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

Author Quentin Aristote

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LIPIcs.STACS.2025.10.pdf
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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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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.

Cite As Get BibTex

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

Author Details

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.

References

  1. Michael Barr. Relational algebras. In S. MacLane, H. Applegate, M. Barr, B. Day, E. Dubuc, Phreilambud, A. Pultr, R. Street, M. Tierney, and S. Swierczkowski, editors, Reports of the Midwest Category Seminar IV, pages 39-55, Berlin, Heidelberg, 1970. Springer Berlin Heidelberg. Google Scholar
  2. Jon Beck. Distributive laws. In H. Appelgate, M. Barr, J. Beck, F. W. Lawvere, F. E. J. Linton, E. Manes, M. Tierney, F. Ulmer, and B. Eckmann, editors, Seminar on Triples and Categorical Homology Theory, Lecture Notes in Mathematics, pages 119-140, Berlin, Heidelberg, 1969. Springer. URL: https://doi.org/10.1007/BFb0083084.
  3. Gabriella Böhm. The weak theory of monads. Advances in Mathematics, 225(1):1-32, September 2010. URL: https://doi.org/10.1016/j.aim.2010.02.015.
  4. Gabriella Böhm. On the iteration of weak wreath products. Theory and Applications of Categories, 26(2):30-59, 2012. URL: http://www.tac.mta.ca/tac/volumes/26/2/26-02abs.html.
  5. Gabriella Böhm, Stephen Lack, and Ross Street. On the 2-Categories of Weak Distributive Laws. Communications in Algebra, 39(12):4567-4583, December 2011. URL: https://doi.org/10.1080/00927872.2011.616436.
  6. Gabriella Böhm, Stephen Lack, and Ross Street. Idempotent splittings, colimit completion, and weak aspects of the theory of monads. Journal of Pure and Applied Algebra, 216(2):385-403, February 2012. URL: https://doi.org/10.1016/j.jpaa.2011.07.003.
  7. Filippo Bonchi and Alessio Santamaria. Convexity via Weak Distributive Laws. Logical Methods in Computer Science, Volume 18, Issue 4, November 2022. URL: https://doi.org/10.46298/lmcs-18(4:8)2022.
  8. Francis Borceux. Handbook of Categorical Algebra: Volume 1: Basic Category Theory, volume 1 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge, 1994. URL: https://doi.org/10.1017/CBO9780511525858.
  9. Francis Borceux. Handbook of Categorical Algebra: Volume 2: Categories and Structures, volume 2 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge, 1994. URL: https://doi.org/10.1017/CBO9780511525865.
  10. Eugenia Cheng. Iterated distributive laws. Mathematical Proceedings of the Cambridge Philosophical Society, 150(3):459-487, May 2011. URL: https://doi.org/10.1017/S0305004110000599.
  11. Maria Manuel Clementino, Eva Colebunders, and Walter Tholen. Lax algebras as spaces. In Dirk Hofmann, Gavin J. Seal, and Walter Tholen, editors, Monoidal Topology: A Categorical Approach to Order, Metric, and Topology, Encyclopedia of Mathematics and Its Applications, pages 375-466. Cambridge University Press, Cambridge, 2014. URL: https://doi.org/10.1017/CBO9781107517288.007.
  12. Fredrik Dahlqvist and Renato Neves. Compositional semantics for new paradigms: Probabilistic, hybrid and beyond, April 2018. URL: https://doi.org/10.48550/arXiv.1804.04145.
  13. Peter John Freyd and Andre Scedrov. Categories, Allegories. North Holland, January 1990. Google Scholar
  14. Richard Garner. The Vietoris Monad and Weak Distributive Laws. Applied Categorical Structures, 28(2):339-354, April 2020. URL: https://doi.org/10.1007/s10485-019-09582-w.
  15. Jean Goubault-Larrecq. Weak Distributive Laws between Monads of Continuous Valuations and of Non-Deterministic Choice, August 2024. URL: https://doi.org/10.48550/arXiv.2408.15977.
