Geodesic
The double powerlocale and exponentiation: A case study in geometric logic
Theory and applications of categories, Tome 12 (2004), pp. 372-422

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If X is a locale, then its double powerlocale PX is defined to be PU(PL(X)) where PU and PL 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 S^X exists (where S is the Sierpinski locale), then PX is an exponential S^(S^X). Second, if in addition W is a locale for which PW is homeomorphic to S^X, then X is an exponential S^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.

Classification : 06D22, 03G30, 03F55, 54C35, 18C15
Keywords: locale, powerlocale, exponentiable, geometric logic 
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     author = {Steven Vickers},
     title = {The double powerlocale and exponentiation: {A} case study in geometric logic},
     journal = {Theory and applications of categories},
     pages = {372--422},
     publisher = {mathdoc},
     volume = {12},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2004_12_a12/}
}
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Steven Vickers. The double powerlocale and exponentiation: A case study in geometric logic. Theory and applications of categories, Tome 12 (2004), pp. 372-422. http://geodesic.mathdoc.fr/item/TAC_2004_12_a12/