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
3 - Finitary Iteration
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 presents initial algebras and terminal coalgebras obtained by the most common method. For the initial algebra, this is by iteration starting from the initial object through the natural numbers. For the terminal coalgebra, this is the dual: the iteration begins with the terminal object. The chapter is mainly concerned with examples drawn from sets, posets, complete partial orders, and metric spaces.
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- Initial Algebras and Terminal CoalgebrasThe Theory of Fixed Points of Functors, pp. 65 - 89Publisher: Cambridge University PressPrint publication year: 2025