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Monotone Weak Distributive Laws over the Lifted Powerset Monad in Categories of Algebras

Author Quentin Aristote

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Quentin Aristote
  • Université Paris Cité, CNRS, Inria, IRIF, F-75013, Paris, France

Acknowledgements

For sometimes small but always fruitful discussions on topics related to this work, the author thanks Gabriella Böhm, Victor Iwaniack, Jean Goubault-Larrecq, Alexandre Goy, Daniela Petrişan and Sam van Gool.

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Quentin Aristote. Monotone Weak Distributive Laws over the Lifted Powerset Monad in Categories of Algebras. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.10

Abstract

In both the category of sets and the category of compact Hausdorff spaces, there is a monotone weak distributive law that combines two layers of non-determinism. Noticing the similarity between these two laws, we study whether the latter can be obtained automatically as a weak lifting of the former. This holds partially, but does not generalize to other categories of algebras. We then characterize when exactly monotone weak distributive laws over powerset monads in categories of algebras exist, on the one hand exhibiting a law combining probabilities and non-determinism in compact Hausdorff spaces and showing on the other hand that such laws do not exist in a lot of other cases.

Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
Keywords
  • weak distributive law
  • weak extension
  • weak lifting
  • iterated distributive law
  • Yang-Baxter equation
  • powerset monad
  • Vietoris monad
  • Radon monad
  • Eilenberg-Moore category
  • regular category
  • relational extension

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