Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 Introduction
- 2 Algebras and Coalgebras
- 3 Finitary Iteration
- 4 Finitary Set Functors
- 5 Finitary Iteration in Enriched Settings
- 6 Transfinite Iteration
- 7 Terminal Coalgebras as Algebras, Initial Algebras as Coalgebras
- 8 Well-Founded Coalgebras
- 9 State Minimality and Well-Pointed Coalgebras
- 10 Fixed Points Determined by Finite Behaviour
- 11 Sufficient Conditions for Initial Algebras and Terminal Coalgebras
- 12 Liftings and Extensions from Set
- 13 Interaction between Initial Algebras and Terminal Coalgebras
- 14 Derived Functors
- 15 Special Topics
- Appendix A Functors with Initial Algebras or Terminal Coalgebras
- Appendix B A Primer on Fixed Points in Ordered and Metric Structures
- Appendix C Set Functors
- References
- List of Categories
- Index
2 - Algebras and Coalgebras
Published online by Cambridge University Press: 30 January 2025
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 Introduction
- 2 Algebras and Coalgebras
- 3 Finitary Iteration
- 4 Finitary Set Functors
- 5 Finitary Iteration in Enriched Settings
- 6 Transfinite Iteration
- 7 Terminal Coalgebras as Algebras, Initial Algebras as Coalgebras
- 8 Well-Founded Coalgebras
- 9 State Minimality and Well-Pointed Coalgebras
- 10 Fixed Points Determined by Finite Behaviour
- 11 Sufficient Conditions for Initial Algebras and Terminal Coalgebras
- 12 Liftings and Extensions from Set
- 13 Interaction between Initial Algebras and Terminal Coalgebras
- 14 Derived Functors
- 15 Special Topics
- Appendix A Functors with Initial Algebras or Terminal Coalgebras
- Appendix B A Primer on Fixed Points in Ordered and Metric Structures
- Appendix C Set Functors
- References
- List of Categories
- Index
Summary
This chapter provides examples of the central concepts of the book: initial algebras and terminal coalgebras. These are mainly for polynomial endofunctors on the category of sets, but also for such functors on sorted sets, nominal sets, and presheaves. We discuss connections to induction and recursion and to their duals, coinduction and corecursion, and also to bisimulation. We present Lambek’s Lemma, a result used throughout the book.
- Type
- Chapter
- Information
- Initial Algebras and Terminal CoalgebrasThe Theory of Fixed Points of Functors, pp. 12 - 64Publisher: Cambridge University PressPrint publication year: 2025