License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CALCO.2019.12
URN: urn:nbn:de:0030-drops-114403
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/11440/
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Adámek, Jirí

On Terminal Coalgebras Derived from Initial Algebras

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LIPIcs-CALCO-2019-12.pdf (0.5 MB)

Abstract

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.

BibTeX - Entry

@InProceedings{admek:LIPIcs:2019:11440,
  author =	{Jir{\'\i} Ad{\'a}mek},
  title =	{{On Terminal Coalgebras Derived from Initial Algebras}},
  booktitle =	{8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019)},
  pages =	{12:1--12:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-120-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{139},
  editor =	{Markus Roggenbach and Ana Sokolova},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/11440},
  URN =		{urn:nbn:de:0030-drops-114403},
  doi =		{10.4230/LIPIcs.CALCO.2019.12},
  annote =	{Keywords: terminal coalgebras, initial algebras, algebraically complete category, finitary functor, fixed points of functors}
}

Keywords: terminal coalgebras, initial algebras, algebraically complete category, finitary functor, fixed points of functors
Collection: 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019)
Issue Date: 2019
Date of publication: 25.11.2019

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