Some Aspects of (Non) functoriality of Natural Discrete Covers of Locales
Frame, Locale, Sublocale, Sublocale lattice, Essential extension, Subfit, Booleanization
College of Natural Science and Mathematics, Mathematics
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.
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Ball, Richard N, et al. “Some Aspects of (Non) Functoriality of Natural Discrete Covers of Locales.” Quaestiones Mathematicae, vol. 42, no. 6, 2019, pp. 701–715. doi: 10.2989/16073606.2018.1485756.