Abstract
Originally, exponentiable maps in the category Top of topological spaces were described by Niefield in terms of certain fibrewise Scott-open sets. This generalizes the first characterization of exponentiable spaces by Day and Kelly, which was improved thereafter by Hofmann and Lawson who described them as core-compact spaces.
Besides various categorical methods, the Sierpinski-space is an essential tool in Niefield's original proof. Therefore, this approach fails to apply to quotient reflective subcategories of Top like Haus, the category of Hausdorff spaces. A recent generalization of the Hofmann–Lawson improvement to exponentiable maps enables now to reprove the characterization in a completely different and very elementary way. This approach works for any nontrivial quotient reflective subcategory of Top or Top/ T , the category of all spaces over a fixed base space T, as well as for exponentiable monomorphisms with respect to epi-reflective subcategories.
An important special case is the category Sep⊂_Top/ T of separated maps, i.e. distinct points in the same fibre can be separated in the total space by disjoint open neighbourhoods. The exponentiable objects in Sep turn out to be the open and fibrewise locally compact maps. The same holds for Haus/ T , T a Hausdorff space. In this case, a similar characterization was obtained by Cagliari and Mantovani.
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Richter, G. An Elementary Approach to Exponentiable Maps. Applied Categorical Structures 12, 287–300 (2004). https://doi.org/10.1023/B:APCS.0000031070.39281.21
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DOI: https://doi.org/10.1023/B:APCS.0000031070.39281.21