The Joy of Abstraction: An Exploration of Math, Category Theory, and Life

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