Applied Categorical Structures

, Volume 12, Issue 2, pp 127–154 | Cite as

One Setting for All: Metric, Topology, Uniformity, Approach Structure

  • Maria Manuel Clementino
  • Dirk Hofmann
  • Walter Tholen


For a complete lattice V which, as a category, is monoidal closed, and for a suitable Set-monad T we consider (T,V)-algebras and introduce (T,V)-proalgebras, in generalization of Lawvere's presentation of metric spaces and Barr's presentation of topological spaces. In this lax-algebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories behave functorially both in T and in V, one establishes a network of functors at the general level which describe the basic connections between the structures mentioned by the title. Categories of (T,V)-algebras and of (T,V)-proalgebras turn out to be topological over Set.

V-matrix V-promatrix (T,V)-algebra (T,V)-proalgebra co-Kleisli composition ordered set metric space topological space uniform space approach space prometric space protopological space proapproach space topological category 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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.
    Börger, R.: Coproducts and ultrafilters, J. Pure Appl. Algebra 46 (1987), 35–47.Google Scholar
  4. 4.
    Clementino, M. M. and Hofmann, D.: Topological features of lax algebras, Appl. Categ. Structures 11(3) (2003), 267–286.Google Scholar
  5. 5.
    Clementino, M. M. and Hofmann, D.: Effective descent morphisms in categories of lax algebras, Preprint 02-20, Department of Mathematics, University of Coimbra, 2002.Google Scholar
  6. 6.
    Clementino, M. M., Hofmann, D. and Tholen, W.: Exponentiability in categories of lax algebras, Preprint 03-02, Department of Mathematics, University of Coimbra, 2003.Google Scholar
  7. 7.
    Clementino, M. M. and Tholen,W.: Metric, topology and multicategory-a common approach, J. Pure Appl. Algebra 179 (2003), 13–47.Google Scholar
  8. 8.
    Hofmann, D.: An algebraic description of regular epimorphisms in topology, Preprint.Google Scholar
  9. 9.
    Eilenberg, S. and Moore, J. C.: Adjoint functors and triples, Illinois J.Math. 9 (1965), 381–398.Google Scholar
  10. 10.
    Herrlich, H.: On the failure of Birkhoff's theorem for small based equational categories of algebras, Cahiers Topologie Géom. Différentielle Catég. 34 (1993), 185–192.Google Scholar
  11. 11.
    Herrlich, H., Lowen-Colebunders, E. and Schwarz, F.: Improving Top: PrTop and PsTop, in Category Theory at Work, Heldermann Verlag, Berlin, 1991, pp. 21–34.Google Scholar
  12. 12.
    Lawvere, F. W.: Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. Fis. Milano 43 (1973), 135–166.Google Scholar
  13. 13.
    Linton, F. E. J.: Some aspects of equational categories, in Proc. Conf. Categorical Algebra (La Jolla 1965), Springer-Verlag, Berlin, 1966, pp. 84–95.Google Scholar
  14. 14.
    Lowen, R.: Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1997.Google Scholar
  15. 15.
    Lowen, R. and Windels, B.: AUnif: A common supercategory of pMet and Unif, Internat. J. Math. Math. Sci. 21 (1998), 1–18.Google Scholar
  16. 16.
    Manes, E. G.: A triple theoretic construction of compact algebras, in Springer Lecture Notes in Math. 80, 1969, pp. 91–118.Google Scholar
  17. 17.
    Tholen, W.: Procategories and multiadjoint functors, Canad. J. Math 36 (1984), 144–155.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Maria Manuel Clementino
    • 1
  • Dirk Hofmann
    • 2
  • Walter Tholen
    • 3
  1. 1.Departamento de MatemáticaUniversidade de CoimbraCoimbraPortugal
  2. 2.Departamento de MatemáticaUniversidade de AveiroAveiroPortugal
  3. 3.Department of Mathematics and StatisticsYork UniversityTorontoCanada

Personalised recommendations