Exponentiability for maps means fibrewise core-compactness

Originally, exponentiable spaces X were characterized by Day and Kelly in terms of Scott-open sets, which form a topology on the topology of X. Later on, Hofmann and Lawson described exponentiability for spaces by standard topological terminology as core-compactness or quasi-local compactness. The primary characterization of exponentiable maps by Niefield is in the spirit of Day–Kelly and entails their result as special case, because spaces may be considered as maps to the one-point space. A map-version for the Hofmann–Lawson description was missing. Now, this paper offers a fibrewise notion of core-compactness which is equivalent to exponentiability and specializes to core-compactness for spaces. Moreover, among separated maps (i.e. distinct points in the same fibre may be separated by disjoint open neighbourhoods), the exponentiable ones are just the restrictions of perfect (i.e. separated and proper) maps to open subspaces. This is the map-version of the Whitehead–Michael characterization of exponentiableHausdorff spaces by local compactness and extends the corresponding result by Clementino–Hofmann–Tholen for Hausdorff spaces to arbitrary ones. It proves their respective conjecture.