Bialgebras in Rel
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From Non-Semisimple Hopf Algebras to Correlation Functions For
ZMP-HH/13-2 Hamburger Beitr¨age zur Mathematik Nr. 468 February 2013 FROM NON-SEMISIMPLE HOPF ALGEBRAS TO CORRELATION FUNCTIONS FOR LOGARITHMIC CFT J¨urgen Fuchs a, Christoph Schweigert b, Carl Stigner c a Teoretisk fysik, Karlstads Universitet Universitetsgatan 21, S–65188 Karlstad b Organisationseinheit Mathematik, Universit¨at Hamburg Bereich Algebra und Zahlentheorie Bundesstraße 55, D–20146 Hamburg c Camatec Industriteknik AB Box 5134, S–65005 Karlstad Abstract We use factorizable finite tensor categories, and specifically the representation categories of factorizable ribbon Hopf algebras H, as a laboratory for exploring bulk correlation functions in arXiv:1302.4683v2 [hep-th] 22 Aug 2013 local logarithmic conformal field theories. For any ribbon Hopf algebra automorphism ω of H we present a candidate for the space of bulk fields and endow it with a natural structure of a commutative symmetric Frobenius algebra. We derive an expression for the corresponding bulk partition functions as bilinear combinations of irreducible characters; as a crucial ingredient this involves the Cartan matrix of the category. We also show how for any candidate bulk state space of the type we consider, correlation functions of bulk fields for closed oriented world sheets of any genus can be constructed that are invariant under the natural action of the relevant mapping class group. 1 1 Introduction Understanding a quantum field theory includes in particular having a full grasp of its correla- tors on various space-time manifolds, including the relation between correlation functions on different space-times. This ambitious goal has been reached for different types of theories to a variable extent. -
On the Completeness of the Traced Monoidal Category Axioms in (Rel,+)∗
Acta Cybernetica 23 (2017) 327–347. On the Completeness of the Traced Monoidal Category Axioms in (Rel,+)∗ Mikl´os Barthaa To the memory of my friend and former colleague Zolt´an Esik´ Abstract It is shown that the traced monoidal category of finite sets and relations with coproduct as tensor is complete for the extension of the traced sym- metric monoidal axioms by two simple axioms, which capture the additive nature of trace in this category. The result is derived from a theorem saying that already the structure of finite partial injections as a traced monoidal category is complete for the given axioms. In practical terms this means that if two biaccessible flowchart schemes are not isomorphic, then there exists an interpretation of the schemes by partial injections which distinguishes them. Keywords: monoidal categories, trace, iteration, feedback, identities in cat- egories, equational completeness 1 Introduction It was in March, 2012 that Zolt´an visited me at the Memorial University in St. John’s, Newfoundland. I thought we would find out a research topic together, but he arrived with a specific problem in mind. He knew about the result found by Hasegawa, Hofmann, and Plotkin [12] a few years earlier on the completeness of the category of finite dimensional vector spaces for the traced monoidal category axioms, and wanted to see if there is a similar completeness statement true for the matrix iteration theory of finite sets and relations as a traced monoidal category. Unfortunately, during the short time we spent together, we could not find the solu- tion, and after he had left I started to work on some other problems. -
Unitary Modular Tensor Categories
