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 ∈ 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 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 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 ≈ 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

In this paper we establish a one-to-one correspondence between S 1-gerbes 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 higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with presently working as a postdoc at the University of Nottingham, UK 1 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 non-simply 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 non-abelian case via the theory of double Lie groupoids. 1

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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 G-categories, recently introduced for the purposes of 3-manifold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G non-trivial objects of grade g exist in C ⋊ S. 1

We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth compact oriented 2-manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a two-dimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an open-closed TQFT with a finite set of D-branes using the example of the groupoid algebra of a finite groupoid.

The aim of the paper is twofold. First, we show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category G−LocA of twisted representations. This category is a braided crossed G-category in the sense of Turaev [60]. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of G-manifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT ❀ modular category ❀ 3-manifold invariant. Secondly, we study the relation between G−LocA and the braided (in the usual sense) representation category Rep AG of the orbifold theory AG. We prove the equivalence Rep AG ≃ (G−LocA) G, which is a rigorous implementation of the insight that one needs to take the twisted representations of A into account in order to determine Rep AG. In the opposite direction we have G−LocA ≃ Rep AG ⋊ S, where S ⊂ Rep AG is the full

We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the Tannaka-Krein reconstruction problem. We show that every concrete semisimple tensor ∗-category with conjugates is equivalent to the category of finite dimensional non-degenerate ∗-representations of a discrete algebraic quantum group. Working in the self-dual framework of algebraic quantum groups, we then relate this to earlier results of S. L. Woronowicz and S. Yamagami. We establish the relation between braidings and R-matrices in this context. Our approach emphasizes the role of the natural transformations of the embedding functor. Thanks to the semisimplicity of our categories and the emphasis on representations rather than corepresentations, our proof is more direct and conceptual than previous reconstructions. As a special case, we reprove the classical Tannaka-Krein result for compact groups. It is only here that analytic aspects enter, otherwise we proceed in a purely algebraic way. In particular, the existence of a Haar functional is reduced to a well known general result concerning discrete multiplier Hopf ∗-algebras. 1

Abstract. We define higher Frobenius-Schur 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 k-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a k-linear semisimple pivotal monoidal category — where both notions are defined —, the Frobenius-Schur indicators can be computed as traces of the Frobenius-Schur endomorphisms.