License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CALCO.2023.21
URN: urn:nbn:de:0030-drops-188189
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18818/
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Adámek, Jiří ; Milius, Stefan ; Moss, Lawrence S.

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

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LIPIcs-CALCO-2023-21.pdf (0.8 MB)

Abstract

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.

BibTeX - Entry

@InProceedings{adamek_et_al:LIPIcs.CALCO.2023.21,
  author =	{Ad\'{a}mek, Ji\v{r}{\'\i} and Milius, Stefan and Moss, Lawrence S.},
  title =	{{On Kripke, Vietoris and Hausdorff Polynomial Functors}},
  booktitle =	{10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)},
  pages =	{21:1--21:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-287-7},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{270},
  editor =	{Baldan, Paolo and de Paiva, Valeria},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18818},
  URN =		{urn:nbn:de:0030-drops-188189},
  doi =		{10.4230/LIPIcs.CALCO.2023.21},
  annote =	{Keywords: Hausdorff functor, Vietoris functor, initial algebra, terminal coalgebra}
}

Keywords: Hausdorff functor, Vietoris functor, initial algebra, terminal coalgebra
Collection: 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)
Issue Date: 2023
Date of publication: 02.09.2023

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