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
4 - Finitary Set Functors
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 discusses terminal coalgebras obtained by methods other than the finitary iteration that we saw in Chapter 3. One way is by taking a quotient of a weakly terminal coalgebra. Another is to use Worrell’s Theorem: the terminal coalgebra of a finitary set functor is obtainable as a limit, using a doubled form of infinite iteration. The chapter also contains a number of presentations of the terminal coalgebra of the finite power-set functor on sets and of the first infinite limit of its terminal-coalgebra chain.
Keywords
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- Initial Algebras and Terminal CoalgebrasThe Theory of Fixed Points of Functors, pp. 90 - 132Publisher: Cambridge University PressPrint publication year: 2025