I have just put a new paper of mine on the web at http://mcs.open.ac.uk/sjv22/PPandExp.ps Steve Vickers. Details: "The double powerlocale and exponentiation: A case study in geometric logic" Steven Vickers. If X is a locale, then its double powerlocale PP(X) is defined to be P_U(P_L(X)) where P_U and P_L are the upper and lower powerlocale constructions. We prove various results relating it to exponentiation of locales, including the following. First, if X is a locale for which the exponential $^X exists (where $ is the Sierpinski locale), then PP(X) is an exponential $^($^X). Second, if in addition W is a locale for which PP(W) is homeomorphic to $^X, then X is an exponential $^W. The work uses geometric reasoning, i.e. reasoning stable under pullback along geometric morphisms, and this enables the locales to be discussed in terms of their points as though they were spaces. It relies on a number of geometricity results including those for locale presentations and for powerlocales.