Penneys Math 8800 Unitary modular tensor categories 5. Unitary modular tensor categories 5.1. Braided fusion categories. Let C be a tensor category. Definition 5.1.1. A braiding on C is a family β of isomorphisms 8 9 > a b > <> => = βa;b : b ⊗ a ! a ⊗ b > > :> b a ;> a;b2C which satisfy the following axioms: • (naturality) for all f 2 C(b ! c) and g 2 C(a ! d), a c a c f = (id ⊗f) ◦ β = β ◦ (f ⊗ id ) = a a;b a;c a f b a b a d b d b g = (g ⊗ id ) ◦ β = β ◦ (id ⊗g) = b a;b d;b b g b a b a • (monoidality) For all b; c 2 C, a b c a b c a b⊗c a⊗b c = and = b⊗c a c a⊗b b c a c a b where we have suppressed the associators. More formally, the following diagrams should commute: id ⊗β b ⊗ (c ⊗ a) b a;c b ⊗ (a ⊗ c) α (b ⊗ a) ⊗ c α βa;b⊗idc (5.1.2) β (b ⊗ c) ⊗ a a;b⊗c a ⊗ (b ⊗ c) α (a ⊗ b) ⊗ c β ⊗id −1 (c ⊗ a) ⊗ b a;c b (a ⊗ c) ⊗ b α a ⊗ (c ⊗ b) α−1 ida ⊗βb;c (5.1.3) β −1 c ⊗ (a ⊗ b) a⊗b;c (a ⊗ b) ⊗ c α a ⊗ (b ⊗ c) 1 When C is unitary, we typically require a braiding to be unitary as well. Remark 5.1.4. By [Gal14], every braiding on a unitary fusion category is automatically unitary. -
A Survey of Graphical Languages for Monoidal Categories
5 Traced categories 32 5.1 Righttracedcategories . 33 5.2 Planartracedcategories. 34 5.3 Sphericaltracedcategories . 35 A survey of graphical languages for monoidal 5.4 Spacialtracedcategories . 36 categories 5.5 Braidedtracedcategories . 36 5.6 Balancedtracedcategories . 39 5.7 Symmetrictracedcategories . 40 Peter Selinger 6 Products, coproducts, and biproducts 41 Dalhousie University 6.1 Products................................. 41 6.2 Coproducts ............................... 43 6.3 Biproducts................................ 44 Abstract 6.4 Traced product, coproduct, and biproduct categories . ........ 45 This article is intended as a reference guide to various notions of monoidal cat- 6.5 Uniformityandregulartrees . 47 egories and their associated string diagrams. It is hoped that this will be useful not 6.6 Cartesiancenter............................. 48 just to mathematicians, but also to physicists, computer scientists, and others who use diagrammatic reasoning. We have opted for a somewhat informal treatment of 7 Dagger categories 48 topological notions, and have omitted most proofs. Nevertheless, the exposition 7.1 Daggermonoidalcategories . 50 is sufficiently detailed to make it clear what is presently known, and to serve as 7.2 Otherprogressivedaggermonoidalnotions . ... 50 a starting place for more in-depth study. Where possible, we provide pointers to 7.3 Daggerpivotalcategories. 51 more rigorous treatments in the literature. Where we include results that have only 7.4 Otherdaggerpivotalnotions . 54 been proved in special cases, we indicate this in the form of caveats. 7.5 Daggertracedcategories . 55 7.6 Daggerbiproducts............................ 56 Contents 8 Bicategories 57 1 Introduction 2 9 Beyond a single tensor product 58 2 Categories 5 10 Summary 59 3 Monoidal categories 8 3.1 (Planar)monoidalcategories . 9 1 Introduction 3.2 Spacialmonoidalcategories . -
Fusion Categories and Modular Tensor Categories Seminar
Fusion categories and modular tensor categories seminar Aaron Mazel-Gee Contents 1 Introduction { Part I (Ehud \Udi" Meir) 1 2 Introduction { Part II (Orit Davidovich) 4 2.1 Recap . .4 2.2 Commutativity . .4 2.3 Graphical calculus . .5 2.4 Modularity . .6 3 Dimensions in fusion categories (Orit) 6 3.1 Dimensions . .6 3.2 Modularity . .8 3.3 Drinfeld centers . .8 4 Frobenius-Perron dimension and module categories (Udi) 8 4.1 Frobenius-Perron dimension . .8 4.2 Module categories and Morita equivalence . .9 5 On Morita equivalence for fusion categories (Udi) 10 6 Ocneanu rigidity (Orit) 12 6.1 Unitary fusion categories . 12 6.2 Algebro-linear data of fusion categories . 13 6.3 Davydov-Yetter cohomology . 14 7 Group theoretical fusion categories (Udi) 15 1 Introduction { Part I (Ehud \Udi" Meir) In order to explain what a fusion category is, we will begin with an example. Example 1. Suppose G is a finite group and k = k is a field with char k = 0. Consider the category C = Rep(G), the finite-dimensional k-vector spaces with a G-action. What do we know about this category? 1. By Maschke's theorem, we know that C is semisimple: that is, every object is a sum of simple (indecomposable) objects. 2. It is k-linear: HomC(X; Y ) is a k-vector space, and it has kernels and cokernels. 3. C is a monoidal category: if V; W 2 C, then we have V ⊗W 2 C (by the diagonal action: g(v ⊗w) = gv ⊗gw)). This determines a functor C ×C ! C, which will satisfy certain axioms which we'll explore later. -
