Some aspects of (non) functoriality of natural discrete covers of locales

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.