The Vietoris Monad and Weak Distributive Laws

  • Richard GarnerEmail author


The Vietoris monad on the category of compact Hausdorff spaces is a topological analogue of the power-set monad on the category of sets. Exploiting Manes’ characterisation of the compact Hausdorff spaces as algebras for the ultrafilter monad on sets, we give precise form to the above analogy by exhibiting the Vietoris monad as induced by a weak distributive law, in the sense of Böhm, of the power-set monad over the ultrafilter monad.


Vietoris hyperspace Monads Distributive laws Weak distributive laws Continuous lattices 


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  1. 1.
    Barr, M.: Relational algebras. In: Reports of the Midwest Category Seminar, IV. Lecture Notes in Mathematics, vol. 137, pp. 39–55. Springer (1970)Google Scholar
  2. 2.
    Beck, J.: Distributive laws. In: Seminar on Triples and Categorical Homology Theory (Zürich, 1966/1967), vol. 80, pp. 119–140. Lecture Notes in Mathematics. Springer (1969)Google Scholar
  3. 3.
    Böhm, G.: The weak theory of monads. Adv. Math. 225, 1–32 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Clementino, M.M., Hofmann, D., Janelidze, G.: The monads of classical algebra are seldom weakly cartesian. J. Homotopy Relat. Struct. 9, 175–197 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Clementino, M.M., Hofmann, D., Tholen, W.: One setting for all: metric, topology, uniformity, approach structure. Appl. Categorical Struct. 12, 127–154 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Day, A.: Filter monads, continuous lattices and closure systems. Can. J. Math. Journal Canadien de Mathématiques 27, 50–59 (1975)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Flagg, B.: Algebraic theories of compact pospaces. Topol. Its Appl. 77, 277–290 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W., Scott, D.S.: A Compendium of Continuous Lattices. Springer, Berlin (1980) CrossRefGoogle Scholar
  9. 9.
    Hofmann, D.: The enriched Vietoris monad on representable spaces. J. Pure Appl. Algebra 218, 2274–2318 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hofmann, D., Seal, G.J., Tholen, W.: Lax algebras. In: Monoidal Topology, vol. 153, pp. 145–283. Encyclopedia of Mathematics and its Applications. CUP (2014)Google Scholar
  11. 11.
    Hofmann, D., Seal, G.J., Tholen, W., Eds.: Monoidal Topology, vol. 153 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge. A categorical approach to order, metric, and topology (2014)Google Scholar
  12. 12.
    Hyland, M., Nagayama, M., Power, J., Rosolini, G.: A category theoretic formulation for engeler-style models of the untyped lambda. Electronic Notes Theor. Comput. Sci. 161, 43–57 (2006)CrossRefGoogle Scholar
  13. 13.
    Johnstone, P., Power, J., Tsujishita, T., Watanabe, H., Worrell, J.: On the structure of categories of coalgebras. Theoret. Comput. Sci. 260, 87–117 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jónsson, B., Tarski, A.: Boolean algebras with operators. I. Am. J. Math. 73, 891–939 (1951)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Joyal, A.: Foncteurs analytiques et espèces de structures. In Combinatoire énumérative (Montreal, 1985), vol. 1234, pp. 126–159. Lecture Notes in Mathematics. Springer (1986)Google Scholar
  16. 16.
    Klin, B., Salamanca, J.: Iterated covariant powerset is not a monad. In: Proceedings of the Thirty-Fourth Conference on the Mathematical Foundations of Programming Semantics (MFPS XXXIV), vol. 341, pp. 261–276. Electron. Notes Theor. Comput. Sci. Elsevier, Amsterdam (2018)Google Scholar
  17. 17.
    Kurz, A., Velebil, J.R.: Relation lifting, a survey. J. Logical Algebraic Methods Program. 85, 475–499 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Manes, E.: A triple theoretic construction of compact algebras. In: Sem. on Triples and Categorical Homology Theory (ETH, Zürich, 1966/1967, pp. 91–118. Springer (1969)Google Scholar
  19. 19.
    Manes, E.: Algebraic Teories, vol. 26 of Graduate Texts in Mathematics. Springer (1976)Google Scholar
  20. 20.
    Meyer, J.-P.: Induced functors on categories of algebras. Math. Z. 142, 1–14 (1975)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Scott, D.: Continuous lattices. In:Toposes, Algebraic Geometry and Logic, vol. 274, pp. 97–136. Lecture Notes in Mathematics, Springer (1972)Google Scholar
  22. 22.
    Smyth, M.B.: Power domains and predicate transformers: a topological view. In: Automata, Languages and Programming (Barcelona, 1983), vol. 154, pp. 662–675. Lecture Notes in Comput. Sci. Springer (1983)Google Scholar
  23. 23.
    Street, R.: Weak distributive laws. Theory Appl. Categories 22, 313–320 (2009)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Tholen, W.: Quantalic topological theories. Tbilisi Math. J. 10(3), 223–237 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tholen, W.: Lax distributive laws for topology, I. Cahiers de Topologie et Géométrie Différentielle Catégoriques 60, 311–364 (2019)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Trnková, V.: Relational automata in a category and their languages. In: Fundamentals of Computation Theory (Proc. Internat. Conf., Poznań-Kórnik, 1977), vol. 56, pp. 340–355. Lecture Notes in Comput. Sci. (1977)Google Scholar
  27. 27.
    Vietoris, L.: Bereiche zweiter Ordnung. Monatshefte für Mathematik und Physik 32, 258–280 (1922)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wyler, O.: Algebraic theories of continuous lattices. In: Banaschewski, B., Hoffmann, R.-E. (eds.) Proceedings of the Conference on Topological and Categorical Aspects of Continuous Lattices. vol. 871 of Lecture Notes in Mathematics, Springer (1981)Google Scholar
  29. 29.
    Wyler, O.: Algebraic theories for continuous semilattices. Arch. Ration. Mech. Anal. 90, 99–113 (1985)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Zhao, X.: Idempotent semirings with a commutative additive reduct. Semigroup Forum 64, 289–296 (2002)MathSciNetCrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Centre of Australian Category TheoryMacquarie UniversitySydneyAustralia

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