The regular-locally compact coreflection of a stably locally compact locale

Abstract The Scott continuous nuclei form a subframe of the frame of all nuclei. We refer to this subframe as the patch frame. We show that the patch construction exhibits (i) the category of regular locally compact locales and perfect maps as a coreflective subcategory of the category of stably locally compact locales and perfect maps, (ii) the category of compact regular locales and continuous maps as a coreflective subcategory of the category of stably compact locales and perfect maps, and (iii) the category of Stone locales and continuous maps as a coreflective subcategory of the category of spectral locales and perfect maps. (Here a stably locally compact locale is not necessarily compact, and a stably compact locale is a compact and stably locally compact locale.) We relate our patch construction to Banaschewski and Brummer's construction of the dual equivalence of the category of stably compact locales and perfect maps with the category of compact regular biframes and biframe homomorphisms.

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