Skip to main content
SpringerLink
Log in

Fraïssé Limits in Comma Categories

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

Fraïssé’s theorem characterizing the existence of universal homogeneous structures is a cornerstone of model theory. A categorical version of these results was developed by Droste and Göbel. Such an abstract version of Fraïssé theory allows to construct unusual objects that are far away from the usual structures. In this paper we are going to derive sufficient conditions for a comma category to contain universal homogeneous objects. Using this criterion, we characterize homogeneous structures that possess universal homogeneous endomorphisms. The existence of such endomorphisms helps to reduce questions about the full endomorphism monoid to the self-embedding monoid of the structure. As another application we characterize the retracts of homogeneous structures that are induced by universal homogeneous retractions. This extends previous results by Bonato, Delić, Mudrinski, Dolinka, and Kubiś.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Banaschewski, B., Herrlich, H.: Subcategories defined by implications. Houston J. Math. 2, 149–171 (1976)

    MathSciNet  MATH  Google Scholar 

  2. Behrisch, M., Truss, J.K., Vargas-García, E.: Reconstructing the topology on monoids and polymorphism clones of the rationals. Stud. Logica. 105(1), 65–91 (2017). https://doi.org/10.1007/s11225-016-9682-z

    Article  MathSciNet  MATH  Google Scholar 

  3. Bodirsky, M., Pinsker, M., Pongrácz, A.: Reconstructing the topology of clones. Trans. Am. Math. Soc. 369(5), 3707–3740 (2017). https://doi.org/10.1090/tran/6937

    Article  MathSciNet  MATH  Google Scholar 

  4. Bonato, A., Delić, D.: The monoid of the random graph. Semigroup Forum 61(1), 138–148 (2000). https://doi.org/10.1007/PL00006009

    Article  MathSciNet  MATH  Google Scholar 

  5. Borceux, F.: Handbook of Categorical Algebra (vol. 1), Encyclopedia of Mathematics and Its Applications, vol. 50. Cambridge University Press, Cambridge (1994). https://doi.org/10.1017/CBO9780511525858

    Google Scholar 

  6. Cabello Sánchez, F., Garbulińska-Wȩgrzyn, J., Kubiś, W.: Quasi-Banach spaces of almost universal disposition. J. Funct. Anal. 267(3), 744–771 (2014). https://doi.org/10.1016/j.jfa.2014.05.005

    Article  MathSciNet  MATH  Google Scholar 

  7. Dolinka, I.: A characterization of retracts in certain Fraïssé limits. Math. Log. Q. 58(1–2), 46–54 (2012). https://doi.org/10.1002/malq.201020084

    Article  MathSciNet  MATH  Google Scholar 

  8. Dolinka, I.: The Bergman property for endomorphism monoids of some Fraïssé limits. Forum Math. 26(2), 357–376 (2014). https://doi.org/10.1515/form.2011.153

    Article  MathSciNet  MATH  Google Scholar 

  9. Dovgoshey, O., Martio, O., Ryazanov, V., Vuorinen, M.: The Cantor function. Expo. Math. 24(1), 1–37 (2006). https://doi.org/10.1016/j.exmath.2005.05.002

    Article  MathSciNet  MATH  Google Scholar 

  10. Droste, M., Göbel, R.: A categorical theorem on universal objects and its application in Abelian group theory and computer science. In: Bokut’, L.A., Ershov, Y.L., Kostrikin, A.I. (eds.) Proceedings of International Conference on Memory A. I. Mal’cev, Novosibirsk/USSR 1989, Contemp. Math., vol. 131(3), pp. 49–74. AMS, Providence (1992)

  11. Erdős, P., Rényi, A.: Asymmetric graphs. Acta Math. Acad. Sci. Hungar 14, 295–315 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fraïssé, R.: Sur certaines relations qui généralisent l’ordre des nombres rationnels. C. R. Acad. Sci. Paris 237, 540–542 (1953)

    MathSciNet  MATH  Google Scholar 

  13. Hodges, W.: A Shorter Model Theory. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  14. Kubiś, W.: Fraïssé sequences: category-theoretic approach to universal homogeneous structures. Ann. Pure Appl. Logic 165(11), 1755–1811 (2014). https://doi.org/10.1016/j.apal.2014.07.004

    Article  MathSciNet  MATH  Google Scholar 

  15. Kubiś, W.: Injective objects and retracts of Fraïssé limits. Forum Math. 27(2), 807–842 (2015). https://doi.org/10.1515/forum-2012-0081

    MathSciNet  MATH  Google Scholar 

  16. Mac Lane, S.: Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5, 2nd edn. Springer, New York (1998)

    MATH  Google Scholar 

  17. Macpherson, D.: A survey of homogeneous structures. Discrete Math. 311(15), 1599–1634 (2011). https://doi.org/10.1016/j.disc.2011.01.024

    Article  MathSciNet  MATH  Google Scholar 

  18. Mudrinski, N.: Notes on endomorphisms of Henson graphs and their complements. Ars Combin. 96, 173–183 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Pech, C., Pech, M.: Universal homogeneous constraint structures and the hom-equivalence classes of weakly oligomorphic structures. ArXiv e-prints (2012)

  20. Pech, C., Pech, M.: On automatic homeomorphicity for transformation monoids. Monatsh. Math. 179(1), 129–148 (2016). https://doi.org/10.1007/s00605-015-0767-y

    Article  MathSciNet  MATH  Google Scholar 

  21. Pech, C., Pech, M.: Polymorphism clones of homogeneous structures (gate coverings and automatic homeomorphicity). Algebra Universalis (2018, To appear)

  22. Rado, R.: Universal graphs and universal functions. Acta Arith. 9, 331–340 (1964)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Radebeul, Germany

    Christian Pech

  2. Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia

    Maja Pech

Authors
  1. Christian Pech
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maja Pech
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maja Pech.

Additional information

Communicated by J. Rosicky.

The second author was supported by the Ministry of Education and Science of the Republic of Serbia through Grant No. 174018.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pech, C., Pech, M. Fraïssé Limits in Comma Categories. Appl Categor Struct 26, 799–820 (2018). https://doi.org/10.1007/s10485-018-9519-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-018-9519-1

Keywords

Mathematics Subject Classification

Search

Navigation