  16. Alexandre Goy. On the Compositionality of Monads via Weak Distributive Laws. PhD thesis, Université Paris-Saclay, October 2021. URL: https://theses.hal.science/tel-03426949.
  17. Alexandre Goy. Weakening and Iterating Laws using String Diagrams. Electronic Notes in Theoretical Informatics and Computer Science, Volume 1 - Proceedings of MFPS XXXVIII, February 2023. URL: https://doi.org/10.46298/entics.10482.
  18. Alexandre Goy and Daniela Petrişan. Combining probabilistic and non-deterministic choice via weak distributive laws. In Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '20, pages 454-464, New York, NY, USA, July 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3373718.3394795.
  19. Alexandre Goy, Daniela Petrişan, and Marc Aiguier. Powerset-Like Monads Weakly Distribute over Themselves in Toposes and Compact Hausdorff Spaces. In DROPS-IDN/v2/Document/10.4230/LIPIcs.ICALP.2021.132. Schloss-Dagstuhl - Leibniz Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.132.
  20. Dirk Hofmann, Gavin J. Seal, and Walter Tholen, editors. Monoidal Topology: A Categorical Approach to Order, Metric, and Topology. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge, 2014. URL: https://doi.org/10.1017/CBO9781107517288.
  21. Klaus Keimel and Gordon D. Plotkin. Mixed powerdomains for probability and nondeterminism. Logical Methods in Computer Science, Volume 13, Issue 1, January 2017. URL: https://doi.org/10.23638/LMCS-13(1:2)2017.
  22. Bartek Klin and Julian Salamanca. Iterated Covariant Powerset is not a Monad. Electronic Notes in Theoretical Computer Science, 341:261-276, December 2018. URL: https://doi.org/10.1016/j.entcs.2018.11.013.
  23. Alexander Kurz and Jiří Velebil. Quasivarieties and varieties of ordered algebras: Regularity and exactness. Mathematical Structures in Computer Science, 27(7):1153-1194, October 2017. URL: https://doi.org/10.1017/S096012951500050X.
  24. Ernest Manes. A triple theoretic construction of compact algebras. In H. Appelgate, M. Barr, J. Beck, F. W. Lawvere, F. E. J. Linton, E. Manes, M. Tierney, F. Ulmer, and B. Eckmann, editors, Seminar on Triples and Categorical Homology Theory, pages 91-118, Berlin, Heidelberg, 1969. Springer Berlin Heidelberg. Google Scholar
  25. Ernie Manes and Philip Mulry. Monad compositions. I: General constructions and recursive distributive laws. Theory and Applications of Categories, 18:172-208, 2007. URL: https://eudml.org/doc/128434.
  26. E. Moggi. Computational lambda-calculus and monads. In [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science, pages 14-23, June 1989. URL: https://doi.org/10.1109/LICS.1989.39155.
  27. Anna B. Romanowska and Jonathan D. H. Smith. On the Structure of Barycentric Algebras. Houston Journal of Mathematics, 16(3):431-448, 1990. URL: https://www.math.uh.edu/~hjm/restricted/archive/v016n3/0431ROMANOWSKA.pdf.
  28. Ross Street. Weak distributive laws. Theory and Applications of Categories, 22(12):313-320, 2009. URL: http://www.tac.mta.ca/tac/volumes/22/12/22-12abs.html.
  29. Daniele Varacca and Glynn Winskel. Distributing probability over non-determinism. Mathematical Structures in Computer Science, 16(1):87-113, February 2006. URL: https://doi.org/10.1017/S0960129505005074.
  30. Harald Woracek and Ana Sokolova. Re: Decomposing convex subsets of simplices along free morphisms, June 2024. Google Scholar
  31. Maaike Zwart and Dan Marsden. No-Go Theorems for Distributive Laws. In 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 1-13, June 2019. URL: https://doi.org/10.1109/LICS.2019.8785707.
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