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

Cover\nHalf-title\nTitle Page\nCopyright\nDedication\nContents\nPrologue\n	The status of mathematics\n	Traditional mathematics: subjects\n	Traditional mathematics: methods\n	The content in this book\n	Audience\nPart One: Building Up to Categories\n	1. Categories: The Idea\n		1.1 Abstraction and analogies\n		1.2 Connections and unification\n		1.3 Context\n		1.4 Relationships\n		1.5 Sameness\n		1.6 Characterizing things by the role they play\n		1.7 Zooming in and out\n		1.8 Framework and techniques\n	2. Abstraction\n		2.1 What is math?\n		2.2 The twin disciplines of logic and abstraction\n		2.3 Forgetting details\n		2.4 Pros and cons\n		2.5 Making analogies into actual things\n		2.6 Different abstractions of the same thing\n		2.7 Abstraction journey through levels of math\n	3. Patterns\n		3.1 Mathematics as pattern spotting\n		3.2 Patterns as analogies\n		3.3 Patterns as signs of structure\n		3.4 Abstract structure as a type of pattern\n		3.5 Abstraction helps us see patterns\n	4. Context\n		4.1 Distance\n		4.2 Worlds of numbers\n		4.3 The zero world\n	5. Relationships\n		5.1 Family relationships\n		5.2 Symmetry\n		5.3 Arithmetic\n		5.4 Modular arithmetic\n		5.5 Quadrilaterals\n		5.6 Lattices of factors\n	6. Formalism\n		6.1 Types of tourism\n		6.2 Why we express things formally\n		6.3 Example: metric spaces\n		6.4 Basic logic\n		6.5 Example: modular arithmetic\n		6.6 Example: lattices of factors\n	7. Equivalence Relations\n		7.1 Exploring equality\n		7.2 The idea of abstract relations\n		7.3 Reflexivity\n		7.4 Symmetry\n		7.5 Transitivity\n		7.6 Equivalence relations\n		7.7 Examples from math\n		7.8 Interesting failures\n	8. Categories: The Definition\n		8.1 Data: objects and relationships\n		8.2 Structure: things we can do with the data\n		8.3 Properties: stipulations on the structure\n		8.4 The formal definition\n		8.5 Size issues\n		8.6 The geometry of associativity\n		8.7 Drawing helpful diagrams\n		8.8 The point of composition\nInterlude: A Tour of Math\n	9. Examples We’ve Already Seen, Secretly\n		9.1 Symmetry\n		9.2 Equivalence relations\n		9.3 Factors\n		9.4 Number systems\n	10. Ordered Sets\n		10.1 Totally ordered sets\n		10.2 Partially ordered sets\n	11. Small Mathematical Structures\n		11.1 Small drawable examples\n		11.2 Monoids\n		11.3 Groups\n		11.4 Points and paths\n	12. Sets and Functions\n		12.1 Functions\n		12.2 Structure: identities and composition\n		12.3 Properties: unit and associativity laws\n		12.4 The category of sets and functions\n	13. Large Worlds of Mathematical Structures\n		13.1 Monoids\n		13.2 Groups\n		13.3 Posets\n		13.4 Topological spaces\n		13.5 Categories\n		13.6 Matrices\nPart Two: Doing Category Theory\n	14. Isomorphisms\n		14.1 Sameness\n		14.2 Invertibility\n		14.3 Isomorphism in a category\n		14.4 Treating isomorphic objects as the same\n		14.5 Isomorphisms of sets\n		14.6 Isomorphisms of large structures\n		14.7 Further topics on isomorphisms\n	15. Monics and Epics\n		15.1 The asymmetry of functions\n		15.2 Injective and surjective functions\n		15.3 Monics: categorical injectivity\n		15.4 Epics: categorical surjectivity\n		15.5 Relationship with isomorphisms\n		15.6 Monoids\n		15.7 Further topics\n	16. Universal Properties\n		16.1 Role vs character\n		16.2 Extremities\n		16.3 Formal definition\n		16.4 Uniqueness\n		16.5 Terminal objects\n		16.6 Ways to fail\n		16.7 Examples\n		16.8 Context\n		16.9 Further topics\n	17. Duality\n		17.1 Turning arrows around\n		17.2 Dual category\n		17.3 Monic and epic\n		17.4 Terminal and initial\n		17.5 An alternative definition of categories\n	18. Products and Coproducts\n		18.1 The idea behind categorical products\n		18.2 Formal definition\n		18.3 Products as terminal objects\n		18.4 Products in Set\n		18.5 Uniqueness of products in Set\n		18.6 Products inside posets\n		18.7 The category of posets\n		18.8 Monoids and groups\n		18.9 Some key morphisms induced by products\n		18.10 Dually: coproducts\n		18.11 Coproducts in Set\n		18.12 Decategorification: relationship with arithmetic\n		18.13 Coproducts in other categories\n		18.14 Further topics\n	19. Pullbacks and Pushouts\n		19.1 Pullbacks\n		19.2 Pullbacks in Set\n		19.3 Pullbacks as terminal objects somewhere\n		19.4 Example: Definition of category using pullbacks\n		19.5 Dually: pushouts\n		19.6 Pushouts in Set\n		19.7 Pushouts in topology\n		19.8 Further topics\n	20. Functors\n		20.1 Making up the definition\n		20.2 Functors between small examples\n		20.3 Functors from small drawable categories\n		20.4 Free and forgetful functors\n		20.5 Preserving and reflecting structure\n		20.6 Further topics\n	21. Categories of Categories\n		21.1 The category Cat\n		21.2 Terminal and initial categories\n		21.3 Products and coproducts of categories\n		21.4 Isomorphisms of categories\n		21.5 Full and faithful functors\n	22. Natural Transformations\n		22.1 Definition by abstract feeling\n		22.2 Aside on homotopies\n		22.3 Shape\n		22.4 Functor categories\n		22.5 Diagrams and cones over diagrams\n		22.6 Natural isomorphisms\n		22.7 Equivalence of categories\n		22.8 Examples of equivalences of large categories\n		22.9 Horizontal composition\n		22.10 Interchange\n		22.11 Totality\n	23. Yoneda\n		23.1 The joy of Yoneda\n		23.2 Revisiting sameness\n		23.3 Representable functors\n		23.4 The Yoneda embedding\n		23.5 The Yoneda Lemma\n		23.6 Further topics\n	24. Higher Dimensions\n		24.1 Why higher dimensions?\n		24.2 Defining 2-categories directly\n		24.3 Revisiting homsets\n		24.4 From underlying graphs to underlying 2-graphs\n		24.5 Monoidal categories\n		24.6 Strictness vs weakness\n		24.7 Coherence\n		24.8 Degeneracy\n		24.9 n and\rinfinity\n		24.10 The moral of the story\nEpilogue: Thinking Categorically\n	Motivations\n	The process of doing category theory\n	The practice of category theory\nAppendices\n	Appendix A: Background on Alphabets\n	Appendix B: Background on Basic Logic\n	Appendix C: Background on Set Theory\n	Appendix D: Background on Topological Spaces\nGlossary\nFurther Reading\nAcknowledgements\nIndex