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Categorial Independence and Lévy Processes
Symmetry, integrability and geometry: methods and applications, Tome 18 (2022) Cet article a éte moissonné depuis la source Math-Net.Ru

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We generalize Franz' independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit object is an initial object, in which case the inclusions can be defined starting from the tensor category alone. The obtained independence for morphisms is called categorial independence. We define categorial Lévy processes on every tensor category with initial unit object and present a construction generalizing the reconstruction of a Lévy process from its convolution semigroup via the Daniell–Kolmogorov theorem. Finally, we discuss examples showing that many known independences from algebra as well as from (noncommutative) probability are special cases of categorial independence.
Keywords: general independence, monoidal categories, synthetic probability, noncommutative probability, quantum stochastic processes. 
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Malte Gerhold; Stephanie Lachs; Michael Schürmann. Categorial Independence and Lévy Processes. Symmetry, integrability and geometry: methods and applications, Tome 18 (2022). http://geodesic.mathdoc.fr/item/SIGMA_2022_18_a74/

[1] Adámek J., Herrlich H., Strecker G., Abstract and concrete categories: the joy of cats, 2006 http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf

[2] Aguiar M., Mahajan S., Monoidal functors, species and Hopf algebras, CRM Monogr. Ser., 29, Amer. Math. Soc., Providence, RI, 2010 | DOI | MR

[3] Albandik S., Meyer R., “Product systems over Ore monoids”, Doc. Math., 20 (2015), arXiv: 1502.07768 | DOI | MR

[4] Ben Ghorbal A., Schürmann M., “Non-commutative notions of stochastic independence”, Math. Proc. Cambridge Philos. Soc., 133 (2002), 531–561 | DOI | MR

[5] Ben Ghorbal A., Schürmann M., “Quantum Lévy processes on dual groups”, Math. Z., 251 (2005), 147–165 | DOI | MR

[6] Bhat B.V.R., Mukherjee M., “Inclusion systems and amalgamated products of product systems”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 13 (2010), 1–26, arXiv: 0907.0095 | DOI | MR

[7] Bhat B.V.R., Skeide M., “Tensor product systems of Hilbert modules and dilations of completely positive semigroups”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 3 (2000), 519–575 | DOI | MR

[8] Bożejko M., Leinert M., Speicher R., “Convolution and limit theorems for conditionally free random variables”, Pacific J. Math., 175 (1996), 357–388, arXiv: funct-an/9410004 | DOI | MR

[9] Franz U., “Lévy processes on quantum groups and dual groups”, Quantum Independent Increment Processes. II, Lecture Notes in Math., 1866, Springer, Berlin, 2006, 161–257 | DOI | MR

[10] Franz U., Skalski A., Noncommutative mathematics for quantum systems, Cambridge-IISc Series, Cambridge University Press, Delhi, 2016 | DOI | MR

[11] Fritz T., “A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics”, Adv. Math., 370 (2020), 107239, 105 pp., arXiv: 1908.07021 | DOI | MR

[12] Gerhold M., On several problems in the theory of comonoidal systems and subproduct systems, Ph.D. Thesis, Universität Greifswald, 2015 https://nbn-resolving.org/urn:nbn:de:gbv:9-002244-6

[13] Gerhold M., “Bimonotone Brownian motion”, Proceedings of QP 38, Special Issue Dedicated to Professor Accardi, Professor Volovich and a Memorial Issue to Professor Hida, QP–PQ: Quantum Probab. White Noise Anal., 32, World Sci. Publ. (to appear) , arXiv: 1708.03510

[14] Gerhold M., Schoenberg correspondence for multifaced independence, arXiv: 2104.02985

[15] Gerhold M., “Additive deformations of Hopf algebras”, J. Algebra, 339 (2011), 114–134, arXiv: 1006.0847 | DOI | MR

[16] Gerhold M., Hasebe T., Ulrich M., Towards a classification of multi-faced independence: a representation-theoretic approach, arXiv: 2111.07649

[17] Gerhold M., Kietzmann S., Lachs S., “Additive deformations of braided Hopf algebras”, Noncommutative Harmonic Analysis with Applications to Probability III, Banach Center Publ., 96, Polish Acad. Sci. Inst. Math., Warsaw, 2012, 175–191, arXiv: 1104.1559 | DOI | MR

[18] Gerhold M., Lachs S., “Classification and GNS-construction for general universal products”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 18 (2015), 1550004, 29 pp. | DOI | MR

[19] Gerhold M., Skeide M., “Subproduct systems and Cartesian systems; new results on factorial languages and their relations with other areas”, J. Stoch. Anal., 1 (2020), 5, 21 pp., arXiv: 1402.0198 | DOI | MR

[20] Grätzer G., Universal algebra, 2nd ed., Springer-Verlag, New York –Heidelberg, 1979 | DOI | MR

