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Terminal Coalgebras for Finitary Functors

Authors Jiří Adámek , Stefan Milius , Lawrence S. Moss

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Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Logic and verification
Keywords
  • terminal coalgebra
  • countable iteration
  • descending chain condition

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Abstract

We present a result that implies that an endofunctor on a category has a terminal coalgebra obtainable as a countable limit of its terminal-coalgebra sequence. It holds for finitary endofunctors preserving nonempty binary intersections on locally finitely presentable categories, assuming that the posets of strong quotients and subobjects of every finitely presentable object satisfy the descending chain condition. This allows one to adapt finiteness arguments that were originally advanced by Worrell concerning terminal coalgebras for finitary set functors. Examples include the categories of sets, posets, vector spaces, graphs, and nominal sets. A similar argument is presented for the category of metric spaces (although it is not locally finitely presentable).

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Jiří Adámek, Stefan Milius, and Lawrence S. Moss. Terminal Coalgebras for Finitary Functors. In 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 342, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CALCO.2025.3

Author Details

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

Funding

  • Adámek, Jiří: Supported by the grant No. 22-02964S of the Czech Grant Agency.
  • Milius, Stefan: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 470467389.
  • Moss, Lawrence S.: Supported by grant #586136 from the Simons Foundation.

References

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