@UNPUBLISHED{publication1, title = "Some aspects of (non)functoriality of natural discrete covers of locales", author = "{BALL, Richard N.} and {PICADO, Jorge} and {PULTR, Ales}", year = "2018-03-26", abstract = "The frame Sc(L) generated by closed sublocales of a locale L is known to be a natural Boolean (``discrete'') extension of a subfit L; also it is known to be its maximal essential extension. In this paper we first show that it is an essential extension of any L and that the maximal essential extensions of L and Sc(L) are isomorphic. The construction Sc is not functorial; this leads to the question of individual liftings of homomorphisms L\→ M to homomorphisms Sc(L)\→Sc(M). This is trivial for Boolean L and easy for a wide class of spatial L,M. Then, we show that one can lift all h:L\→ 2 for weakly Hausdorff L (and hence the spectra of L and Sc(L) are naturally isomorphic), and finally present liftings of h:L\→ M for regular L and arbitrary Boolean M. ", url = "http://www.mat.uc.pt/preprints/eng\_2018.html", note = "" }