Title
Some Aspects of (Non) functoriality of Natural Discrete Covers of Locales
Document Type
Article
Publication Date
8-7-2019
Keywords
Frame, Locale, Sublocale, Sublocale lattice, Essential extension, Subfit, Booleanization
Organizational Units
College of Natural Science and Mathematics, Mathematics
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.
Publication Statement
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Recommended Citation
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.