Sobriety for equilogical spaces

The category of equilogical spaces, as well as the exact completions of the category of T0T0-spaces and of the category of topological spaces, offers locally cartesian closed extensions of the category of topological spaces. Hence in any one of such categories, it is straightforward to consider spaces of continuous functions without bothering about ensuring that they be topological spaces.We test this fact with the notion of sober topological space, producing a synthetic characterization of those topological spaces which are sober in terms of a construction on equilogical spaces of functions.