Applied Categorical Structures

, Volume 7, Issue 1–2, pp 209–226 | Cite as

Categorical Generalization of a Universal Domain

  • JiřÍ Velebil
Article
  • 41 Downloads

Abstract

J. Adámek defined SC categories as a categorical generalization of Scott domains. Namely, an SC category is finitely accessible, has an initial object and is boundedly cocomplete (each diagram with a compatible cocone has a colimit). SC categories are proved to serve well as a basis for the computer language semantics.

The purpose of this paper is to generalize the concept of a universal Scott domain to a universal SC category. We axiomatize properties of subcategories of finitely presentable objects of SC categories (generalizing thus semilattices of compact elements of Scott domains). The categories arising are called FCC (finitely consistently cocomplete) categories. It is shown that there exists a universal FCC category, i.e., such that every FCC category may be FCC embedded into it. The result is an application of a general procedure introduced 30 years ago by V. Trnková.

Scott domain Scott complete category universal domain 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adámek, J.: A categorical generalization of Scott domains, to appear in Math. Structures Comput. Sci. (1995).Google Scholar
  2. 2.
    Adámek, J., Herrlich, H. and Strecker, G.: Abstract and Concrete Categories, Wiley Interscience, New York, 1990.Google Scholar
  3. 3.
    Adámek, J. and Rosický, J.: Locally Presentable and Accessible Categories, Cambridge University Press, Cambridge, 1994.Google Scholar
  4. 4.
    Borceux, F.: Handbook of Categorical Algebra (in three volumes), Cambridge University Press, Cambridge, 1994.Google Scholar
  5. 5.
    Jech, T.: Set Theory, Academic Press, London, New York, San Francisco, 1978.Google Scholar
  6. 6.
    MacLane, S.: Categories for the Working Mathematician, Springer-Verlag, New York, 1971.Google Scholar
  7. 7.
    Makkai, M. and Paré, R.: Accessible Categories: The Foundations of Categorical Model Theory, Contemporary Mathematics (AMS), Vol. 104, 1989.Google Scholar
  8. 8.
    Schubert, H.: Kategorien I, II, Springer-Verlag, Berlin, 1970.Google Scholar
  9. 9.
    Scott, D.: Data types as lattices, SIAM J. Comput. 5 (1976), 522–587.Google Scholar
  10. 10.
    Scott, D.: Domains for denotational semantics, in: M. Nielsen and E. M. Schmidt (eds.), Proceedings ICALP 1982, Springer Lecture Notes in Computer Science 140, Springer-Verlag, Berlin, 1982, pp. 577–613.Google Scholar
  11. 11.
    Stoltenberg-Hansen, V., Lindström, I. and Griffor, E.: Mathematical Theory of Domains, Cambridge Tracts in Theoretical Computer Science 22, Cambridge University Press, 1994.Google Scholar
  12. 12.
    Trnková, V.: Universal categories, Comment. Math. Univ. Carolin. 7 (1966), 143–206.Google Scholar
  13. 13.
    Trnková, V.: Universal category with limits of finite diagrams, Comment. Math. Univ. Carolin. 7 (1966), 447–456.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • JiřÍ Velebil
    • 1
  1. 1.Department of MathematicsFEL ČVUTPrague 6Czech Republic. e-mail

Personalised recommendations