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 TopD of TD-spaces and their continuous maps, and the category LocD of all locales and localic maps which preserve coveredness  of primes. Despite the fact that LocD plays an important role in the TD-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 LocD 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 LocD are precisely  the D-sublocales — the notion analogue to sublocale in the TD-duality — a situation akin to the standard fact that sublocales are precisely  regular monomorphisms in Loc. The results are then applied to obtain the TD-analogues of some familiar results for sober  spaces and some new characterizations of TD-spatiality of localic squares in terms of certain discrete covers of locales.