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
20 - Functors
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
In this chapter we look at the concept of morphisms between categories, using the principle of preserving structure. We give the definition in two ways, one for each of our two approaches to defining categories (by homsets or by graphs). We look at functors between small examples of categories, including functors between posets, monoids, and groups, expressed as categories. We consider functors from small drawable categories and show that they produce a diagram of that shape in the target category. We think about the category consisting of a single non-trivial isomorphism, and see that a functor out of it picks out an isomorphism in the target category. We describe free and forgetful functors, including the free monoid functor. We define the concept of functors preserving and reflecting structure, and show that not all functors preserve epics, but they all preserve split epics. We consider whether the above forgetful functors preserve terminal and initial objects. Further topics include the fundamental group functor, and Van Kampen’s theorem reframed as preservation of pushouts under certain circumstances. We introduce contravariant functors.
- Type
- Chapter
- Information
- The Joy of AbstractionAn Exploration of Math, Category Theory, and Life, pp. 290 - 308Publisher: Cambridge University PressPrint publication year: 2022