On the categorical behaviour of locales and D-localic maps

Abstract

It was shown by Banaschewski and Pultr that the classical adjunction between Top and Loc restricts to an adjunction between the category Top _D of T_D -spaces and their continuous maps, and the category Loc _D of all locales and localic maps which preserve coveredness of primes. Despite the fact that Loc _D plays an important role in the T_D -duality, not much is known about its categorical structure, and it is the aim of this paper to fill this gap. In particular, we show that Loc _D is closed under finite products in Loc and moreover we characterize the existence of equalizers. As a consequence, it is proved that regular monomorphisms in Loc _D are precisely the D-sublocales — the notion analogue to sublocale in the T_D -duality — a situation akin to the standard fact that sublocales are precisely regular monomorphisms in Loc. The results are then applied to obtain the T_D -analogues of some familiar results for sober spaces and some new characterizations of T_D -spatiality of localic squares in terms of certain discrete covers of locales.