Internal Reshetikhin-Turaev Topological Quantum Field Theories Mickaël Lallouche
Internal Reshetikhin-Turaev Topological Quantum Field Theories Mickaël Lallouche To cite this version: Mickaël Lallouche. Internal Reshetikhin-Turaev Topological Quantum Field Theories. General Topol- ogy [math.GN]. Université Montpellier, 2016. English. NNT : 2016MONTS015. tel-01681854v2 HAL Id: tel-01681854 https://tel.archives-ouvertes.fr/tel-01681854v2 Submitted on 12 Jan 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Délivré par l’ Université de Montpellier Préparée au sein de l’école doctorale Information Structures Systèmes et de l’unité de recherche Institut Montpelliérain Alexander Grothendieck Spécialité: Mathématiques et modélisation Présentée par Mickaël Lallouche Théories des champs quantiques topologiques internes de type Reshetikhin-Turaev Soutenue le 31 octobre 2016 devant le jury composé de M. Stéphane BASEILHAC Université de Montpellier Président du jury M. Christian BLANCHET UP7D / UMPC Rapporteur (absent) M. Alain BRUGUIERES Université de Montpellier Directeur de thèse M. Louis FUNAR Université Grenoble Alpes Examinateur M. Gwénaël MASSUYEAU Université de Strasbourg Examinateur M. Christoph SCHWEIGERT Universität Hamburg Rapporteur M. Alexis VIRELIZIER Université de Lille I Co-directeur de thèse 2 Remerciements À chaque fois que je me mets à écrire, c’est à 1 000 mains. -
Computing F-Matrices for Modular Tensor Categories
Computing F-Matrices for Modular Tensor Categories Galit Anikeeva Advisors: Daniel Bump, Guillermo Antonio Aboumrad August 29, 2020 Abstract In this paper, we find algorithms to compute the F-matrices for modular tensor categories, and in particular relevant anyon theories. Moreover, we implement this algorithm in SageMath. Contents 1 Introduction 2 2 Background 3 2.1 Modular Tensor Categories . .3 2.1.1 Fusion category . .3 2.1.2 Braided Monoidal Category . .5 2.1.3 Ribbon Category . .6 2.1.4 Monoidal Tensor Category . .8 2.1.5 Interpretation in TQC . .8 2.1.6 R-coefficients . .8 2.1.7 Ocneanu rigidity . .9 bc 2.2 Theories with Na 2 f0; 1g ................................9 2.3 Groebner basis . 10 3 Proposal 10 3.1 Eliminating leftover gauges . 11 3.2 Optimizations . 11 3.2.1 Creating the ideals . 11 3.2.2 Reducing and recomputing the ideal . 11 4 Future Work 12 5 Acknowledgements 12 A Some F matrices. 13 A.1 A1.............................................. 13 A.1.1 k =1 ........................................ 13 A.1.2 k =2 ........................................ 14 A.1.3 k =3 ........................................ 14 1 A.2 A2.............................................. 16 A.3 B2.............................................. 17 A.3.1 k =1 ........................................ 17 A.4 D3.............................................. 17 A.4.1 k =1 ........................................ 17 A.5 G2.............................................. 18 A.5.1 k =1 ........................................ 18 A.6 E8.............................................. 18 A.6.1 k =2 ........................................ 18 1 Introduction Topological quantum computing [1] is a proposal for a quantum computer that is theoretically distinct from (but computationally equivalent [2] to) the usual gate/circuit model of quantum computing. This proposal deals with topological states of matter. -
A Survey of Graphical Languages for Monoidal Categories
