Mathematician and popular science author Eugenia Cheng is on a mission to show you that mathematics can be flexible, creative, and visual. This joyful journey through the world of abstract mathematics into category theory will demystify mathematical thought processes and help you develop your own thinking, with no formal mathematical background needed. The book brings abstract mathematical ideas down to earth using examples of social justice, current events, and everyday life – from privilege to COVID-19 to driving routes. The journey begins with the ideas and workings of abstract mathematics, after which you will gently climb toward more technical material, learning everything needed to understand category theory, and then key concepts in category theory like natural transformations, duality, and even a glimpse of ongoing research in higher-dimensional category theory. For fans of How to Bake Pi, this will help you dig deeper into mathematical concepts and build your mathematical background.
Author(s): Eugenia Cheng
Edition: 1
Publisher: Cambridge University Press
Year: 2022
Language: English
Commentary: Publisher's PDF
Pages: 438
City: Cambridge, UK
Tags: Popular Science; Mathematics; Category Theory
Cover
Endorsements
Half-title
Title page
Copyright information
Dedication
Contents
Prologue
The status of mathematics
Traditional mathematics: subjects
Traditional mathematics: methods
The content in this book
Audience
PART ONE
BUILDING UP TO CATEGORIES
1
Categories: the idea
1.1 Abstraction and analogies
1.2 Connections and unification
1.3 Context
1.4 Relationships
1.5 Sameness
1.6 Characterizing things by the role they play
1.7 Zooming in and out
1.8 Framework and techniques
2
Abstraction
2.1 What is math?
2.2 The twin disciplines of logic and abstraction
2.3 Forgetting details
2.4 Pros and cons
2.5 Making analogies into actual things
2.6 Different abstractions of the same thing
2.7 Abstraction journey through levels of math
3
Patterns
3.1 Mathematics as pattern spotting
3.2 Patterns as analogies
3.3 Patterns as signs of structure
3.4 Abstract structure as a type of pattern
3.5 Abstraction helps us see patterns
4
Context
4.1 Distance
4.2 Worlds of numbers
4.3 The zero world
5
Relationships
5.1 Family relationships
5.2 Symmetry
5.3 Arithmetic
5.4 Modular arithmetic
5.5 Quadrilaterals
5.6 Lattices of factors
6
Formalism
6.1 Types of tourism
6.2 Why we express things formally
6.3 Example: metric spaces
6.4 Basic logic
6.5 Example: modular arithmetic
6.6 Example: lattices of factors
7
Equivalence relations
7.1 Exploring equality
7.2 The idea of abstract relations
7.3 Reflexivity
7.4 Symmetry
7.5 Transitivity
7.6 Equivalence relations
7.7 Examples from math
7.8 Interesting failures
8
Categories: the definition
8.1 Data: objects and relationships
8.2 Structure: things we can do with the data
8.3 Properties: stipulations on the structure
8.4 The formal definition
8.5 Size issues
8.6 The geometry of associativity
8.7 Drawing helpful diagrams
8.8 The point of composition
INTERLUDE
A TOUR OF MATH
9
Examples we’ve already seen, secretly
9.1 Symmetry
9.2 Equivalence relations
9.3 Factors
9.4 Number systems
10
Ordered sets
10.1 Totally ordered sets
10.2 Partially ordered sets
11
Small mathematical structures
11.1 Small drawable examples
11.2 Monoids
11.3 Groups
11.4 Points and paths
12
Sets and functions
12.1 Functions
12.2 Structure: identities and composition
12.3 Properties: unit and associativity laws
12.4 The category of sets and functions
13
Large worlds of mathematical structures
13.1 Monoids
13.2 Groups
13.3 Posets
13.4 Topological spaces
13.5 Categories
13.6 Matrices
PART TWO
DOING CATEGORY THEORY
14
Isomorphisms
14.1 Sameness
14.2 Invertibility
14.3 Isomorphism in a category
14.4 Treating isomorphic objects as the same
14.5 Isomorphisms of sets
14.6 Isomorphisms of large structures
14.7 Further topics on isomorphisms
15
Monics and epics
15.1 The asymmetry of functions
15.2 Injective and surjective functions
15.3 Monics: categorical injectivity
15.4 Epics: categorical surjectivity
15.5 Relationship with isomorphisms
15.6 Monoids
15.7 Further topics
16
Universal properties
16.1 Role vs character
16.2 Extremities
16.3 Formal definition
16.4 Uniqueness
16.5 Terminal objects
16.6 Ways to fail
16.7 Examples
16.8 Context
16.9 Further topics
17
Duality
17.1 Turning arrows around
17.2 Dual category
17.3 Monic and epic
17.4 Terminal and initial
17.5 An alternative definition of categories
18
Products and coproducts
18.1 The idea behind categorical products
18.2 Formal definition
18.3 Products as terminal objects
18.4 Products in Set
18.5 Uniqueness of products in Set
18.6 Products inside posets
18.7 The category of posets
18.8 Monoids and groups
18.9 Some key morphisms induced by products
18.10 Dually: coproducts
18.11 Coproducts in Set
18.12 Decategorification: relationship with arithmetic
18.13 Coproducts in other categories
18.14 Further topics
19
Pullbacks and pushouts
19.1 Pullbacks
19.2 Pullbacks in Set
19.3 Pullbacks as terminal objects somewhere
19.4 Example: Definition of category using pullbacks
19.5 Dually: pushouts
19.6 Pushouts in Set
19.7 Pushouts in topology
19.8 Further topics
20
Functors
20.1 Making up the definition
20.2 Functors between small examples
20.3 Functors from small drawable categories
20.4 Free and forgetful functors
20.5 Preserving and reflecting structure
20.6 Further topics
21
Categories of categories
21.1 The category Cat
21.2 Terminal and initial categories
21.3 Products and coproducts of categories
21.4 Isomorphisms of categories
21.5 Full and faithful functors
22
Natural transformations
22.1 Definition by abstract feeling
22.2 Aside on homotopies
22.3 Shape
22.4 Functor categories
22.5 Diagrams and cones over diagrams
22.6 Natural isomorphisms
22.7 Equivalence of categories
22.8 Examples of equivalences of large categories
22.9 Horizontal composition
22.10 Interchange
22.11 Totality
23
Yoneda
23.1 The joy of Yoneda
23.2 Revisiting sameness
23.3 Representable functors
23.4 The Yoneda embedding
23.5 The Yoneda Lemma
23.6 Further topics
24
Higher dimensions
24.1 Why higher dimensions?
24.2 Defining 2-categories directly
24.3 Revisiting homsets
24.4 From underlying graphs to underlying 2-graphs
24.5 Monoidal categories
24.6 Strictness vs weakness
24.7 Coherence
24.8 Degeneracy
24.9 n and
infinity
24.10 The moral of the story
Epilogue
Thinking categorically
Motivations
The process of doing category theory
The practice of category theory
APPENDICES
Appendix A
Background on alphabets
Appendix B
Background on basic logic
Appendix C
Background on set theory
Appendix D
Background on topological spaces
Glossary
Further reading
Acknowledgements
Index
