Initial Algebras and Terminal Coalgebras
Initial Algebras and Terminal Coalgebras
Stefan Milius, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Lawrence S. Moss, Indiana University, Bloomington
£155.00
GBPProviding an in-depth treatment of an exciting research area, this text's central topics are initial algebras and terminal coalgebras, primary objects of study in all areas of theoretical computer science connected to semantics. It contains a thorough presentation of iterative constructions, giving both classical and new results on terminal coalgebras obtained by limits of canonical chains, and initial algebras obtained by colimits. These constructions are also developed in enriched settings, especially those enriched over complete partial orders and complete metric spaces, connecting the book to topics like domain theory. Also included are an extensive treatment of set functors, and the first book-length presentation of the rational fixed point of a functor, and of lifting results which connect fixed points of set functors with fixed points of endofunctors on other categories. Representing more than fifteen years of work, this will be the leading text on the subject for years to come.
- Represents more than 15 years of work by this team of authors and will be the leading text on its topics for years to come
- Covers topics including the iterative method of constructing initial algebras and terminal coalgebras, transfinite constructions, the limit-colimit coincidence in enriched settings, corecursive algebras and well-founded coalgebras, on the rational fixed point and on liftings
- Includes appendices on the most common fixed point theorems and on properties of set functors
- Contains nearly all of the needed definitions and pointers to the literature on basic material to remain accessible to a wide readership
Reviews & endorsements
'The ultimate book explaining the state-of-the-art in the field of initial algebras and terminal coalgebras. Highly recommended to readers seeking both an introduction as well as an in-depth treatment of a general theory of iteration, with applications in algebraic specification, state-based modelling and behavioural equivalence.' Barbara König, Universität Duisburg-Essen
'Fixed points are central to computer science. The book is therefore a must for every categorically minded computer scientist.' Ichiro Hasuo, National Institute of Informatics
Product details
February 2025Hardback
9781108835466
640 pages
229 × 152 × 35 mm
1.136kg
Available
Table of Contents
- 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
- A. Functors with initial algebras or terminal coalgebras
- B. A primer on fixed points in ordered and metric structures
- C. Set functors
- References
- Index of categories
- Subject index.
