Basic Category Theory 1st Edition by Tom Leinster, ISBN-13: 978-1107044241

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Basic Category Theory 1st Edition by Tom Leinster, ISBN-13: 978-1107044241

[PDF eBook eTextbook]

• Publisher: ‎ Cambridge University Press; 1st edition (September 22, 2014)
• Language: ‎ English
• 190 pages
• ISBN-10: ‎ 1107044243
• ISBN-13: ‎ 978-1107044241

A short introduction ideal for students learning category theory for the first time.

At the heart of this short introduction to category theory is the idea of a universal property, important throughout mathematics. After an introductory chapter giving the basic definitions, separate chapters explain three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties all three together. The book is suitable for use in courses or for independent study. Assuming relatively little mathematical background, it is ideal for beginning graduate students or advanced undergraduates learning category theory for the first time. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations. Copious exercises are included.

Table of Contents:

Note to the reader page vii
Introduction 1
1 Categories, functors and natural transformations 9
1.1 Categories 10
1.2 Functors 17
1.3 Natural transformations 27
2 Adjoints 41
2.1 Definition and examples 41
2.2 Adjunctions via units and counits 50
2.3 Adjunctions via initial objects 58
3 Interlude on sets 65
3.1 Constructions with sets 66
3.2 Small and large categories 73
3.3 Historical remarks 78
4 Representables 83
4.1 Definitions and examples 84
4.2 The Yoneda lemma 93
4.3 Consequences of the Yoneda lemma 99
5 Limits 107
5.1 Limits: definition and examples 107
5.2 Colimits: definition and examples 126
5.3 Interactions between functors and limits 136
6 Adjoints, representables and limits 141
6.1 Limits in terms of representables and adjoints 141
6.2 Limits and colimits of presheaves 145
6.3 Interactions between adjoint functors and limits 157
Appendix Proof of the general adjoint functor theorem 171
Further reading 174
Index of notation 177
Index 178

Tom Leinster has held postdoctoral positions at Cambridge and the Institut des Hautes Études Scientifiques (France), and held an EPSRC Advanced Research Fellowship at the University of Glasgow. He is currently a Chancellor’s Fellow at the University of Edinburgh. He is also the author of Higher Operads, Higher Categories (Cambridge University Press, 2004), and one of the hosts of the research blog, The n-Category Café.

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