Book contents
- Frontmatter
- Dedication
- Contents
- Prologue
- Part One Building up to Categories
- 1 Categories: the idea
- 2 Abstraction
- 3 Patterns
- 4 Context
- 5 Relationships
- 6 Formalism
- 7 Equivalence relations
- 8 Categories: the definition
- Interlude A Tour of Math
- Part Two Doing Category Theory
- Epilogue Thinking categorically
- Appendices
- Glossary
- Further Reading
- Acknowledgements
- Index
8 - Categories: the definition
from Part One - Building up to Categories
Published online by Cambridge University Press: 13 October 2022
- Frontmatter
- Dedication
- Contents
- Prologue
- Part One Building up to Categories
- 1 Categories: the idea
- 2 Abstraction
- 3 Patterns
- 4 Context
- 5 Relationships
- 6 Formalism
- 7 Equivalence relations
- 8 Categories: the definition
- Interlude A Tour of Math
- Part Two Doing Category Theory
- Epilogue Thinking categorically
- Appendices
- Glossary
- Further Reading
- Acknowledgements
- Index
Summary
In this chapter we finally give the formal definition of a category. We build up to each part of the definition first, also explaining that definitions of abstract mathematical structures often take the form of data, structure, and properties. We start with the data, which is objects and arrows. Then we give the structure, which is composition and identities. We discuss the fact that arrows can be drawn in many different ways while still giving the same abstract structure. We see that identities are a generalization of reflexivity, and composition is a generalization of transitivity. Finally, we give the properties or axioms, which are the unit laws and associativity of composition. We include a brief mention of size issues, and also describe how associativity can be encapsulated geometrically as a tetrahedron. We discuss the idea of drawing diagrams in category theory, in which we omit identities and composites for efficiency, and introduce the notion of commutative diagrams. The chapter concludes with a discussion of the importance of composition in a category.
- Type
- Chapter
- Information
- The Joy of AbstractionAn Exploration of Math, Category Theory, and Life, pp. 95 - 108Publisher: Cambridge University PressPrint publication year: 2022