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Communication Dans Un Congrès Année : 2021

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

Résumé

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.
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hal-04491472 , version 1 (29-03-2024)

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Alexandre Goy, Daniela Petrişan, Marc Aiguier. Powerset-Like Monads Weakly Distribute over Themselves in Toposes and Compact Hausdorff Spaces. 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), Jul 2022, Glasgow (SCOTLAND), France. ⟨10.4230/LIPIcs.ICALP.2021.132⟩. ⟨hal-04491472⟩
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