Abstract
We develop a theory of Katětov functors which provide a uniform way of constructing Fraïssé limits. Among applications, we present short proofs and improvements of several recent results on the structure of the group of automorphisms and the semigroup of endomorphisms of some Fraïssé limits.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adámek, J., Herrlich, H., Rosický, J., Tholen, W.: On a generalized small-object argument for the injective subcategory problem. Cah. Topol. Géom. Différ. Catég. 43(2), 83–106 (2002)
Ben Yaacov, I.: Fraïssé limits of metric structures. J. Symb. Log. 80, 100–115 (2015)
Ben Yaacov, I.: The linear isometry group of the Gurarij space is universal. Proc. Amer. Math. Soc. 142, 2459–2467 (2014)
Bilge, D., Melleray, J.: Elements of finite order in automorphism groups of homogeneous structures. Contrib. Discrete Math. 8, 88–119 (2013)
Bonato, A., Delić, D., Dolinka, I.: All countable monoids embed into the monoid of the infinite random graph. Discrete Math. 310, 373–375 (2010)
Cameron, P., Nešetřil, J.: Homomorphism-homogeneous relational structures. Combin. Probab. Comput. 15, 91–103 (2006)
Caramello, O.: Fraïssé’s construction from a topos-theoretic perspective. Log. Univers. 8, 261–281 (2014)
Dolinka, I.: The Bergman property for endomorphism monoids of some Fraïssé limits. Forum Math. 26, 357–376 (2014)
Dolinka, I.: The endomorphism monoid of the random poset contains all countable semigroups. Algebra Univers. 56, 469–474 (2007)
Dolinka, I., Mašulović, D.: A universality result for endomorphism monoids of some ultrahomogeneous structures. Proc. Edinb. Math. Soc. 55, 635–656 (2012)
Droste, M., Göbel, R.: A categorical theorem on universal objects and its application in abelian group theory and computer science. In: Proceedings of the International Conference on Algebra, Part 3, Novosibirsk, 1989, Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, pp. 49–74 (1992)
Fraïssé, R.: Sur l’extension aux relations de quelques propriétés des ordres. Ann. Sci. Ecole Norm. Sup. 71(3), 363–388 (1954)
Grätzer, G.: Universal algebra, 2nd ed. Springer-Verlag, New York (2008)
Grebík, J.: An example of a Fraïssé class without a Katětov functor. arXiv:1604.00358
Hall, P.: Some constructions for locally finite groups. J. London Math. Soc. 34, 305–319 (1959)
Henson, C.W.: A family of countable homogeneous graphs. Pacific J. Math. 38, 69–83 (1971)
Herwig, B., Lascar, D.: Extending partial automorphisms and the profinite topology on free groups. Trans. Amer. Math. Soc 352, 1985–2021 (2000)
Irwin, T., Solecki, S.: Projective Fraïssé limits and the pseudo-arc. Trans. Amer. Math. Soc. 358, 3077–3096 (2006)
Katětov, M.: On universal metric spaces. General topology and its relations to modern analysis and algebra. VI (Prague, 1986), Res. Exp. Math., vol. 16, pp. 323–330. Heldermann, Berlin (1988)
Kechris, A.S., Pestov, V.G., Todorcevic, S.: Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups. Geom. Funct. Anal. 15, 106–189 (2005)
Koubek, V., Reiterman, J.: Categorical constructions of free algebras, colimits, and completions of partial algebras. J. Pure Appl. Algebra 14(2), 195–231 (1979)
Kubiś, W.: Injective objects and retracts of Fraïssé limits. Forum Math. 27, 807–842 (2015)
Kubiś, W.: Fraïssé sequences: category-theoretic approach to universal homogeneous structures. Ann. Pure Appl. Logic 165, 1755–1811 (2014)
Mac Lane, S.: Categories for the working mathematician, 2nd ed. Springer (1978)
Maltcev, V., Mitchell, J.D., Ruškuc, N.: The Bergman property for semigroups. J. London Math. Soc. 80, 212–232 (2009)
Péresse, Y.: Generating uncountable transformation semigroups. Ph.D. thesis, University of St Andrews (2009)
Solecki, S.: Notes on a strengthening of the Herwig-Lascar extension theorem. 2009. Unpublished note, available at http://www.math.uiuc.edu/ssolecki/papers/HervLascfin.pdf
Urysohn, P.S.: Sur un espace métrique universel, I, II. Bull. Sci. Math. 51(2), 43–64, 74–90 (1927). JFM 53.0556.01
Uspenskij, V.V.: On the group of isometries of the Urysohn universal metric space. Commentat. Math. Univ. Carolinae 31, 181–182 (1990)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kubiś, W., Mašulović, D. Katětov Functors. Appl Categor Struct 25, 569–602 (2017). https://doi.org/10.1007/s10485-016-9461-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-016-9461-z