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Locales as spectral spaces

Authors
  • Schwartz, Niels1
  • 1 Universität Passau, Fakultät für Informatik und Mathematik, Passau, 94030, Germany , Passau (Germany)
Type
Published Article
Journal
Algebra universalis
Publisher
Springer Basel
Publication Date
Jun 15, 2013
Volume
70
Issue
1
Pages
1–42
Identifiers
DOI: 10.1007/s00012-013-0241-4
Source
Springer Nature
Keywords
License
Yellow

Abstract

A frame is a complete distributive lattice that satisfies the infinite distributive law b∧⋁i∈Iai=⋁i∈Ib∧ai\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${b \wedge \bigvee_{i \in I} a_i = \bigvee_{i \in I} b \wedge a_i}$$\end{document} . The lattice of open sets of a topological space is a frame. The frames form a category Fr. The category of locales is the opposite category Frop. The category BDLat of bounded distributive lattices contains Fr as a subcategory. The category BDLat is anti-equivalent to the category of spectral spaces, Spec (via Stone duality). There is a subcategory of Spec that corresponds to the subcategory Fr under the anti-equivalence. The objects of this subcategory are called locales, the morphisms are the localic maps; the category is denoted by Loc. Thus locales are spectral spaces. The category Loc is equivalent to the category Frop. A topological approach to locales is initiated via the systematic study of locales as spectral spaces. The first task is to characterize the objects and the morphisms of the category Spec that belong to the subcategory Loc. The relationship between the categories Top (topological spaces), Spec and Loc is studied. The notions of localic subspaces and localic points of a locale are introduced and studied. The localic subspaces of a locale X form an inverse frame, which is anti-isomorphic to the assembly associated with the frame of open and quasi-compact subsets of X.

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