Initial Algebras and Terminal Coalgebras

Jiří Adámek is Professor in the Department of Mathematics at Czech Technical University Prague and Professor Emeritus in the Department of Computer Science at Technical University Braunschweig. He has authored and co-authored ten books, including 'Locally Presentable and Accessible Categories' (1994), 'Abstract and Concrete Categories' (1990), and 'Algebraic Theories' (2011). He is an EATCS Fellow. Stefan Milius is Professor in the Department of Computer Science at Friedrich-Alexander Universität Erlangen-Nürnberg. An expert in the theory of coalgebras, he is also well known for his work on the category-theoretic approach to the semantics of iteration and recursion, for which he has won the prestigious Ackermann Award, and he is one of the inventors of the categorical approach to algebraic language theory. Lawrence S. Moss is Professor in the Mathematics Department at Indiana University. He is President of the Association for Logic, Language, and Information, and co-authored 'Vicious Circles' (1996) and 'Mathematical Structures in Language' (2016). He is known for work on dynamic epistemic logic, non-wellfounded sets and circularity, coalgebra, natural logic, and other areas of logic and mathematics.

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.