Representation and Character Theory of Finite Categorical Groups
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Diagrammatics in Categorification and Compositionality
Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 ABSTRACT Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 Copyright c 2019 by Dmitry Vagner All rights reserved Abstract In the present work, I explore the theme of diagrammatics and their capacity to shed insight on two trends|categorification and compositionality|in and around contemporary category theory. The work begins with an introduction of these meta- phenomena in the context of elementary sets and maps. Towards generalizing their study to more complicated domains, we provide a self-contained treatment|from a pedagogically novel perspective that introduces almost all concepts via diagrammatic language|of the categorical machinery with which we may express the broader no- tions found in the sequel. The work then branches into two seemingly unrelated disciplines: dynamical systems and knot theory. In particular, the former research defines what it means to compose dynamical systems in a manner analogous to how one composes simple maps. The latter work concerns the categorification of the slN link invariant. In particular, we use a virtual filtration to give a more diagrammatic reconstruction of Khovanov-Rozansky homology via a smooth TQFT. -
Duality and Traces for Indexed Monoidal Categories
DUALITY AND TRACES FOR INDEXED MONOIDAL CATEGORIES KATE PONTO AND MICHAEL SHULMAN Abstract. By the Lefschetz fixed point theorem, if an endomorphism of a topological space is fixed-point-free, then its Lefschetz number vanishes. This necessary condition is not usually sufficient, however; for that we need a re- finement of the Lefschetz number called the Reidemeister trace. Abstractly, the Lefschetz number is a trace in a symmetric monoidal cate- gory, while the Reidemeister trace is a trace in a bicategory. In this paper, we show that for any symmetric monoidal category with an associated indexed symmetric monoidal category, there is an associated bicategory which produces refinements of trace analogous to the Reidemeister trace. This bicategory also produces a new notion of trace for parametrized spaces with dualizable fibers, which refines the obvious “fiberwise" traces by incorporating the action of the fundamental group of the base space. Our abstract framework lays the foundation for generalizations of these ideas to other contexts. Contents 1. Introduction2 2. Indexed monoidal categories5 3. Indexed coproducts8 4. Symmetric monoidal traces 12 5. Shadows from indexed monoidal categories 15 6. Fiberwise duality 22 7. Base change objects 27 8. Total duality 29 9. String diagrams for objects 39 10. String diagrams for morphisms 42 11. Proofs for fiberwise duality and trace 49 12. Proofs for total duality and trace 63 References 65 Date: Version of October 31, 2011. Both authors were supported by National Science Foundation postdoctoral fellowships during the writing of this paper. This version of this paper contains diagrams that use dots and dashes to distinguish different features, and is intended for printing on a black and white printer. -
Arxiv:1604.06393V2 [Math.RA]
PARTIAL ACTIONS: WHAT THEY ARE AND WHY WE CARE ELIEZER BATISTA Abstract. We present a survey of recent developments in the theory of partial actions of groups and Hopf algebras. 1. Introduction The history of mathematics is composed of many moments in which an apparent dead end just opens new doors and triggers further progress. The origins of partial actions of groups can be considered as one such case. The notion of a partial group action appeared for the first time in Exel’s paper [38]. The problem was to calculate the K-theory of some C∗-algebras which have an action by automorphisms of the circle S1. By a known result due to Paschke, under suitable conditions about a C∗-algebra carrying an action of the circle group, it can be proved that this C∗- algebra is isomorphic to the crossed product of its fixed point subalgebra by an action of the integers. The main hypothesis of Paschke’s theorem is that the action has large spectral subspaces. A typical example in which this hypothesis does not occur is the action of S1 on the Toeplitz algebra by conjugation by the diagonal unitaries diag(1,z,z2,z3,...), for z ∈ S1. Therefore a new approach was needed to explore the internal structure of those algebras that carry circle actions but don’t have large enough spectral subspaces. Using some techniques coming from dynamical systems, these algebras could be characterized as crossed products by partial automorphisms. By a partial automorphism of an algebra A one means an isomorphism between two ideals in the algebra A. -
The Joy of String Diagrams Pierre-Louis Curien
The joy of string diagrams Pierre-Louis Curien To cite this version: Pierre-Louis Curien. The joy of string diagrams. Computer Science Logic, Sep 2008, Bertinoro, Italy. pp.15-22. hal-00697115 HAL Id: hal-00697115 https://hal.archives-ouvertes.fr/hal-00697115 Submitted on 14 May 2012 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. The Joy of String Diagrams Pierre-Louis Curien Preuves, Programmes et Syst`emes,CNRS and University Paris 7 May 14, 2012 Abstract In the past recent years, I have been using string diagrams to teach basic category theory (adjunctions, Kan extensions, but also limits and Yoneda embedding). Us- ing graphical notations is undoubtedly joyful, and brings us close to other graphical syntaxes of circuits, interaction nets, etc... It saves us from laborious verifications of naturality, which is built-in in string diagrams. On the other hand, the language of string diagrams is more demanding in terms of typing: one may need to introduce explicit coercions for equalities of functors, or for distinguishing a morphism from a point in the corresponding internal homset. So that in some sense, string diagrams look more like a language ”`ala Church", while the usual mathematics of, say, Mac Lane's "Categories for the working mathematician" are more ”`ala Curry". -
Arxiv:2004.02472V1 [Quant-Ph] 6 Apr 2020 1.2 the Problem of Composition
Schwinger’s Picture of Quantum Mechanics IV: Composition and independence F. M. Ciaglia1,5 , F. Di Cosmo2,3,6 , A. Ibort2,3,7 , G. Marmo4,8 1 Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany 2 ICMAT, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) 3 Depto. de Matemáticas, Univ. Carlos III de Madrid, Leganés, Madrid, Spain 4 Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II, Napoli, Italy 5 e-mail: florio.m.ciaglia[at]gmail.com , 6 e-mail: fabio[at] 7 e-mail: albertoi[at]math.uc3m.es 8 e-mail: marmo[at]na.infn.it April 7, 2020 Abstract The groupoids description of Schwinger’s picture of quantum mechanics is continued by discussing the closely related notions of composition of systems, subsystems, and their independence. Physical subsystems have a neat algebraic description as subgroupoids of the Schwinger’s groupoid of the system. The groupoids picture offers two natural notions of composition of systems: Direct and free products of groupoids, that will be analyzed in depth as well as their universal character. Finally, the notion of independence of subsystems will be reviewed, finding that the usual notion of independence, as well as the notion of free independence, find a natural realm in the groupoids formalism. The ideas described in this paper will be illustrated by using the EPRB experiment. It will be observed that, in addition to the notion of the non-separability provided by the entangled state of the system, there is an intrinsic ‘non-separability’ associated to the impossibility of identifying the entangled particles as subsystems of the total system. -
Arxiv:1603.02237V2 [Math.RA]
ARTINIAN AND NOETHERIAN PARTIAL SKEW GROUPOID RINGS PATRIK NYSTEDT University West, Department of Engineering Science, SE-46186 Trollh¨attan, Sweden JOHAN OINERT¨ Blekinge Institute of Technology, Department of Mathematics and Natural Sciences, SE-37179 Karlskrona, Sweden HECTOR´ PINEDO Universidad Industrial de Santander, Escuela de Matem´aticas, Carrera 27 Calle 9, Edificio Camilo Torres Apartado de correos 678, Bucaramanga, Colombia Abstract. Let α = {αg : Rg−1 → Rg}g∈mor(G) be a partial action of a groupoid G on a non-associative ring R and let S = R⋆α G be the associated partial skew groupoid ring. We show that if α is global and unital, then S is left (right) artinian if and only if R is left (right) artinian and Rg = {0}, for all but finitely many g ∈ mor(G). We use this result to prove that if α is unital and R is alternative, then S is left (right) artinian if and only if R is left (right) artinian and Rg = {0}, for all but finitely many g ∈ mor(G). Both of these results apply to partial skew group rings, and in particular they generalize a result by J. K. Park for classical skew group rings, i.e. the case when R is unital and associative, and G is a group which acts globally on R. Moreover, we provide two applications of our main result. Firstly, we generalize I. G. Connell’s classical result for group rings by giving a characterization of artinian (non-associative) groupoid rings. This result is in turn arXiv:1603.02237v2 [math.RA] 11 Oct 2016 applied to partial group algebras. -
A Groupoid Approach to Noncommutative T-Duality
