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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 108 (9 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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Holonomy and parallel transport for Abelian gerbes
, 2001
"... In this paper we establish a onetoone correspondence between S 1gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higherorder analogue of the familiar equivalence ..."
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Cited by 62 (8 self)
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In this paper we establish a onetoone correspondence between S 1gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higherorder analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with group S 1 on a simply connected manifold M is a group morphism from the thin second homotopy group to S 1, satisfying a smoothness condition, where a homotopy between maps from [0,1] 2 to M is thin when its derivative is of rank ≤ 2. For the nonsimply connected case, holonomy is replaced by a parallel transport functor between two monoidal Lie groupoids. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case. Our approach to abelian gerbes with connections holds out prospects for generalizing to the nonabelian case via the theory of double Lie groupoids.
Topological quantum field theories from compact Lie groups. arXiv:0905.0731
, 2009
"... Let G be a compact Lie group and BG a classifying space for G. Then a class in H n 1 BG; Z leads to an ndimensional topological quantum field theory (TQFT), at least for n�1,2,3. The theory for n�1is trivial, but we include it for completeness. The theory for n�2has some infinities if G is not a fi ..."
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Cited by 41 (2 self)
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Let G be a compact Lie group and BG a classifying space for G. Then a class in H n 1 BG; Z leads to an ndimensional topological quantum field theory (TQFT), at least for n�1,2,3. The theory for n�1is trivial, but we include it for completeness. The theory for n�2has some infinities if G is not a finite group; it is a topological limit of 2dimensional YangMills theory. The most direct analog for n�3 is an L 2 version of the topological quantum field theory based on the classical ChernSimons invariant, which is only partially defined. The TQFT constructed by Witten and ReshetikhinTuraev which goes by the name ‘ChernSimons theory ’ (sometimes ‘holomorphic ChernSimons theory ’ to distinguish it from the L 2 theory) is completely finite. The theories we construct here are extended, or multitiered, TQFTs which go all the way down to points. For the n�3 ChernSimons theory, which we term a ‘0123 theory ’ to emphasize the extension down to points, we only treat the cases where G is finite or G is a torus, the latter being one of the main novelties in this paper. In other words, for toral theories we provide an answer to the longstanding question: What does ChernSimons theory attach to a point? The answer is a bit subtle as ChernSimons is an anomalous field theory of oriented manifolds. 1 This framing anomaly was already flagged in Witten’s seminal paper [Wi]. Here we interpret the anomaly as an invertible
HIGHER FROBENIUSSCHUR INDICATORS FOR PIVOTAL CATEGORIES
, 2005
"... Abstract. We define higher FrobeniusSchur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a klinear semisimple rigid ..."
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Cited by 33 (11 self)
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Abstract. We define higher FrobeniusSchur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a klinear semisimple rigid monoidal category, which we call the FrobeniusSchur endomorphisms. For a klinear semisimple pivotal monoidal category — where both notions are defined —, the FrobeniusSchur indicators can be computed as traces of the FrobeniusSchur endomorphisms.
On Galois extensions of braided tensor categories, mapping class group representations and simple current extensions
 In preparation
"... We show that the author’s notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed Gcategories, recently introduced for the purposes of 3manifold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G non ..."
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Cited by 23 (6 self)
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We show that the author’s notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed Gcategories, recently introduced for the purposes of 3manifold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G nontrivial objects of grade g exist in C ⋊ S. 1