{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T21:42:34Z","timestamp":1769118154763,"version":"3.49.0"},"reference-count":16,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2020,8,11]],"date-time":"2020-08-11T00:00:00Z","timestamp":1597104000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Compositionality"],"abstract":"<jats:p>Markov categories are a recent category-theoretic approach to the foundations of probability and statistics. Here we develop this approach further by treating infinite products and the Kolmogorov extension theorem. This is relevant for all aspects of probability theory in which infinitely many random variables appear at a time. These infinite tensor products <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:munder><mml:mo>\u2a02<\/mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>i<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:munder><mml:msub><mml:mi>X<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><\/mml:math> come in two versions: a weaker but more general one for families of objects <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msub><mml:mi>X<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:msub><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>i<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>J<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math> in semicartesian symmetric monoidal categories, and a stronger but more specific one for families of objects in Markov categories.As a first application, we state and prove versions of the zero--one laws of Kolmogorov and Hewitt--Savage for Markov categories. This gives general versions of these results which can be instantiated not only in measure-theoretic probability, where they specialize to the standard ones in the setting of standard Borel spaces, but also in other contexts.<\/jats:p>","DOI":"10.32408\/compositionality-2-3","type":"journal-article","created":{"date-parts":[[2020,8,11]],"date-time":"2020-08-11T22:06:34Z","timestamp":1597183594000},"page":"3","source":"Crossref","is-referenced-by-count":20,"title":["Infinite products and zero-one laws in categorical probability"],"prefix":"10.46298","volume":"2","author":[{"given":"Tobias","family":"Fritz","sequence":"first","affiliation":[{"name":"Deparment of Mathematical Sciences, University of Copenhagen, 2100 Denmark"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eigil Fjeldgren","family":"Rischel","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, N2L 2Y5, Waterloo, Ontario, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2020,8,11]]},"reference":[{"key":"0","unstructured":"Erik Sparre Andersen and B\u00f8rge Jessen. On the introduction of measures in infinite product sets. Kongelige Danske Videnskabernes Selskab, Matematiske-Fysiske meddelelser, 25(4):8, 1948."},{"key":"1","doi-asserted-by":"crossref","unstructured":"Bruce E. Blackadar. Infinite tensor products of $C\\sp*$-algebras. Pacific Journal of Mathematics, 72(2):313-334, 1977.","DOI":"10.2140\/pjm.1977.72.313"},{"key":"2","doi-asserted-by":"publisher","unstructured":"Kenta Cho and Bart Jacobs. Disintegration and Bayesian inversion via string diagrams. Mathematical Structures in Computer Science, 29:938-971, 2019. https:\/\/doi.org\/10.1017\/S0960129518000488.","DOI":"10.1017\/S0960129518000488"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih. Hodge cycles, motives, and Shimura varieties, volume 900 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1982. https:\/\/doi.org\/10.1007\/978-3-540-38955-2.","DOI":"10.1007\/978-3-540-38955-2"},{"key":"4","unstructured":"D. H. Fremlin. Measure theory. Vol. 4. Torres Fremlin, Colchester, 2006. Topological measure spaces. Part I, II, Corrected second printing of the 2003 original. https:\/\/www1.essex.ac.uk\/maths\/people\/fremlin\/mt.htmwww1.essex.ac.uk\/maths\/people\/fremlin\/mt.htm."},{"key":"5","doi-asserted-by":"publisher","unstructured":"Tobias Fritz. A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. https:\/\/doi.org\/10.1016\/j.aim.2020.107239.","DOI":"10.1016\/j.aim.2020.107239"},{"key":"6","unstructured":"Tobias Fritz, Paolo Perrone, and Sharwin Rezagholi. Probability, valuations, hyperspace: Three monads on top and the support as a morphism. ArXiv, abs\/1910.03752, 2019."},{"key":"7","unstructured":"Malte Gerhold, Stephanie Lachs, and Michael Sch\u00fcrmann. Categorial L\u00e9vy processes. https:\/\/arxiv.org\/abs\/1612.05139arXiv:1612.05139."},{"key":"8","doi-asserted-by":"publisher","unstructured":"Mich\u00e8le Giry. A categorical approach to probability theory. In Categorical aspects of topology and analysis (Ottawa, Ont., 1980), volume 915 of Lecture Notes in Mathematics, pages 68-85. Springer, 1982. https:\/\/doi.org\/10.1007\/BFb0092872.","DOI":"10.1007\/BFb0092872"},{"key":"9","unstructured":"Peter V. Golubtsov. Axiomatic description of categories of information converters. Problemy Peredachi Informatsii, 35(3):80-98, 1999. In Russian. English translation in Problems of Information Transmission 35(3):259-274, 1999."},{"key":"10","doi-asserted-by":"publisher","unstructured":"Edwin Hewitt and Leonard J. Savage. Symmetric measures on cartesian products. Transactions of the American Mathematical Society, 80(2):470-470, February 1955. https:\/\/doi.org\/10.1090\/s0002-9947-1955-0076206-8.","DOI":"10.1090\/s0002-9947-1955-0076206-8"},{"key":"11","doi-asserted-by":"publisher","unstructured":"Georg Karner. Continuous monoids and semirings. Theoretical Computer Science, 318(3):355-372, 2004. https:\/\/doi.org\/10.1016\/j.tcs.2004.01.020.","DOI":"10.1016\/j.tcs.2004.01.020"},{"key":"12","doi-asserted-by":"publisher","unstructured":"Achim Klenke. Probability theory, a comprehensive course. Universitext. Springer, London, second edition, 2014. https:\/\/doi.org\/10.1007\/978-1-4471-5361-0.","DOI":"10.1007\/978-1-4471-5361-0"},{"key":"13","doi-asserted-by":"crossref","unstructured":"A. N. Kolmogorov. Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin, 1933.","DOI":"10.1007\/978-3-642-49888-6"},{"key":"14","doi-asserted-by":"publisher","unstructured":"D. Ramachandran. Perfect mixtures of perfect measures. Annals of Probability, 7(3):444-452, 1979. https:\/\/doi.org\/10.1214\/aop\/1176995045.","DOI":"10.1214\/aop\/1176995045"},{"key":"15","unstructured":"Andrea Schalk. Algebras for generalized power constructions. PhD thesis, University of Darmstadt, 1993. Available at www.cs.man.ac.uk\/ schalk\/publ\/diss.ps.gzwww.cs.man.ac.uk\/ schalk\/publ\/diss.ps.gz."}],"container-title":["Compositionality"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/compositionality-journal.org\/papers\/compositionality-2-3\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2024,6,3]],"date-time":"2024-06-03T20:22:42Z","timestamp":1717446162000},"score":1,"resource":{"primary":{"URL":"https:\/\/compositionality.episciences.org\/13509"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,11]]},"references-count":16,"URL":"https:\/\/doi.org\/10.32408\/compositionality-2-3","relation":{},"ISSN":["2631-4444"],"issn-type":[{"value":"2631-4444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,11]]},"article-number":"3"}}