Topology: A Categorical Approach

by Tai-Danae Bradley, Tyler Bryson, John Terilla

Overview

A graduate-level textbook that presents basic topology from the perspective of category theory.

This graduate-level textbook on topology takes a unique approach: it reintroduces basic, point-set topology from a more modern, categorical perspective. Many graduate students are familiar with the ideas of point-set topology and they are ready to learn something new about them. Teaching the subject using category theory—a contemporary branch of mathematics that provides a way to represent abstract concepts—both deepens students' understanding of elementary topology and lays a solid foundation for future work in advanced topics.

After presenting the basics of both category theory and topology, the book covers the universal properties of familiar constructions and three main topological properties—connectedness, Hausdorff, and compactness. It presents a fine-grained approach to convergence of sequences and filters; explores categorical limits and colimits, with examples; looks in detail at adjunctions in topology, particularly in mapping spaces; and examines additional adjunctions, presenting ideas from homotopy theory, the fundamental groupoid, and the Seifert van Kampen theorem. End-of-chapter exercises allow students to apply what they have learned. The book expertly guides students of topology through the important transition from undergraduate student with a solid background in analysis or point-set topology to graduate student preparing to work on contemporary problems in mathematics.

Product Details

ISBN-13:	9780262539357
Publisher:	MIT Press
Publication date:	08/18/2020
Pages:	166
Product dimensions:	6.90(w) x 8.90(h) x 0.50(d)

About the Author

Tai-Danae Bradley is a PhD candidate in mathematics at the CUNY Graduate Center and the creator of the popular math blog, Math3ma.

Tyler Bryson is a PhD candidate in mathematics at the CUNY Graduate Center.

John Terilla is Professor of Mathematics at Queens College and on the Doctoral Faculty at the CUNY Graduate Center.

Table of Contents

Preface ix

0 Preliminaries 1

0.1 Basic Topology 1

0.2 Basic Category Theory 3

0.2.1 Categories 3

0.2.2 Functors 9

0.2.3 Natural Transformations and the Yoneda Lemma 11

0.3 Basic Set Theory 14

0.3.1 Functions 14

0.3.2 The Empty Set and One-Point Set 15

0.3.3 Products and Coproducts in Set 15

0.3.4 Products and Coproducts in Any Category 17

0.3.5 Exponentiation in Set 17

0.3.6 Partially Ordered Sets 18

Exercises 19

1 Examples and Constructions 21

1.1 Examples and Terminology 21

1.1.1 Examples of Spaces 21

1.1.2 Examples of Continuous Functions 23

1.2 The Subspace Topology 25

1.2.1 The First Characterization 25

1.2.2 The Second Characterization 26

1.3 The Quotient Topology 28

1.3.1 The First Characterization 28

1.3.2 The Second Characterization 29

1.4 The Product Topology 30

1.4.1 The First Characterization 30

1.4.2 The Second Characterization 31

1.5 The Coproduct Topology 32

1.5.1 The First Characterization 32

1.5.2 The Second Characterization 33

1.6 Homotopy and the Homotopy Category 34

Exercises 36

2 Connectedness and Compactness 39

2.1 Connectedness 39

2.1.1 Definitions, Theorems, and Examples 39

2.1.2 The Functor π₀ 43

2.1.3 Constructions and Connectedness 44

2.1.4 Local (Path) Connectedness 46

2.2 Hausdorff Spaces 47

2.3 Compactness 48

2.3.1 Definitions, Theorems, and Examples 48

2.3.2 Constructions and Compactness 50

2.3.3 Local Compactness 51

Exercises 53

3 Limits of Sequences and Filters 55

3.1 Closure and Interior 55

3.2 Sequences 56

3.3 Filters and Convergènce 60

3.4 Tychonoff's Theorem 64

3.4.1 Ultrafilters and Compactness 64

3.4.2 A Proof of Tychonoff's Theorem 68

3.4.3 A Little Set Theory 69

Exercises 71

4 Categorical Limits and Colimits 75

4.1 Diagrams Are Functors 75

4.2 Limits and Colimits 77

4.3 Examples 79

4.3.1 Terminal and Initial Objects 79

4.3.2 Products and Coproducts 80

4.3.3 Pullbacks and Pushouts 81

4.3.4 Inverse and Direct Limits 83

4.3.5 Equalizers and Coequalizers 85

4.4 Completeness and Cocompleteness 86

Exercises 88

5 Adjunctions and the Compact-Open Topology 91

5.1 Adjunctions 92

5.1.1 The Unit and Counit of an Adjunction 93

5.2 Free-Forgetful Adjunction in Algebra 94

5.3 The Forgetful Functor U: Top → Set and Its Adjoints 96

5.4 Adjoint Functor Theorems 97

5.5 Compactifications 98

5.5.1 The One-Point Compactification 98

5.5.2 The Stone-Cech Compactification 99

5.6 The Exponential Topology 101

5.6.1 The Compact-Open Topology 104

5.6.2 The Theorems of Ascoli and Arzela 108

5.6.3 Enrich the Product-Horn Adjunction in Top 109

5.7 Compactly Generated Weakly Hausdorff Spaces 110

Exercises 114

6 Paths, Loops, Cylinders, Suspensions,… 115

6.1 Cylinder-Free Path Adjunction 116

6.2 The Fundamental Groupoid and Fundamental Group 118

6.3 The Categories of Pairs and Pointed Spaces 121

6.4 The Smash-Horn Adjunction 122

6.5 The Suspension-Loop Adjunction 124

6.6 Fibrations and Based Path Spaces 127

6.6.1 Mapping Path Space and Mapping Cylinder 129

6.6.2 Examples and Results 131

6.6.3 Applications of π₁S¹ 137

6.7 The Seifert van Kampen Theorem 139

6.7.1 Examples 141

Exercises 145

Glossary of Symbols 147

Bibliography 149

Index 153