SOME ASPECTS OF (NON)FUNCTORIALITY OF NATURAL DISCRETE COVERS OF LOCALES RICHARD N. BALL, JORGE PICADO, AND ALE ˇ S PULTR Abstract. The frame S c (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 S c (L) are isomorphic. The construction S c is not functorial; this leads to the question of individual liftings of homomorphisms L → M to homomorphisms S c (L) → S c (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 S c (L) are naturally isomorphic), and finally present liftings of h : L → M for regular L and arbitrary Boolean M . Introduction The trivial discretization (X, P(X )) → (X, τ ) of a topological space allows dealing, inside the category of spaces, with general maps X → Y (or maps with a weaker continuity) in parallel with continuous maps (X, τ ) → (Y,θ). This has in the point-free context a not quite so trivial counterpart in the extension L → S c (L) of a subfit frame to a certain Boolean algebra. Let us explain in some detail what S c (L) is about. It is the lattice of sublocales of L join-generated by the closed ones, which appeared, first, in connection with comparison of fitness and subfitness and in the study of scattered frames ([4]), and later turned out to play a basic role as a Date : May 13, 2018. 2010 Mathematics Subject Classification. 06D22, 54D10, 54D35. Key words and phrases. Frame, locale, sublocale, sublocale lattice, essential ex- tension, subfit, Booleanization. The authors gratefully acknowledge support from Projects P202/12/G061 (Grant Agency of the Czech Republic) and MTM2015-63608-P (Ministry of Econ- omy and Competitiveness of Spain) and from the Department of Mathematics of the University of Denver and the Centre for Mathematics of the University of Coimbra (UID/MAT/00324/2013 funded by FCT/MCTES and FEDER through the Part- nership Agreement PT2020). 1