A GROUPOID APPROACH TO NONCOMMUTATIVE T-DUALITY Calder Daenzer A Dissertation in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2007 Jonathan Block Supervisor of Dissertation Ching-Li Chai Graduate Group Chairperson Acknowledgments I would like to thank Oren Ben-Bassat, Tony Pantev, Michael Pimsner, Jonathan Rosenberg, Jim Stasheff, and most of all my advisor Jonathan Block, for the advice and helpful discussions. I would also like to thank Kathryn Schu and my whole family for all of the support. ii ABSTRACT A GROUPOID APPROACH TO NONCOMMUTATIVE T-DUALITY Calder Daenzer Jonathan Block, Advisor Topological T-duality is a transformation taking a gerbe on a principal torus bundle to a gerbe on a principal dual-torus bundle. We give a new geometric con- struction of T-dualization, which allows the duality to be extended to the follow- ing situations: bundles of groups other than tori, even bundles of some nonabelian groups, can be dualized; bundles whose duals are families of noncommutative groups (in the sense of noncommutative geometry) can be treated; and the base manifold parameterizing the bundles may be replaced by a topological stack. Some meth- ods developed for the construction may be of independent interest: these are a Pontryagin type duality between commutative principal bundles and gerbes, non- abelian Takai duality for groupoids, and the computation of certain equivariant Brauer groups. iii Contents 1 Introduction 1 2 Preliminaries 5 2.1 Groupoids and G-Groupoids . 5 2.2 Modules and Morita equivalence of groupoids . -
NONCOMMUTATIVE GEOMETRY and T-DUALITY SEMINAR Lecture 1: Introduction the Main Things We Will Be Talking About in the First Part
NONCOMMUTATIVE GEOMETRY AND T-DUALITY SEMINAR CALDER DAENZER Lecture 1: Introduction The main things we will be talking about in the first part of this course are groupoids and stacks. In the second part of the course we will use this knowledge to study T-duality. As an introduction and motivation, let me quickly point out how groupoids, stacks, and T-duality are related to noncommutative geometry. 0.1. Groupoids and Noncommutative geometry. A major part of Connes' style noncom- mutative geometry can be viewed through the lens of groupoids, via the correspondence groupoids groupoid C*-algebras: More precisely, a locally compact Hausdorff groupoid with an appropriate notion of measure gives rise to a convolution C∗-algebra, and there is a notion of Morita equivalence for groupoids which induces Morita equivalence of the corresponding C∗-algebras. The reason that this correspondence covers a \major part" of noncommutative geometry is that most of the well-known C∗-algebras are in fact Morita equivalent to (twisted) groupoid algebras 1. This includes for example all AF-algebras, noncommutative tori, and Cuntz algebras [Ren], and the continuous trace algebras as well [?]. In fact, at least to me, it is an open question to find a separable C∗-algebra which is not Morita equivalent to a twisted groupoid algebra. There are some genuine advantages to working directly with groupoids rather than C∗-algebras, of which I will name three: (1) The passage to the groupoid algebra can lose information, for example C∗-algebras cannot 1 1 distinguish the non-equivalent groupoids S ⇒ S and Z ⇒ ∗. -
Category Theory Using String Diagrams
Category Theory Using String Diagrams Dan Marsden November 11, 2014 Abstract In [Fokkinga, 1992a,b] and [Fokkinga and Meertens, 1994] a calcula- tional approach to category theory is developed. The scheme has many merits, but sacrifices useful type information in the move to an equational style of reasoning. By contrast, traditional proofs by diagram pasting re- tain the vital type information, but poorly express the reasoning and development of categorical proofs. In order to combine the strengths of these two perspectives, we propose the use of string diagrams, common folklore in the category theory community, allowing us to retain the type information whilst pursuing a calculational form of proof. These graphi- cal representations provide a topological perspective on categorical proofs, and silently handle functoriality and naturality conditions that require awkward bookkeeping in more traditional notation. Our approach is to proceed primarily by example, systematically ap- plying graphical techniques to many aspects of category theory. We de- velop string diagrammatic formulations of many common notions, includ- ing adjunctions, monads, Kan extensions, limits and colimits. We describe representable functors graphically, and exploit these as a uniform source of graphical calculation rules for many category theoretic concepts. These graphical tools are then used to explicitly prove many standard results in our proposed diagrammatic style. 1 Introduction arXiv:1401.7220v2 [math.CT] 9 Nov 2014 This work develops in some detail many aspects of basic category theory, with the aim of demonstrating the combined effectiveness of two key concepts: 1. The calculational reasoning approach to mathematics. 2. The use of string diagrams in category theory. -
