On Terminal Coalgebras Derived from Initial Algebras

Author Jiří Adámek

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Jiří Adámek
  • Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Czech Republic


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Jiří Adámek. On Terminal Coalgebras Derived from Initial Algebras. In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 12:1-12:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


A number of important set functors have countable initial algebras, but terminal coalgebras are uncountable or even non-existent. We prove that the countable cardinality is an anomaly: every set functor with an initial algebra of a finite or uncountable regular cardinality has a terminal coalgebra of the same cardinality. We also present a number of categories that are algebraically complete and cocomplete, i.e., every endofunctor has an initial algebra and a terminal coalgebra. Finally, for finitary set functors we prove that the initial algebra mu F and terminal coalgebra nu F carry a canonical ultrametric with the joint Cauchy completion. And the algebra structure of mu F determines, by extending its inverse continuously, the coalgebra structure of nu F.

Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
  • terminal coalgebras
  • initial algebras
  • algebraically complete category
  • finitary functor
  • fixed points of functors


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