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7 - Terminal Coalgebras as Algebras, Initial Algebras as Coalgebras

Published online by Cambridge University Press:  30 January 2025

Jiří Adámek ,
Stefan Milius  and
Lawrence S. Moss
Jiří Adámek
Affiliation:
Czech Technical University in Prague
Stefan Milius
Affiliation:
Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Lawrence S. Moss
Affiliation:
Indiana University, Bloomington
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Summary

Corecursive algebras are algebras that admit unique solutions of recursive equation systems. We study these and a generalization: completely iterative algebras. The terminal coalgebra turns out to be the initial corecursive algebra as well as the initial completely iterative algebra. Dually, the initial algebra is the initial (parametrically) recursive coalgebra. These results explain the title of the chapter. We apply recursive coalgebras in order to obtain a new proof of the Initial Algebra Theorem from Chapter 6.

Keywords

recursive coalgebrascorecursive algebrascompletely iterative algebras
Type
Chapter
Information
Initial Algebras and Terminal Coalgebras
The Theory of Fixed Points of Functors
 , pp. 215 - 252
Publisher: Cambridge University Press
Print publication year: 2025

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