A survey of graphical languages for monoidal categories Peter Selinger Dalhousie University Abstract This article is intended as a reference guide to various notions of monoidal cat- egories and their associated string diagrams. It is hoped that this will be useful not just to mathematicians, but also to physicists, computer scientists, and others who use diagrammatic reasoning. We have opted for a somewhat informal treatment of topological notions, and have omitted most proofs. Nevertheless, the exposition is sufficiently detailed to make it clear what is presently known, and to serve as a starting place for more in-depth study. Where possible, we provide pointers to more rigorous treatments in the literature. Where we include results that have only been proved in special cases, we indicate this in the form of caveats. Contents 1 Introduction 2 2 Categories 5 3 Monoidal categories 8 3.1 (Planar)monoidalcategories . 9 3.2 Spacialmonoidalcategories . 14 3.3 Braidedmonoidalcategories . 14 3.4 Balancedmonoidalcategories . 16 arXiv:0908.3347v1 [math.CT] 23 Aug 2009 3.5 Symmetricmonoidalcategories . 17 4 Autonomous categories 18 4.1 (Planar)autonomouscategories. 18 4.2 (Planar)pivotalcategories . 22 4.3 Sphericalpivotalcategories. 25 4.4 Spacialpivotalcategories . 25 4.5 Braidedautonomouscategories. 26 4.6 Braidedpivotalcategories. 27 4.7 Tortilecategories ............................ 29 4.8 Compactclosedcategories . 30 1 5 Traced categories 32 5.1 Righttracedcategories . 33 5.2 Planartracedcategories. 34 5.3 Sphericaltracedcategories . 35 5.4 Spacialtracedcategories . 36 5.5 Braidedtracedcategories . 36 5.6 Balancedtracedcategories . 39 5.7 Symmetrictracedcategories . 40 6 Products, coproducts, and biproducts 41 6.1 Products................................. 41 6.2 Coproducts ............................... 43 6.3 Biproducts................................ 44 6.4 Traced product, coproduct, and biproduct categories . -
Some Aspects of Categories in Computer Science
This is a technical rep ort A mo died version of this rep ort will b e published in Handbook of Algebra Vol edited by M Hazewinkel Some Asp ects of Categories in Computer Science P J Scott Dept of Mathematics University of Ottawa Ottawa Ontario CANADA Sept Contents Intro duction Categories Lamb da Calculi and FormulasasTyp es Cartesian Closed Categories Simply Typ ed Lamb da Calculi FormulasasTyp es the fundamental equivalence Some Datatyp es Polymorphism Polymorphic lamb da calculi What is a Mo del of System F The Untyp ed World Mo dels and Denotational Semantics CMonoids and Categorical Combinators Church vs Curry Typing Logical Relations and Logical Permutations Logical Relations and Syntax Example ReductionFree Normalization Categorical Normal Forms P category theory and normalization algorithms Example PCF PCF Adequacy Parametricity Dinaturality Reynolds Parametricity -
Traced Monoidal Categories As Algebraic Structures in Prof
Traced monoidal categories as algebraic structures in Prof Nick Hu Jamie Vicary University of Oxford University of Cambridge [email protected] [email protected] Abstract We define a traced pseudomonoid as a pseudomonoid in a monoidal bicategory equipped with extra structure, giving a new characterisation of Cauchy complete traced monoidal cat- egories as algebraic structures in Prof, the monoidal bicategory of profunctors. This enables reasoning about the trace using the graphical calculus for monoidal bicategories, which we illustrate in detail. We apply our techniques to study traced ∗-autonomous categories, prov- ing a new equivalence result between the left ⊗-trace and the right -trace, and describing a new condition under which traced ∗-autonomous categories become` autonomous. 1 Introduction One way to interpret category theory is as a theory of systems and processes, whereby monoidal structure naturally lends itself to enable processes to be juxtaposed in parallel. Following this analogy, the presence of a trace structure embodies the notion of feedback: some output of a process is directly fed back in to one of its inputs. For instance, if we think of processes as programs, then feedback is some kind of recursion [11]. This becomes clearer still when we consider how tracing is depicted in the standard graphical calculus [16, § 5], as follows: A X A f f X (1) B X B Many important algebraic structures which are typically defined as sets-with-structure, like mon- oids, groups, or rings, may be described abstractly as algebraic structure, which when interpreted in Set yield the original definition. -
