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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Cagliari, F., Mantovani, S. & Vitale, E.M. Regularity of the category of Kelley spaces. Appl Categor Struct 3, 357–361 (1995). https://doi.org/10.1007/BF00872904
Mathematics Subject Classifications (1991)
- compactly generated spaces
- regular and exact categories
- locally cartesian closed categories