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A categorical interpretation of partial actions 20 2.2.1

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Contents

Table of Contents i

Acknowledgements iii

Introduction v

Chapter 1. Categories and monoidal structures 1

1.1. Categories and limits 1

1.2. Monoidal structures 9

1.2.1. Monoidal categories and monoidal functors 9

1.2.2. Lax monoidal categories 13

Chapter 2. Partial actions of groups 19

2.1. The classical definition of partial group actions 19 2.2. A categorical interpretation of partial actions 20

2.2.1. Lax and quasi partial actions 20

2.2.2. Partial actions and spans 21

Chapter 3. Hopf algebras and their partial (co)actions 31

3.1. Algebras and coalgebras 31

3.2. Bialgebras and Hopf algebras 34

3.3. Modules and comodules 38

3.4. Partial actions and coactions of Hopf algebras 42 3.5. Partial representations and partial modules 49

Chapter 4. Geometric partial comodules over a coalgbra 53

4.1. Geometrically partial comodules 53

4.2. Partial comodule morphisms 63

4.3. Coassociativity 66

4.4. Completeness and cocompleteness of the category of

partial comodules 70

Chapter 5. Partial comodules over a Hopf algebra and

Hopf-Galois theory 81

5.1. The lax monoidal category of geometric partial comodules

over a bialgebra 81

i

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ii CONTENTS

5.2. Partial comodule algebras 88

5.2.1. Algebras in the category of partial comodules 89 5.2.2. Partial comodules in the category of algebras 91 5.3. Partial Hopf modules and partial Hopf-Galois theory 94

5.3.1. Partial Hopf modules 94

5.3.2. Hopf-Galois theory 96

Bibliography 109

Références

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