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Published in: Applied Categorical Structures 1/2021

28-09-2020

The Kechris–Pestov–Todorčević Correspondence from the Point of View of Category Theory

Author: Dragan Mašulović

Published in: Applied Categorical Structures | Issue 1/2021

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Abstract

The Kechris–Pestov–Todorčević correspondence (KPT-correspondence for short) is a surprising correspondence between model theory, combinatorics and topological dynamics. In this paper we present a categorical re-interpretation of (a part of) the KPT-correspondence with the aim of proving a dual statement. Our strategy is to take a “direct” result and then analyze the necessary infrastructure that makes the result true by providing a purely categorical proof of the categorical version of the result. We can then capitalize on the Duality Principle to obtain the dual statements almost for free. We believe that the dual version of the KPT-correspondence can not only provide the new insights into the interplay of combinatorial, model-theoretic and topological phenomena this correspondence binds together, but also explores the limits to which categorical treatment of combinatorial phenomena can take us.

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Literature
1.
go back to reference Bartošová, D.: Universal minimal flows of groups of automorphisms of uncountable structures. Canadian Mathematical Bulletin (2012) Bartošová, D.: Universal minimal flows of groups of automorphisms of uncountable structures. Canadian Mathematical Bulletin (2012)
2.
go back to reference Bartošová, D.: More universal minimal flows of groups of automorphisms of uncountable structures. Asymptotic geometric analysis. In: Ludwig, M., Milman, V., Pestov, V., Tomczak-Jaegermann, N. (eds.)Proceedings of the Fall 2010 Fields Institute Thematic Program, vol. 68. Fields Institute Communications Springer (2013) Bartošová, D.: More universal minimal flows of groups of automorphisms of uncountable structures. Asymptotic geometric analysis. In: Ludwig, M., Milman, V., Pestov, V., Tomczak-Jaegermann, N. (eds.)Proceedings of the Fall 2010 Fields Institute Thematic Program, vol. 68. Fields Institute Communications Springer (2013)
4.
go back to reference Ferenczi, L., Lopez-Abad, J., Mbombo, B., Todorcevic, S.: Amalgamation and Ramsey properties of \(L_p\) spaces. Preprint (arXiv:1903.05504) Ferenczi, L., Lopez-Abad, J., Mbombo, B., Todorcevic, S.: Amalgamation and Ramsey properties of \(L_p\) spaces. Preprint (arXiv:​1903.​05504)
5.
go back to reference Graham, R.L., Leeb, K., Rothschild, B.L.: Ramsey’s theorem for a class of categories. Adv. Math. 8, 417–443 (1972). errata 10, 326–327 (1973)MathSciNetCrossRef Graham, R.L., Leeb, K., Rothschild, B.L.: Ramsey’s theorem for a class of categories. Adv. Math. 8, 417–443 (1972). errata 10, 326–327 (1973)MathSciNetCrossRef
6.
go back to reference Irwin, T., Solecki, S.: Projective Fraïssé limits and the pseudo-arc. Trans. Am. Math. Soc. 358(7), 3077–3096 (2006)CrossRef Irwin, T., Solecki, S.: Projective Fraïssé limits and the pseudo-arc. Trans. Am. Math. Soc. 358(7), 3077–3096 (2006)CrossRef
7.
go back to reference Kechris, A.S., Pestov, V.G., Todorčević, S.: Fraïssé limits, Ramsey theory and topological dynamics of automorphism groups. GAFA Geom. Funct. Anal. 15, 106–189 (2005)CrossRef Kechris, A.S., Pestov, V.G., Todorčević, S.: Fraïssé limits, Ramsey theory and topological dynamics of automorphism groups. GAFA Geom. Funct. Anal. 15, 106–189 (2005)CrossRef
8.
go back to reference Krupiński, K., Pillay, A.: On the topological dynamics of automorphism groups, a model-theoretic perspective. Preprint arXiv:1909.13285 Krupiński, K., Pillay, A.: On the topological dynamics of automorphism groups, a model-theoretic perspective. Preprint arXiv:​1909.​13285
9.
go back to reference Leeb, K.: The Categories of Combinatorics. Combinatorial Structures and Their Applications. Gordon and Breach, New York (1970)MATH Leeb, K.: The Categories of Combinatorics. Combinatorial Structures and Their Applications. Gordon and Breach, New York (1970)MATH
10.
go back to reference Mašulović, D., Scow, L.: Categorical equivalence and the Ramsey property for finite powers of a primal algebra. Algebra Universalis 78, 159–179 (2017)MathSciNetCrossRef Mašulović, D., Scow, L.: Categorical equivalence and the Ramsey property for finite powers of a primal algebra. Algebra Universalis 78, 159–179 (2017)MathSciNetCrossRef
11.
12.
go back to reference Nguyen, L., Thé, V.: More on the Kechris–Pestov–Todorcevic correspondence: precompact expansions. Fund. Math. 222, 19–47 (2013) Nguyen, L., Thé, V.: More on the Kechris–Pestov–Todorcevic correspondence: precompact expansions. Fund. Math. 222, 19–47 (2013)
13.
14.
15.
go back to reference Prömel, H.J., Voigt, B.: Hereditary attributes of surjections and parameter sets. Eur. J. Combin. 7, 161–170 (1986)MathSciNetCrossRef Prömel, H.J., Voigt, B.: Hereditary attributes of surjections and parameter sets. Eur. J. Combin. 7, 161–170 (1986)MathSciNetCrossRef
16.
go back to reference Zucker, A.: Topological dynamics of automorphism groups, ultrafilter combinatorics, and the generic point problem. Trans. Am. Math. Soc. 368, 6715–6740 (2016)MathSciNetCrossRef Zucker, A.: Topological dynamics of automorphism groups, ultrafilter combinatorics, and the generic point problem. Trans. Am. Math. Soc. 368, 6715–6740 (2016)MathSciNetCrossRef
Metadata
Title
The Kechris–Pestov–Todorčević Correspondence from the Point of View of Category Theory
Author
Dragan Mašulović
Publication date
28-09-2020
Publisher
Springer Netherlands
Published in
Applied Categorical Structures / Issue 1/2021
Print ISSN: 0927-2852
Electronic ISSN: 1572-9095
DOI
https://doi.org/10.1007/s10485-020-09611-z

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