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

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Type: Preprint
National /International: International
Title: Some aspects of (non)functoriality of natural discrete covers of locales
Publication Date: 2018-03-26
Authors: - Richard N. Ball
- Jorge Picado
- Ales Pultr
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.
Institution: DMUC 18-10
Online version: http://www.mat.uc.pt...prints/eng_2018.html
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