Preliminary Version On the compact-regular coreflection of a stably compact locale Mart́ın

A nucleus on a frame is a finite-meet preserving closure operator. The nuclei on a frame form themselves a frame, with the Scott continuous nuclei as a subframe. We refer to this subframe as the patch frame. We show that the patch construction exhibits the category of compact regular locales and continuous maps as a coreflective subcategory of the category of stably compact locales and perfect maps, and the category of Stone locales and continuous maps as a coreflective subcategory of the category of spectral locales and spectral maps. We relate our patch construction to Banaschewski and Brümmer’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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