U Ottawa L'universite Canadienne Canada's University FACULTE DES ETUDES SUPERIEURES ^=1 FACULTY of GRADUATE and ET POSTOCTORALES U Ottawa POSDOCTORAL STUDIES
nm u Ottawa L'Universite canadienne Canada's university FACULTE DES ETUDES SUPERIEURES ^=1 FACULTY OF GRADUATE AND ET POSTOCTORALES U Ottawa POSDOCTORAL STUDIES L'Universite canadienne Canada's university Octavio Malherbe ^_^^-^^^„„„^______ Ph.D. (Mathematics) _.__...„___„„ Department of Mathematics and Statistics Categorical Models of Computation: Partially Traced Categories and Presheaf Models of Quantum Computation TITRE DE LA THESE / TITLE OF THESIS Philip Scott „„„„„„„„_^ Peter Selinger Richard Bulte Pieter Hofstra Robin Cockett Benjamin Steinberg University of Calgary Gary W. Slater Le Doyen de la Faculte des etudes superieures et postdoctorales / Dean of the Faculty of Graduate and Postdoctoral Studies CATEGORICAL MODELS OF COMPUTATION: PARTIALLY TRACED CATEGORIES AND PRESHEAF MODELS OF QUANTUM COMPUTATION By Octavio Malherbe, B.Sc, M.Sc. Thesis submitted to the Faculty of Graduate and Postdoctoral Studies University of Ottawa in partial fulfillment of the requirements for the PhD degree in the awa-Carleton Institute for Graduate Studies and Research in Mathematics and Statistics © Octavio Malherbe, Ottawa, Canada, 2010 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington OttawaONK1A0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-73903-7 Our file Notre reference ISBN: 978-0-494-73903-7 NOTICE: AVIS: The author has granted a non L'auteur a accorde une licence non exclusive exclusive license -
The Uniformity Principle on Traced Monoidal Categories†
Publ. RIMS, Kyoto Univ. 40 (2004), 991–1014 The Uniformity Principle on Traced Monoidal Categories† By Masahito Hasegawa∗ Abstract The uniformity principle for traced monoidal categories has been introduced as a natural generalization of the uniformity principle (Plotkin’s principle) for fixpoint operators in domain theory. We show that this notion can be used for constructing new traced monoidal categories from known ones. Some classical examples like the Scott induction principle are shown to be instances of these constructions. We also characterize some specific cases of our constructions as suitable enriched limits. §1. Introduction Traced monoidal categories were introduced by Joyal, Street and Verity [16] as the categorical structure for cyclic phenomena arising from various areas of mathematics, most notably knot theory [25]. They are (balanced) monoidal categories [15] enriched with a trace, which is a natural generalization of the traditional notion of trace in linear algebra, and can be regarded as an operator to create a “loop”. For example, the braid closure operation — e.g. the con- struction of the trefoil knot from a braid depicted below — can be considered as the trace operator on the category of tangles. ⇒ = Communicated by R. Nakajima. Received December 16, 2003. 2000 Mathematics Subject Classification(s): 18D10, 19D23 †This article is an invited contribution to a special issue of Publications of RIMS com- memorating the fortieth anniversary of the founding of the Research Institute for Math- ematical Sciences. ∗Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan; PRESTO, Japan Science and Technology Agency, Japan c 2004 Research Institute for Mathematical Sciences, Kyoto University.