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
19 - Pullbacks and pushouts
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
Pullbacks and pushouts are more advanced universal properties, showing some more of the general features that are missing from the universal properties we’ve seen so far. We describe the general idea of a cone over a diagram, and define a limit over a diagram as a universal cone over that diagram, and show that products are in fact limits over a discrete diagram. We define pullbacks as limits over a particular diagram which includes some arrows. We unravel pullbacks in Set, and show that intersections are an example. We also describe the category in which pullbacks are terminal objects, and show how pullbacks can be used to define composable pairs of arrows in the definition of a category. We define pushouts as the dual of pullbacks, and then unravel that as a direct definition. We examine pushouts in Set and show that, given two sets, the square involving their intersection and their union is both a pullback and a pushout. We briefly discuss pushouts of topological spaces as a way of glueing spaces together.
- Type
- Chapter
- Information
- The Joy of AbstractionAn Exploration of Math, Category Theory, and Life, pp. 270 - 289Publisher: Cambridge University PressPrint publication year: 2022