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
14 - Derived Functors
Published online by Cambridge University Press: 30 January 2025
- 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
Given an endofunctor F we can form various derived endofunctors whose initial algebras and terminal coalgebras are related to those of F. The most prominent example are coproducts of F with constant functors, yielding free F-algebras, cofree F-coalgebras, and free completely iterative F-algebras. An initial algebra exists for a composite functor FG if and only if it does for GF. We also present Freyd’s Iterated Square Theorem and its converse: A functor F on category with finite coproducts has an initial algebra precisely when FF does. The chapter also studies functors on slice categories and product categories, coproducts of functors, double-algebras, and coproducts of monads.
Keywords
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- Chapter
- Information
- Initial Algebras and Terminal CoalgebrasThe Theory of Fixed Points of Functors, pp. 477 - 509Publisher: Cambridge University PressPrint publication year: 2025