Some Aspects of (Non) functoriality of Natural Discrete Covers of Locales

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College of Natural Science and Mathematics, Mathematics


Frame, Locale, Sublocale, Sublocale lattice, Essential extension, Subfit, Booleanization


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 LM 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 : L2 for weakly Hausdorff L (and hence the spectra of L and Sc(L) are naturally isomorphic), and finally present liftings of h : LM for regular L and arbitrary Boolean M.

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