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An Australian conspectus of higher categories

, 2004
"... Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional wo ..."
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Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences
Braided doubles and rational Cherednik algebras
 Adv. Math
"... Abstract. We introduce and study a large class of algebras with triangular decomposition which we call braided doubles. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of quasi ..."
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Abstract. We introduce and study a large class of algebras with triangular decomposition which we call braided doubles. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of quasiYetterDrinfeld (QYD) modules over Hopf algebras which turn out to be a generalisation of the ordinary YetterDrinfeld modules. To each braiding (a solution to the braid equation) we associate a QYDmodule and the corresponding braided Heisenberg double — this is a quantum deformation of the Weyl algebra where the role of polynomial algebras is played by NicholsWoronowicz algebras. Our main result is that any rational Cherednik algebra canonically embeds into the braided Heisenberg double attached to the corresponding complex reflection group.
N.: Constructing the extended Haagerup planar algebra. http://arxiv.org/abs/0909.4099v1 [math.OA
, 2009
"... Abstract We construct a subfactor planar algebra, and as a corollary a subfactor, with the ‘extended Haagerup ’ principal graph pair. This is the last open case from Haagerup’s 1993 list of potential principal graphs of subfactors with index in the range (4, 3 + √ 3). We prove that the subfactor pla ..."
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Abstract We construct a subfactor planar algebra, and as a corollary a subfactor, with the ‘extended Haagerup ’ principal graph pair. This is the last open case from Haagerup’s 1993 list of potential principal graphs of subfactors with index in the range (4, 3 + √ 3). We prove that the subfactor planar algebra with these principal graphs is unique. We give a skein theoretic description, and a description as a subalgebra generated by a certain element in the graph planar algebra of its principal graph. We give an explicit algorithm for evaluating closed diagrams using the skein theoretic description. This evaluation algorithm is unusual because intermediate steps may increase the number of generators in a diagram. AMS Classification 46L37; 18D10
Graded extensions of monoidal categories
 J. ALGEBRA
, 2001
"... The longknown results of SchreierEilenbergMac Lane on group extensions are raised to a categorical level, for the classification and construction of the manifold of all graded monoidal categories, the type being given group � with 1component a given monoidal category. Explicit application is mad ..."
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The longknown results of SchreierEilenbergMac Lane on group extensions are raised to a categorical level, for the classification and construction of the manifold of all graded monoidal categories, the type being given group � with 1component a given monoidal category. Explicit application is made to the classification of strongly graded bialgebras over commutative rings.
Dynamical reflection equation
"... We construct a dynamical reflection equation algebra, ˜ K, via a dynamical twist of the ordinary reflection equation algebra. A dynamical version of the reflection equation is deduced as a corollary. We show that ˜ K is a right comodule algebra over a dynamical analog of the FaddeevReshetikhinTak ..."
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We construct a dynamical reflection equation algebra, ˜ K, via a dynamical twist of the ordinary reflection equation algebra. A dynamical version of the reflection equation is deduced as a corollary. We show that ˜ K is a right comodule algebra over a dynamical analog of the FaddeevReshetikhinTakhtajan algebra equipped with a structure of right bialgebroid. We introduce dynamical trace and use it for constructing central elements of ˜ K.
ON ENDOMORPHISM ALGEBRAS OF SEPARABLE MONOIDAL FUNCTORS
"... Abstract. We show that the (co)endomorphism algebra of a sufficiently separable “fibre ” functor into Vectk, for k a field of characteristic 0, has the structure of what we call a “unital ” von Neumann core in Vectk. For Vectk, this particular notion of algebra is weaker than that of a Hopf algebra, ..."
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Abstract. We show that the (co)endomorphism algebra of a sufficiently separable “fibre ” functor into Vectk, for k a field of characteristic 0, has the structure of what we call a “unital ” von Neumann core in Vectk. For Vectk, this particular notion of algebra is weaker than that of a Hopf algebra, although the corresponding concept in Set is again that of a group. 1.
TANNAKAKREIN DUALITY FOR COMPACT GROUPOIDS II, FOURIER TRANSFORM
, 2003
"... Abstract. In a series of papers, we have shown that from the representationtheory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the TannakaKrein duality for compact groups. In this part we study the Fourier and FourierPlancherel transforms and prove the Plan ..."
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Abstract. In a series of papers, we have shown that from the representationtheory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the TannakaKrein duality for compact groups. In this part we study the Fourier and FourierPlancherel transforms and prove the Plancherel theorem for compact groupoids. We also study the central functions in the algebra of square integrable functions on the isotropy groups. 1. introduction In a series of papers, we have generalized the TannakaKrein duality to compact groupoids. In [A1] we studied the representationtheory of compact groupoids. In particular, we showed that irreducible representationshave finite dimensional fibres. We also proved the Schur’s lemma, GelfandRaikov theorem and PeterWeyl theorem for compact groupoids. In this part we study the Fourier and FourierPlancherel transforms on compact groupoids. In section two we develop the theory of Fourier transforms on the Banach algebra bundle L 1 (G) of a compact groupoid G. As in the group case, a parallel theory of FourierPlancherel transform on the Hilbert space bundle L 2 (G) is constructed. This provides a surjective isometric linear isomorphism from L 2 (G) to L 2 ( ˆ G), in an appropriate sense. Also the relation between ˆ G and the conjugacy groupoid G G is studied. The results of this section are effectively used in [A2] to show that the natural homomorphism from G to its Tannaka groupoid T (G) is surjective. Section three considers the inverse Fourier and FourierPlancherel transforms. In this section we prove Plancherel theorem for compact groupoids. In section four we study the class functions and central elements in the algebras of functions on fibres of G and prove a diagonal version of the Plancherel theorem. All over this paper we assume that G is compact and the Haar system on G is normalized.
Balanced Coalgebroids
, 2000
"... A balanced coalgebroid is a V op category with extra structure ensuring that its category of representations is a balanced monoidal category. We show, in a sense to be made precise, that a balanced structure on a coalgebroid may be reconstructed from the corresponding structure on its category of ..."
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A balanced coalgebroid is a V op category with extra structure ensuring that its category of representations is a balanced monoidal category. We show, in a sense to be made precise, that a balanced structure on a coalgebroid may be reconstructed from the corresponding structure on its category of representations. This includes the reconstruction of dual quasibialgebras, quasitriangular dual quasibialgebras, and balanced quasitriangular dual quasibialgebras; the latter of which is a quantum group when equipped with a compatible antipode. As an application we construct a balanced coalgebroid whose category of representations is equivalent to the symmetric monoidal category of chain complexes. The appendix provides the definitions of a braided monoidal bicategory and sylleptic monoidal bicategory.