1 Introd Uction 2 Partial Representations and Partial Group
Resenhas IME-USP 2002, Vol. 5, No. 4,305 - 327. Partial actions, crossed products and partial representations 1 M. Dokuchaev Abstract: We give a survey of algebraic results on partial representations of groups, partial actions and related concepts. Key words: Partial actions, crossed products, partia! representations . 1 Introd uction Partial actions of groups, partial representations and the notions related to them appeared in the theory of operator algebras as usefull tools of their study (see [8], [9], [10], [17], [19]) . In this survey we give an account ofthe algebraic results on this topic. We start Section 2 by showing how partial actions of groups on abstract sets naturally appear as restrictions of usual actions, giving at the same time a motivation fo r the concept of partial representation. Then we proceed by discussing the structure of partial group rings of finite groups, considering aIs o the corresponding isomorphism problem. Section 3 deals with the structure of partial representations of arbitary groups. In Section 4 we investigate the question when a given partial action of a group on an algebra can be viewed as a restriction of a usual action and study the associativity question of crossed products defined by partial actions. In the final Section 5 we try to explore the interaction between these concepts. By a unital ring we shall understand an associative ring with unity elemento Given a unital commutative ring R, by an R-algebra we mean an associative R-algebra. 2 Partial representations and partial group rings For an abstract set X consider the set I(X) of partially defined bijections of X , that is bijections <p : A --+ B, where the domain dom( <p) = A and the range ran(<p) = B are subsets of X . -
Semantics for a Lambda Calculus for String Diagrams
Semantics for a Lambda Calculus for String Diagrams Bert Lindenhovius Michael Mislove Department of Computer Science Department of Computer Science Tulane University Tulane University Vladimir Zamdzhiev Université de Lorraine, CNRS, Inria, LORIA Linear/non-linear (LNL) models, as described by Benton, soundly model a LNL term calculus and LNL logic closely related to intuitionistic linear logic. Every such model induces a canonical en- richment that we show soundly models a LNL lambda calculus for string diagrams, introduced by Rios and Selinger (with primary application in quantum computing). Our abstract treatment of this language leads to simpler concrete models compared to those presented so far. We also extend the language with general recursion and prove soundness. Finally, we present an adequacy result for the diagram-free fragment of the language which corresponds to a modified version of Benton and Wadler’s adjoint calculus with recursion. In keeping with the purpose of the special issue, we also describe the influence of Samson Abram- sky’s research on these results, and on the overall project of which this is a part. 1 Dedication We are pleased to contribute to this volume honoring SAMSON ABRAMSKY’s many contributions to logic and its applications in theoretical computer science. This contribution is an extended version of the paper [28], which concerns the semantics of high-level functional quantum programming languages. This research is part of a project on the same general theme. The project itself would not exist if it weren’t for Samson’s support, guidance and participation. And, as we document in the final section of this contribution, Samson’s research has had a direct impact on this work, and also points the way for further results along the general line we are pursuing. -
Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics
entropy Article Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics Florio M. Ciaglia 1,*,† , Fabio Di Cosmo 2,3,† , Alberto Ibort 2,3,† and Giuseppe Marmo 4,† 1 Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany 2 ICMAT, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Nicolás Cabrera, 13-15, Campus de Cantoblanco, UAM, 28049 Madrid, Spain; [email protected] (F.D.C.); [email protected] (A.I.) 3 Departemento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain 4 Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II, 80126 Napoli, Italy; [email protected] * Correspondence: [email protected] † These authors contributed equally to this work. Received: 18 September 2020; Accepted: 11 November 2020; Published: 13 November 2020 Abstract: The evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio’s theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one.