Category Theory in Context

Category theory has provided the foundations for many of the twentieth century's greatest advances in pure mathematics. This concise, original text for a one-semester course on the subject is derived from courses that author Emily Riehl taught at Harvard and Johns Hopkins Universities. The treatment introduces the essential concepts of category theory: categories, functors, nat...

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Category theory has provided the foundations for many of the twentieth century's greatest advances in pure mathematics. This concise, original text for a one-semester course on the subject is derived from courses that author Emily Riehl taught at Harvard and Johns Hopkins Universities. The treatment introduces the essential concepts of category theory: categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads, and other topics.

Suitable for advanced undergraduates and graduate students in mathematics, the text provides tools for understanding and attacking difficult problems in algebra, number theory, algebraic geometry, and algebraic topology. Drawing upon a broad range of mathematical examples from the categorical perspective, the author illustrates how the concepts and constructions of category theory arise from and illuminate more basic mathematical ideas. Prerequisites are limited to familiarity with some basic set theory and logic.

Preface ix
Sample corollaries x
A tour of basic categorical notions xi
Note to the reader xv
Notational conventions xvi
Acknowledgments xvi
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Preface ix
Sample corollaries x
A tour of basic categorical notions xi
Note to the reader xv
Notational conventions xvi
Acknowledgments xvi
Chapter 1 Categories, Functors, Natural Transformations 1
1.1 Abstract and concrete categories 3
1.2 Duality 9
1.3 Functoriality 13
1.4 Naturality 23
1.5 Equivalence of categories 29
1.6 The art of the diagram chase 36
1.7 The 2-category of categories 44
Chapter 2 Universal Properties, Representability, and the Yoneda Lemma 49
2.1 Representable functors 50
2.2 The Yoneda lemma 55
2.3 Universal properties and universal elements 62
2.4 The category of elements 66
Chapter 3 Limits and Colimits 73
3.1 Limits and colimits as universal cones 74
3.2 Limits in the category of sets 84
3.3 Preservation, reflection, and creation of limits and colimits 90
3.4 The representable nature of limits and colimits 93
3.5 Complete and cocomplete categories 99
3.6 Functoriality of limits and colimits 106
3.7 Size matters 109
3.8 Interactions between limits and colimits 110
Chapter 4 Adjunctions 115
4.1 Adjoint functors 116
4.2 The unit and counit as universal arrows 122
4.3 Contravariant and multivariable adjoint functors 126
4.4 The calculus of adjunctions 132
4.5 Adjunctions, limits, and colimits 136
4.6 Existence of adjoint functors 143
Chapter 5 Monads and their Algebras 153
5.1 Monads from adjunctions 154
5.2 Adjunctions from monads 158
5.3 Monadic functors 166
5.4 Canonical presentations via free algebras 168
5.5 Recognizing categories of algebras 173
5.6 Limits and colimits in categories of algebras 180
Chapter 6 All Concepts are Kan Extensions 189
6.1 Kan extensions 190
6.2 A formula for Kan extensions 193
6.3 Pointwise Kan extensions 199
6.4 Derived functors as Kan extensions 204
6.5 All concepts 209
Epilogue: Theorems in Category Theory 217
E.1 Theorems in basic category theory 217
E.2 Coherence for symmetric monoidal categories 219
E.3 The universal property of the unit interval 221
E.4 A characterization of Grothendieck toposes 222
E.5 Embeddings of abelian categories 223
Bibliography 225
Catalog of Categories 229
Glossary of Notation 231
Index 233
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