{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T22:49:35Z","timestamp":1769122175324,"version":"3.49.0"},"reference-count":59,"publisher":"Cambridge University Press (CUP)","issue":"8","license":[{"start":{"date-parts":[[2022,3,8]],"date-time":"2022-03-08T00:00:00Z","timestamp":1646697600000},"content-version":"unspecified","delay-in-days":188,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2021,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider three monads on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129521000414_inline1.png\"\/><jats:tex-math>\n$\\mathsf{Top}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad <jats:italic>H<\/jats:italic>, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second one is the monad <jats:italic>V<\/jats:italic> of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129521000414_inline2.png\"\/><jats:tex-math>\n$V \\to H$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. In particular, this implies that every <jats:italic>H<\/jats:italic>-algebra (topological complete semilattice) is also a <jats:italic>V<\/jats:italic>-algebra. We show that <jats:italic>V<\/jats:italic> can be restricted to a submonad of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129521000414_inline3.png\"\/><jats:tex-math>\n$\\tau$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-smooth probability measures on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129521000414_inline4.png\"\/><jats:tex-math>\n$\\mathsf{Top}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. By composing these morphisms of monads, we obtain that taking the supports of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129521000414_inline5.png\"\/><jats:tex-math>\n$\\tau$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-smooth probability measures is also a morphism of monads.<\/jats:p>","DOI":"10.1017\/s0960129521000414","type":"journal-article","created":{"date-parts":[[2022,3,8]],"date-time":"2022-03-08T02:44:25Z","timestamp":1646707465000},"page":"850-897","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":6,"title":["Probability, valuations, hyperspace: Three monads on top and the support as a morphism"],"prefix":"10.1017","volume":"31","author":[{"given":"Tobias","family":"Fritz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Paolo","family":"Perrone","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1090-0240","authenticated-orcid":false,"given":"Sharwin","family":"Rezagholi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2022,3,8]]},"reference":[{"key":"S0960129521000414_ref9","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1972.42.569"},{"key":"S0960129521000414_ref32","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2011.04.005"},{"key":"S0960129521000414_ref24","article-title":"Non-Hausdorff Topology and Domain Theory, Cambridge University","author":"Goubault-Larrecq","year":"2013","journal-title":"Press."},{"key":"S0960129521000414_ref14","unstructured":"Escard\u00d3, M. and Heckmann, R. 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In: Proceedings of the Fourth Annual Symposium of Logics in Computer Science."},{"key":"S0960129521000414_ref36","doi-asserted-by":"publisher","DOI":"10.1007\/s00233-008-9078-0"},{"key":"S0960129521000414_ref21","article-title":"Continuous Lattices and Domains, Cambridge University","author":"Gierz","year":"2003","journal-title":"Press."},{"key":"S0960129521000414_ref30","doi-asserted-by":"publisher","DOI":"10.1007\/BF01507294"},{"key":"S0960129521000414_ref47","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(82)90004-4"},{"key":"S0960129521000414_ref10","author":"Cohen","year":"2006"},{"key":"S0960129521000414_ref26","unstructured":"Hausdorff, F. (1914). Grundz\u00dcge der Mengenlehre, Veit und Co., German."},{"key":"S0960129521000414_ref52","author":"Smyth","year":"1983"},{"key":"S0960129521000414_ref29","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1978-0515540-7"},{"key":"S0960129521000414_ref59","doi-asserted-by":"publisher","DOI":"10.1007\/BF01214522"},{"key":"S0960129521000414_ref35","doi-asserted-by":"publisher","DOI":"10.1016\/j.entcs.2004.10.001"},{"key":"S0960129521000414_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/j.entcs.2018.11.007"},{"key":"S0960129521000414_ref54","first-page":"39","article-title":"Monadic functors and convexity","volume":"22","author":"Swirszcz","year":"1974","journal-title":"Bulletin de l\u2019Aca\u00c9mie Polonaise des Sciences: S\u00c9rie des sciences mathematiques, astronomique et physique"},{"key":"S0960129521000414_ref19","first-page":"170","article-title":"A probability monad as the colimit of spaces of finite samples","volume":"34","author":"Fritz","year":"2019","journal-title":"Theory and Applications of Categories"},{"key":"S0960129521000414_ref40","first-page":"97","article-title":"Commutative monads as a theory of distributions","volume":"26","author":"Kock","year":"2012","journal-title":"Theory and Applications of Categories"},{"key":"S0960129521000414_ref39","unstructured":"Kirch, O. (1993). Bereiche und Bewertungen. German, MA thesis. Technische Hochschule, Darmstadt."},{"key":"S0960129521000414_ref27","unstructured":"Heckmann, R. (1995). Spaces of valuations. Working paper version. Available at http:\/\/citeseerx.ist.psu.edu\/viewdoc\/download?doi=10.1.1.45.5845."},{"key":"S0960129521000414_ref46","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1951-0042109-4"},{"key":"S0960129521000414_ref45","author":"Manes","year":"2003"},{"key":"S0960129521000414_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-8641(01)00249-8"},{"key":"S0960129521000414_ref51","unstructured":"Schalk, A. (1993). Algebras for Generalized Power Constructions. Available at www.cs.man.ac.uk\/schalk\/publ\/diss.ps.gz. Phd thesis. University of Darmstadt."},{"key":"S0960129521000414_ref56","unstructured":"van Breugel, F. (2005). The Metric Monad for Probabilistic Nondeterminism. 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