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Powerset-Like Monads Weakly Distribute over Themselves in Toposes and Compact Hausdorff Spaces

Authors Alexandre Goy , Daniela Petrişan , Marc Aiguier



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Author Details

Alexandre Goy
  • Université Paris-Saclay, CentraleSupélec, MICS, France
Daniela Petrişan
  • Université de Paris, IRIF, France
Marc Aiguier
  • Université Paris-Saclay, CentraleSupélec, MICS, France

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Alexandre Goy, Daniela Petrişan, and Marc Aiguier. Powerset-Like Monads Weakly Distribute over Themselves in Toposes and Compact Hausdorff Spaces. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 132:1-132:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.132

Abstract

The powerset monad on the category of sets does not distribute over itself. Nevertheless a weaker form of distributive law of the powerset monad over itself exists and it essentially stems from the canonical Egli-Milner extension of the powerset to the category of relations. On the other hand, any regular category yields a category of relations, and some regular categories also possess a powerset-like monad, as is the Vietoris monad on compact Hausdorff spaces. We derive the Egli-Milner extension in three different frameworks : sets, toposes, and compact Hausdorff spaces. We prove that it corresponds to a monotone weak distributive law in each case by showing that the multiplication extends to relations but the unit does not. We provide an application to coalgebraic determinization of alternating automata.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • Egli-Milner relation
  • weak extension
  • weak distributive law
  • weak lifting
  • powerset monad
  • Vietoris monad
  • topos
  • alternating automaton
  • generalized determinization

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