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Higher dimensional algebra V: 2groups
 Theory Appl. Categ
"... A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to tw ..."
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Cited by 25 (2 self)
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A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call ‘weak ’ and ‘coherent ’ 2groups. A weak 2group is a weak monoidal category in which every morphism has an inverse and every object x has a ‘weak inverse’: an object y such that x ⊗ y ∼ = 1 ∼ = y ⊗ x. A coherent 2group is a weak 2group in which every object x is equipped with a specified weak inverse ¯x and isomorphisms ix: 1 → x ⊗ ¯x, ex: ¯x ⊗ x → 1 forming an adjunction. We describe 2categories of weak and coherent 2groups and an ‘improvement ’ 2functor that turns weak 2groups into coherent ones, and prove that this 2functor is a 2equivalence of 2categories. We internalize the concept of coherent 2group, which gives a quick way to define Lie 2groups. We give a tour of examples, including the ‘fundamental 2group ’ of a space and various Lie 2groups. We also explain how coherent 2groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simplyconnected compact simple Lie group G a family of 2groups G � ( � ∈ Z) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2groups are built using Chern–Simons theory, and are closely related to the Lie 2algebras g � ( � ∈ R) described in a companion paper. 1 1
Spherical categories
 Adv. Math
, 1999
"... Abstract. This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras. We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a ..."
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Cited by 25 (2 self)
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Abstract. This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras. We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a monoidal category with duals following [MacLane 1963]. In the second section we give the definition of a spherical category, and construct a natural quotient which is also spherical. In the
Interacting quantum observables
 of Lecture Notes in Computer Science
, 2008
"... Abstract. We formalise the constructive content of an essential feature of quantum mechanics: the interaction of complementary quantum observables, and information flow mediated by them. Using a general categorical formulation, we show that pairs of mutually unbiased quantum account on the quantum d ..."
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Cited by 23 (13 self)
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Abstract. We formalise the constructive content of an essential feature of quantum mechanics: the interaction of complementary quantum observables, and information flow mediated by them. Using a general categorical formulation, we show that pairs of mutually unbiased quantum account on the quantum data encoded in complex phases, and prove a normal form theorem for it. Together these enable us to describe all observables of finite dimensional Hilbert space quantum mechanics. The resulting equations suffice to perform computations with elementary quantum gates, translate between distinct quantum computational models, establish the equivalence of entangled quantum states, and simulate quantum algorithms such as the quantum Fourier transform. All these computations moreover happen within an intuitive diagrammatic calculus. 1
Duality and defects in rational conformal field theory
, 2006
"... We study topological defect lines in twodimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We sh ..."
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Cited by 20 (7 self)
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We study topological defect lines in twodimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We show how the resulting onedimensional phase boundaries can be used to extract symmetries and orderdisorder dualities of the CFT. The case of central charge c = 4/5, for which there are two inequivalent world sheet phases corresponding to the tetracritical Ising model and the critical threestates
Equational Reasoning With 2Dimensional Diagrams (preliminary Version)
 Term Rewriting, volume 909 of LNCS
, 1992
"... The significance of the 2dimensional calculus, which goes back to Penrose, has already been pointed out by Joyal and Street in [JoS91]. Independently, Burroni has introduced a general notion of ndimensional presentation in [Bur91] and he has shown that the equational logic of terms is a special ca ..."
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Cited by 20 (1 self)
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The significance of the 2dimensional calculus, which goes back to Penrose, has already been pointed out by Joyal and Street in [JoS91]. Independently, Burroni has introduced a general notion of ndimensional presentation in [Bur91] and he has shown that the equational logic of terms is a special case of 2dimensional calculus. Here, we propose a combinatorial definition of 2dimensional diagrams and a simple method for proving that certain monoidal categories are finitely 2presentable. We illustrate the translation of terms into diagrams and we explain the change from groups to quantum groups in a purely syntactical way. This paper should serve as a reference for our future work on symbolic computation, including a theory of 2dimensional rewriting and the design of software for interactive diagrammatic reasoning. New address: CNRS  Laboratoire de math'ematiques discr`etes, 163 avenue de Luminy  Case 930, 13288 Marseille Cedex 9, France. Email: lafont@lmd.univmrs.fr 2 1 FROM T...
Frobenius monads and pseudomonoids
 2CATEGORIES COMPANION 73
, 2004
"... Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalenc ..."
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Cited by 19 (4 self)
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Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to "strongly separable " Frobenius algebras and "weak monoidal Morita equivalence". Wreath products of Frobenius algebras are discussed.
On Matrix Quantum Groups Of Type A_n
, 1997
"... To a Hecke symmetry R there associate a matrix bialgebra ER and a matrix Hopf algebra HR , which are called function rings on the matrix quantum semigroup and matrix quantum groups associated to R. We show that for an even Hecke symmetry, the rational representations of the corresponding quantum gr ..."
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Cited by 19 (4 self)
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To a Hecke symmetry R there associate a matrix bialgebra ER and a matrix Hopf algebra HR , which are called function rings on the matrix quantum semigroup and matrix quantum groups associated to R. We show that for an even Hecke symmetry, the rational representations of the corresponding quantum group are absolutely reducible and compute the integral on the function ring of the quantum group, i.e, on HR . Further, we show that the fusion coefficients of simple representations depend only on the rank of the symmetry, and give the explicit formula for the rank or 8dim of HRcomodules. In the general case, we show that the quantum semigroup is "Zariski" dense in the quantum group. This enables us to study the semisimplicity of the associated quantum group in some case. 1.
Integrals for braided Hopf algebras
 J. Pure Appl. Algebra
, 2000
"... Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object IntH is invertible. The fully braided ve ..."
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Cited by 17 (3 self)
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Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object IntH is invertible. The fully braided version of Radford’s formula for the fourth power of the antipode is obtained. Connections of integration with crossproduct and transmutation are studied. 1991 Mathematics Subject Classification. Primary 16W30, 18D15, 17B37; Secondary 18D35.