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Applied Categorical Structures

, Volume 11, Issue 3, pp 267–286 | Cite as

Topological Features of Lax Algebras

  • Maria Manuel Clementino
  • Dirk Hofmann
Article

Abstract

Having as starting point Barr's description of topological spaces as lax algebras for the ultrafilter monad, in this paper we present further topological examples of lax algebras – such as quasi-metric spaces, approach spaces and quasi-uniform spaces – and show that, in a suitable setting, the categories of lax algebras have indeed a topological nature. Furthermore, we generalize to this setting known properties of special categories of lax algebras and, extending the construction of Manes, we describe the Čech–Stone compactification of lax algebras.

(lax) monad (lax) algebra ultrafilter monad topological space approach space quasi-uniform space Čech–Stone compactification 

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References

  1. 1.
    Adámek, J., Herrlich, H. and Strecker, G. E.: Abstract and Concrete Categories, Wiley Interscience, New York, 1990.Google Scholar
  2. 2.
    Barr, M.: Relational algebras, in Springer Lecture Notes in Math. 137, 1970, pp. 39–55.Google Scholar
  3. 3.
    Clementino, M. M. and Hofmann, D.: Triquotient maps via ultrafilter convergence, Proceedings of the American Mathematical Society 130 (2002), 3423–3431.Google Scholar
  4. 4.
    Clementino, M. M., Hofmann, D. and Tholen, W.: One setting for all: Metric, topology, uniformity, approach structure, in preparation.Google Scholar
  5. 5.
    Clementino, M. M., Hofmann, D. and Tholen, W.: The convergence approach to exponentiable maps, Portugaliae Mathematica 60 (2003), 139–159.Google Scholar
  6. 6.
    Freyd, P. J. and Scedrov, A.: Categories, Allegories, North-Holland Mathematical Library, Elsevier Science Publishers B.V., 1990.Google Scholar
  7. 7.
    Isbell, J.: Uniform Spaces, Math. Surveys 12, American Mathematical Society, 1964.Google Scholar
  8. 8.
    Johnstone, P. T.: Stone Spaces, Cambridge Studied in Advanced Mathematics 3, Cambridge University Press, Cambridge, 1982.Google Scholar
  9. 9.
    Lawvere, F. W.: Metric spaces, generalized logic, and closed categories, Rendiconti del Seminario Matematico e Fisico di Milano 43 (1973), 135–166.Google Scholar
  10. 10.
    Lowen, R.: Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1997.Google Scholar
  11. 11.
    Mac Lane, S.: Categories for the Working Mathematician, 2nd edn, Springer, New York, 1998.Google Scholar
  12. 12.
    Manes, E. G.: Compact Hausdorff objects, Topology and its Applications 4 (1974), 341–360.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Maria Manuel Clementino
    • 1
  • Dirk Hofmann
    • 2
  1. 1.Departamento de MatemáticaUniversidade de CoimbraCoimbraPortugal
  2. 2.Departamento de MatemáticaUniversidade de AveiroAveiroPortugal

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