[21] Gu Y., Hasebe T., Skoufranis P., “Bi-monotonic independence for pairs of algebras”, J. Theoret. Probab., 33 (2020), 533–566, arXiv: 1708.05334 | DOI | MR

[22] Gu Y., Skoufranis P., “Conditionally bi-free independence for pairs of faces”, J. Funct. Anal., 273 (2017), 1663–1733, arXiv: 1609.07475 | DOI | MR

[23] Gu Y., Skoufranis P., “Bi-Boolean independence for pairs of algebras”, Complex Anal. Oper. Theory, 13 (2019), 3023–3089, arXiv: 1703.03072 | DOI | MR

[24] Hasebe T., New associative product of three states generalizing free, monotone, anti-monotone, Boolean, conditionally free and conditionally monotone products, arXiv: 1009.1505

[25] Hasebe T., “Differential independence via an associative product of infinitely many linear functionals”, Colloq. Math., 124 (2011), 79–94 | DOI | MR

[26] Johnson R.E., “Unique factorization monoids and domains”, Proc. Amer. Math. Soc., 28 (1971), 397–404 | DOI | MR

[27] Kwaśniewski B.K., Szymański W., “Topological aperiodicity for product systems over semigroups of Ore type”, J. Funct. Anal., 270 (2016), 3453–3504, arXiv: 1312.7472 | DOI | MR

[28] Lachs S., A new family of universal products and aspects of a non-positive quantum probability theory, Ph.D. Thesis, Universität Greifswald, 2015 https://nbn-resolving.org/urn:nbn:de:gbv:9-002242-7

[29] Liu W., Free-free-Boolean independence for triples of algebras, arXiv: 1801.03401

[30] Liu W., “Free-Boolean independence for pairs of algebras”, J. Funct. Anal., 277 (2019), 994–1028, arXiv: 1710.01374 | DOI | MR

[31] Mac Lane S., Categories for the working mathematician, Grad. Texts in Math., 5, 2nd ed., Springer-Verlag, New York, 1998 | DOI | MR

[32] Manzel S., Schürmann M., “Non-commutative stochastic independence and cumulants”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 20 (2017), 1750010, 38 pp., arXiv: 1601.06779 | DOI | MR

[33] Marczewski E., “A general scheme of the notions of independence in mathematics”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys., 6 (1958), 731–736 | MR

[34] McLarty C., Elementary categories, elementary toposes, Oxford Logic Guides, 21, The Clarendon Press, Oxford University Press, New York, 1992 | MR

[35] Métivier M., “Limites projectives de mesures. Martingales. Applications”, Ann. Mat. Pura Appl., 63 (1963), 225–352 | DOI | MR

[36] Muraki N., “The five independences as natural products”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 6 (2003), 337–371 | DOI | MR

[37] nLab: Semicartesian monoidal category: Definition in terms of projections https://ncatlab.org/nlab/show/semicartesian+monoidal+category#definition_in_terms_of_projections

[38] Online discussion on golem: Monoidal categories with projections https://golem.ph.utexas.edu/category/2016/08/monoidal_categories_with_proje.html#c056710

[39] Rischel E.F., Fritz T., “Infinite products and zero-one laws in categorical probability”, Compositionality, 2 (2020), 20 pp., arXiv: 1912.02769 | DOI | MR

[40] Schürmann M., White noise on bialgebras, Lecture Notes in Math., 1544, Springer-Verlag, Berlin, 1993 | DOI | MR

[41] Shalit O., Skeide M., CP-Semigroups, dilations, and subproduct systems: The multi-parameter case and beyond, arXiv: 2003.05166

[42] Simpson A., “Category-theoretic structure for independence and conditional independence”, The Thirty-Third Conference on the Mathematical Foundations of Programming Semantics (MFPS XXXIII), Electron. Notes Theor. Comput. Sci., 336, Elsevier Sci. B.V., Amsterdam, 2018, 281–297 | DOI | MR

[43] Speicher R., “On universal products”, Free Probability Theory (Waterloo, ON, 1995), Fields Inst. Commun., 12, Amer. Math. Soc., Providence, RI, 1997, 257–266 | DOI | MR

[44] Voiculescu D.V., “Free probability for pairs of faces I”, Comm. Math. Phys., 332 (2014), 955–980, arXiv: 1306.6082 | DOI | MR

[45] Voiculescu D.V., “Free probability for pairs of faces II: 2-variables bi-free partial $R$-transform and systems with rank $\leq 1$ commutation”, Ann. Inst. Henri Poincaré Probab. Stat., 52 (2016), 1–15, arXiv: 1308.2035 | DOI | MR

[46] Voiculescu D.V., “Free probability for pairs of faces III: 2-variables bi-free partial $S$- and $T$-transforms”, J. Funct. Anal., 270 (2016), 3623–3638, arXiv: 1504.03765 | DOI | MR

[47] Wirth J., Formule de Lévy Khintchine et deformations d'algebres, Ph.D. Thesis, Universite Paris VI, 2002