Skip to main content
SpringerLink
Log in

Universality of Categories of Coalgebras

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

Under GCH, a set functor F does not preserve finite unions of non-empty sets if and only if the category Coalg F of all F-coalgebras is universal. Independently of GCH, we show that for any non-accessible functor F preserving intersections, the category Coalg F has a large discrete full subcategory, and we give an example of a category of F-coalgebras that is not universal, yet has a large discrete full subcategory.

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. Adámek, J., Rosický, J.: Locally Presentable and Accessible Categories. Cambridge University Press, Cambridge (1994)

    Book  MATH  Google Scholar 

  2. Adámek, J., Trnková, V.: Automata and Algebras in Categories. Kluwer, Dordrecht (1989)

    Google Scholar 

  3. Adams, M.E., Dziobiak, W.: Endomorphisms of monadic Boolean algebras. Algebra Univers. 57, 131–142 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Birkhoff, G.: On groups of automorphisms (in Spanish). Rev. Unión Mat. Argent. 11, 155–157 (1946)

    MathSciNet  MATH  Google Scholar 

  5. Dow, A., Hart, K.P.: The Čech–Stone compactification of \(\Bbb N\) and \(\Bbb R\). In: Hart, K.P., Nagata, J., Vaughan, J.P. (eds.) Encyclopedia of General Topology, pp. 213–217. Elsevier, Amsterdam (2004)

    Google Scholar 

  6. Frucht, R.: Herstellung von Graphen mit vorgegebener abstrakter Gruppe. Compos. Math. 6, 239–250 (1938)

    MathSciNet  MATH  Google Scholar 

  7. Gillman, L., Jerison, M.: Rings of Continuous Functions. Springer, New York (1976)

    MATH  Google Scholar 

  8. deGroot, J.: Groups represented by homeomorphism groups. Math. Ann. 138, 80–102 (1959)

    Article  MathSciNet  Google Scholar 

  9. Gumm, H.-P., Schröder, T.: Covarieties and complete covarieties. In: Jacobs, B., Moss, L., Reichel, H., Rutten, J.J.M.M. (eds.) Proc. 1st Internat. Workshop on Coalgebraic Methods in Computer Science (CMCS ’98). Electronic Notes in Theoretical Computer Science, vol. 11. Elsevier, Amsterdam (1998)

    Google Scholar 

  10. Hedrlín, Z., Sichler, J.: Any boundable binding category contains a proper class of mutually disjoint copies of itself. Algebra Univers. 1, 97–103 (1971)

    Article  MATH  Google Scholar 

  11. Isbell, J.R.: Subobjects, adequacy completeness and categories of algebras. Rozpr. Mat. 36, 1–33 (1964)

    MathSciNet  Google Scholar 

  12. Kahawara, Y., Mori, M.: A small final coalgebra theorem. Theor. Comput. Sci. 233, 129–145 (2000)

    Article  Google Scholar 

  13. Katětov, M.: A theorem on mappings. Comment. Math. Univ. Carol. 8, 431–433 (1967)

    MATH  Google Scholar 

  14. Koubek, V.: Set functors. Comment. Math. Univ. Carol. 12, 175–195 (1971)

    MathSciNet  MATH  Google Scholar 

  15. Koubek, V.: On categories into which each concrete category can be embedded. Cah. Topol. Géom. Differ. 17, 33–57 (1976)

    MathSciNet  MATH  Google Scholar 

  16. Makkai, M., Paré, R.: Accessible Categories; The Foundation of Categorical Model Theory. Contemporary Math., vol. 104. American Math. Society, Providence, RI (1989)

    Google Scholar 

  17. Pultr, A., Trnková, V.: Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories. North-Holland, Amsterdam (1980)

  18. Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. Theor. Comp. Sci. 249, 3–80 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sabidussi, G.: Graphs with given infinite groups. Monatsh. Math. 64, 64–67 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shelah, S., Rudin, M.E.: Unordered types of ultrafilters. Topol. Proc. 3, 199–204 (1978)

    MathSciNet  Google Scholar 

  21. Trnková, V.: On some properties of set functors. Comment. Math. Univ. Carol. 10, 323–352 (1969)

    MATH  Google Scholar 

  22. Trnková, V., Sichler, J.: On universal categories of coalgebras. Algebra Univers. (to appear)

Download references

Author information

Authors and Affiliations

  1. Department of Theoretical Computer Science and Mathematical Logic, and Institute of Theoretical Computer Science, The Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Václav Koubek

  2. Department of Mathematics, University of Manitoba, Winnipeg, Canada, R3T 2N2

    Jiří Sichler

  3. Mathematical Institute of Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Věra Trnková

Authors
  1. Václav Koubek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiří Sichler
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Věra Trnková
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiří Sichler.

Additional information

The first author gratefully acknowledges the support of the project 1M0022162080 of the Czech Ministry of Education. The second author gratefully acknowledges the support provided by the NSERC of Canada. The third author gratefully acknowledges the support of the project MSM 0021620839 of the Czech Ministry of Education.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Koubek, V., Sichler, J. & Trnková, V. Universality of Categories of Coalgebras. Appl Categor Struct 19, 939–957 (2011). https://doi.org/10.1007/s10485-010-9232-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-010-9232-1

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation