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Title: Some aspects of (non) functoriality of natural discrete covers of locales
Authors: Ball, Richard N. 
Picado, Jorge 
Pultr, Aleš 
Keywords: Frame, locale, sublocale, sublocale lattice, essential extension, subfit, Booleanization
Issue Date: 2019
Publisher: Taylor & Francis
Project: UID/MAT/00324/2013 
Serial title, monograph or event: Quaestiones Mathematicae
Volume: 42
Issue: 6
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.
DOI: 10.2989/16073606.2018.1485756
Rights: embargoedAccess
Appears in Collections:I&D CMUC - Artigos em Revistas Internacionais

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