abstract. We provide an analog of the Drinfeld quantum double construction in the context of crossed Hopf group coalgebras introduced by Turaev. We prove that, provided the base group is finite, the double of a semisimple crossed ..."
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Abstract. We provide an analog of the Drinfeld quantum double construction in the context of crossed Hopf group coalgebras introduced by Turaev. We prove that, provided the base group is finite, the double of a semisimple crossed
Tensor categories: A selective guided tour ∗
, 2008
"... These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively nontechnical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something int ..."
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These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively nontechnical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something interesting in them. Once the basic definitions are given, the focus is mainly on klinear categories with finite dimensional homspaces. Connections with quantum groups and low dimensional topology are pointed out, but these notes have no pretension to cover the latter subjects at any depth. Essentially, these notes should be considered as annotations to the extensive bibliography. We also recommend the recent review [33], which covers less ground in a deeper way. 1 Tensor categories 1.1 Strict tensor categories