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A construction of an A∞category
, 2002
"... We construct an A∞category D(CB) from a given A∞category C and its full subcategory B. The construction resembles a particular case of Drinfeld’s quotient of differential graded categories [Dri02]. We use D(CB) to construct an A∞functor of Kinjective resolutions of a complex. The conventional ..."
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We construct an A∞category D(CB) from a given A∞category C and its full subcategory B. The construction resembles a particular case of Drinfeld’s quotient of differential graded categories [Dri02]. We use D(CB) to construct an A∞functor of Kinjective resolutions of a complex. The conventional derived category is obtained as the 0th cohomology of the quotient of differential graded category of complexes over acyclic complexes.
Bottom tangles and universal invariants
, 2006
"... A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite ..."
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A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of B, which provides an especially convenient way to generate all the bottom tangles, and hence all the framed, oriented links, via closure. We also define a kind of “braided Hopf algebra action ” on the set of bottom tangles. Using the universal invariant of bottom tangles associated to each ribbon Hopf algebra H, we define a braided functor J from B to the category ModH of left H–modules. The functor J, together with the set of generators of B, provides an algebraic method to study the range of quantum invariants of links. The braided Hopf algebra action on bottom tangles is mapped by J to the standard braided Hopf algebra structure for H in ModH. Several notions in knot theory, such as genus, unknotting number, ribbon knots, boundary links, local moves, etc are given algebraic interpretations in the setting involving the category B. The functor J provides a convenient way to study the relationships between these notions and quantum invariants.
Towards an Algebraic Characterization of 3dimensional Cobordisms. ArXiv: math.GT/0106253
"... (To appear in Contemp. Math.) Abstract: The goal of this paper is to find a close to isomorphic presentation of 3manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of Hopf algebras and 3dim ..."
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(To appear in Contemp. Math.) Abstract: The goal of this paper is to find a close to isomorphic presentation of 3manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of Hopf algebras and 3dimensional topology. The first is the category Cob of connected surfaces with one boundary component and 3dimensional relative cobordisms, the second is a
Double categories and quantum groupoids
, 2003
"... Abstract. We give the construction of a class of weak Hopf algebras (or quantum groupoids) associated to a matched pair of groupoids and certain cocycle data. This generalizes a now wellknown construction for Hopf algebras, first studied by G. I. Kac in the sixties. Our approach is based on the not ..."
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Abstract. We give the construction of a class of weak Hopf algebras (or quantum groupoids) associated to a matched pair of groupoids and certain cocycle data. This generalizes a now wellknown construction for Hopf algebras, first studied by G. I. Kac in the sixties. Our approach is based on the notion of double groupoids, as introduced by Ehresmann.
Model structures on the category of small double categories, Algebraic and Geometric Topology 8
, 2008
"... Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak ..."
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Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves
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.
TOWARD LOGARITHMIC EXTENSIONS OF ̂ sℓ(2)k CONFORMAL FIELD MODELS
, 2007
"... For positive integer p=k+2, we construct a logarithmic extension of the ̂sℓ(2)k conformal field theory of integrable representations by taking the kernel of two fermionic screening operators in a threeboson realization of ̂ sℓ(2)k. The currents W − (z) and W + (z) of a Walgebra acting in the ker ..."
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Cited by 6 (2 self)
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For positive integer p=k+2, we construct a logarithmic extension of the ̂sℓ(2)k conformal field theory of integrable representations by taking the kernel of two fermionic screening operators in a threeboson realization of ̂ sℓ(2)k. The currents W − (z) and W + (z) of a Walgebra acting in the kernel are determined by a highestweight state of dimension 4p−2 and charge 2p−1, and a (θ =1)twisted highestweight state of the same dimension 4p−2 and charge −2p+1. We construct 2p Walgebra representations, evaluate their characters, and show that together with the p−1 integrable representation characters they generate a modular group representation whose structure is described as a deformation of the (9p−3)dimensional representation Rp+1 ⊕ C 2 ⊗Rp+1 ⊕Rp−1 ⊕ C 2 ⊗ Rp−1 ⊕ C 3 ⊗ Rp−1, where Rp−1 is the SL(2,Z) representation on integrable representation characters and Rp+1 is a (p+1)dimensional SL(2,Z) representation known from the logarithmic (p,1) model. The dimension 9p−3 is conjecturally the dimension of the space of torus amplitudes, and the C n with n = 2 and 3 suggest the Jordan cell sizes in indecomposable Walgebra modules. Under Hamiltonian reduction, the Walgebra currents map into the currents of the triplet Walgebra of the logarithmic (p,1) model.
Tensor products of finitely cocomplete and abelian categories
 Journal of Algebra
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Extended TQFT’s and Quantum Gravity
, 2007
"... Abstract. This paper gives a definition of an extended topological quantum field theory (TQFT) as a weak 2functor Z: nCob2→2Vect, by analogy with the description of a TQFT as a functor Z: nCob→Vect. We also show how to obtain such a theory from any finite group G. This theory is related to a topolo ..."
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Abstract. This paper gives a definition of an extended topological quantum field theory (TQFT) as a weak 2functor Z: nCob2→2Vect, by analogy with the description of a TQFT as a functor Z: nCob→Vect. We also show how to obtain such a theory from any finite group G. This theory is related to a topological gauge theory, the DijkgraafWitten model. To give this definition rigorously, we first define a bicategory of cobordisms between cobordisms. We also give some explicit description of a highercategorical version of Vect, denoted 2Vect, a bicategory of 2vector spaces. Along the way, we prove several results showing how to construct 2vector spaces of Vectvalued presheaves on certain kinds of groupoids. In particular, we use the case when these are groupoids whose objects are connections, and whose morphisms are gauge transformations, on the manifolds on which the extended TQFT is to be defined. On cobordisms between these manifolds, we show how a construction of “pullback and pushforward ” of presheaves gives both the morphisms and 2morphisms in 2Vect for the extended TQFT, and that these