Book contents
- Frontmatter
- Dedication
- Contents
- Prologue
- Part One Building up to Categories
- Interlude A Tour of Math
- Part Two Doing Category Theory
- 14 Isomorphisms
- 15 Monics and epics
- 16 Universal properties
- 17 Duality
- 18 Products and coproducts
- 19 Pullbacks and pushouts
- 20 Functors
- 21 Categories of categories
- 22 Natural transformations
- 23 Yoneda
- 24 Higher dimensions
- Epilogue Thinking categorically
- Appendices
- Glossary
- Further Reading
- Acknowledgements
- Index
22 - Natural transformations
from Part Two - Doing Category Theory
Published online by Cambridge University Press: 13 October 2022
- Frontmatter
- Dedication
- Contents
- Prologue
- Part One Building up to Categories
- Interlude A Tour of Math
- Part Two Doing Category Theory
- 14 Isomorphisms
- 15 Monics and epics
- 16 Universal properties
- 17 Duality
- 18 Products and coproducts
- 19 Pullbacks and pushouts
- 20 Functors
- 21 Categories of categories
- 22 Natural transformations
- 23 Yoneda
- 24 Higher dimensions
- Epilogue Thinking categorically
- Appendices
- Glossary
- Further Reading
- Acknowledgements
- Index
Summary
Natural transformations are an appropriate notion of relationship between functors, so this chapter gives us our first glimpse of two-dimensional structures. We first give the definition by feeling our way through abstractly, and then by analogy with homotopies. We define identities and composition for natural transformations, and thus define functor categories. We define the category of presheaves on a category as a particular functor category. We show how to use natural transformations to define cones over a diagram formally. We look at isomorphisms in functor categories, and prove that these correspond to componentwise isomorphisms. We show how to read commutative diagrams “dynamically”. We define equivalence of categories via pseudo-inverses, and briefly mention the relationship with pointwise equivalence. We define horizontal composition of natural transformations, and whiskering, and prove the interchange law, so that we are ready for the concept of a 2-category.
- Type
- Chapter
- Information
- The Joy of AbstractionAn Exploration of Math, Category Theory, and Life, pp. 328 - 350Publisher: Cambridge University PressPrint publication year: 2022