An Introduction to Groups, Groupoids and Their Representations /
This book offers an introduction to the theory of groupoids and their representations encompassing the standard theory of groups. Using a categorical language, developed from simple examples, the theory of finite groupoids is shown to knit neatly with that of groups and their structure as well as th...
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| Format: | eBook |
| Language: | English |
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Milton :
CRC Press LLC,
2019.
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| Online Access: | Connect to the full text of this electronic book |
Table of Contents:
- Cover; Title Page; Copyright Page; Dedication; Preface; Acknowledgements; Table of Contents; Introduction; PART I: WORKING WITH CATEGORIES AND GROUPOIDS; 1: Categories: Basic Notions and Examples; 1.1 Introducing the main characters; 1.1.1 Connecting dots, graphs and quivers; 1.1.2 Drawing simple quivers and categories; 1.1.3 Relations, inverses, and some interesting problems; 1.2 Categories: Formal definitions; 1.2.1 Finite categories; 1.2.2 Abstract categories; 1.3 A categorical definition of groupoids and groups; 1.4 Historical notes and additional comments
- 1.4.1 Groupoids: A short history1.4.2 Categories; 1.4.3 Groupoids and physics; 1.4.4 Groupoids and other points of view; 2: Groups; 2.1 Groups, subgroups and normal subgroups: Basic notions; 2.1.1 Groups: Definitions and examples; 2.1.2 Subgroups and cosets; 2.1.3 Normal subgroups; 2.2 A family of board games: The symmetric group; 2.3 Group homomorphisms and Cayley's theorem; 2.3.1 Group homomorphisms: First properties; 2.3.2 Cayley's theorem for groups; 2.4 The alternating group; 2.4.1 Conjugacy classes: Young diagrams; 2.4.2 Homomorphisms and exact sequences
- 2.4.3 The group of automorphisms of a group2.5 Products of groups; 2.5.1 Direct product of groups; 2.5.2 Classification of finite Abelian groups; 2.5.3 Semidirect product of groups; 2.6 Historical notes and additional comments; 3: Groupoids; 3.1 Groupoids: Basic concepts; 3.1.1 Groupoids and subgroupoids; 3.1.2 Disjoint union of groupoids; 3.1.3 The groupoid of pairs revisited: Equivalence relations and subgroupoids; 3.1.4 Product of groupoids; 3.2 Puzzles and groupoids; 3.2.1 The "15 puzzle"; 3.2.2 The four squares puzzle: The groupoid 2; 3.2.3 Cyclic puzzles and cyclic groupoids
- 3.2.4 Rubik's 'pocket cube'4: Actions of Groups and Groupoids; 4.1 Symmetries, groups and groupoids; 4.1.1 Groups and symmetries; 4.1.2 Actions of groups; 4.2 The action groupoid; 4.3 Symmetries and groupoids; 4.3.1 Groupoids and generalised actions; 4.3.2 Groupoids and symmetries: The restriction of an action groupoid; 4.4 Weinstein's tilings; 4.4.1 Tilings and groupoids; 4.4.2 Local symmetries; 4.5 Cayley's theorem for groupoids; 5: Functors and Transformations; 5.1 Functors; 5.1.1 Functors: Definitions and first examples; 5.1.2 Functors and realisations of categories
- 5.2 An interlude: Categories and databases5.2.1 A simple database: Classes and courses; 5.2.2 Databases and functors; 5.3 Homomorphisms of groupoids; 5.3.1 Homomorphisms of groupoids: Basic notions; 5.3.2 Exact sequences of homomorphisms of groupoids; 5.3.3 Homomorphisms of groupoids, direct unions and products of groupoids; 5.3.4 Groupoids of automorphisms; 5.4 Equivalence: Natural transformations; 5.4.1 Equivalence of categories; 5.4.2 The notion of natural transformation; 6. The Structure of Groupoids; 6.1 Normal subgroupoids; 6.2 Simple groupoids