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Regularity of the category of Kelley spaces


We show that the cartesian closed category of compactly generated Hausdorff spaces is regular, but is neither exact, nor locally cartesian closed. In fact we find a coequalizer of an equivalence relation which is not stable under pullback.

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  1. 1.

    Barr, M.: Exact categories,Exact Categories and Categories of Sheaves, LNM 236, Springer-Verlag, New York, Heidelberg, Berlin, 1985.

    Google Scholar 

  2. 2.

    Barr, M. and Wells, C.:Toposes, Triples and Theories, Springer-Verlag, New York, Heidelberg, Berlin, 1971.

    Google Scholar 

  3. 3.

    Carboni, A. and Walters, R. F. C.: Cartesian bicategories I,J. Pure Appl. Algebra 45 (1987), 11–32.

    Google Scholar 

  4. 4.

    Cagliari, F. and Mantovani, S.: Local homeomorphisms as the exponentiable morphisms in compact Hausdorff spaces,Topology Appl. 41 (1991), 263–272.

    Google Scholar 

  5. 5.

    Day, B. J. and Kelly, G. M.: On topological quotient maps preserved by pullbacks or products,Proc. Camb. Phil. Soc. 67 (1970), 555–558.

    Google Scholar 

  6. 6.

    Gabriel, P. and Zisman, M.:Categories of Fractions and Homotopy Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1967.

    Google Scholar 

  7. 7.

    Jonhstone, P. T.: On a topological topos,Proc. London Math. Soc. 38 (1979), 237–271.

    Google Scholar 

  8. 8.

    Lawvere, F. W.:Theory of Categories over a Base Topos, Ist. Mat. Univ. Perugia, 1972–1973.

  9. 9.

    Lewis, L. G.: Open maps, colimits and a convenient category of fibred spaces,Topology Appl. 19 (1985), 75–89.

    Google Scholar 

  10. 10.

    Steenrod, N. E.: A convenient category of topological spaces,Michigan Math. J. 14 (1967), 133–152.

    Google Scholar 

  11. 11.

    Vitale, E. M.: Left covering functors, PhD Thesis, Louvain-la-Neuve, 1994.

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Cagliari, F., Mantovani, S. & Vitale, E.M. Regularity of the category of Kelley spaces. Appl Categor Struct 3, 357–361 (1995).

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Mathematics Subject Classifications (1991)

  • 18B30
  • 54D50
  • 18D15

Key words

  • compactly generated spaces
  • regular and exact categories
  • locally cartesian closed categories