On Kripke, Vietoris and Hausdorff Polynomial Functors ((Co)algebraic pearls)

Authors Jiří Adámek, Stefan Milius , Lawrence S. Moss

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Jiří Adámek
  • Czech Technical University in Prague, Czech Republic
  • Technische Universität Braunschweig, Germany
Stefan Milius
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Lawrence S. Moss
  • Indiana University, Bloomington, IN, USA


We are grateful to Pedro Nora for discussions on the proof of Prop. 4.7.

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Jiří Adámek, Stefan Milius, and Lawrence S. Moss. On Kripke, Vietoris and Hausdorff Polynomial Functors ((Co)algebraic pearls). In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 21:1-21:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


The Vietoris space of compact subsets of a given Hausdorff space yields an endofunctor V on the category of Hausdorff spaces. Vietoris polynomial endofunctors on that category are built from V, the identity and constant functors by forming products, coproducts and compositions. These functors are known to have terminal coalgebras and we deduce that they also have initial algebras. We present an analogous class of endofunctors on the category of extended metric spaces, using in lieu of V the Hausdorff functor ℋ. We prove that the ensuing Hausdorff polynomial functors have terminal coalgebras and initial algebras. Whereas the canonical constructions of terminal coalgebras for Vietoris polynomial functors takes ω steps, one needs ω + ω steps in general for Hausdorff ones. We also give a new proof that the closed set functor on metric spaces has no fixed points.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Logic and verification
  • Hausdorff functor
  • Vietoris functor
  • initial algebra
  • terminal coalgebra


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