TUM Library

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 that of their representations is described. The book comprises numerous examples and applications, including well-known games and puzzles, databases and physics applications. Key concepts have been presented using only basic notions so that it can be used both by students and researchers interested in the subject.

Description based upon print version of record.

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

6.3 The structure of groupoids: Second structure theorem

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