Essential completeness in categories of completely regular frames.
type: Item
instance of: scholarly article
title: Essential completeness in categories of completely regular frames.
zbMATH Open document ID: 1307.06004
author: Anthony W. Hager
author: Bernhard Banaschewski
published in: Applied Categorical Structures
publication date: 2013-03-25T00:00:00Z
review text: This paper is, in some sense, a sequel to the authors' earlier one [Algebra Univers. 62, No. 1, 113-123 (2009; Zbl 1192.06013)] in which they showed that various subcategories of the category \(\mathbf{CRFrm}\) of completely regular frames have only trivial injectives. Let \(\mathbf S\) be a full and isomorphism-closed coreflective subcategory of \(\mathbf{CRFrm}\). The authors call such a subcategory \textit{strongly monocoreflective} if the coreflection maps (denoted by \(s_L\colon SL\to L\)) and its monomorphisms are dense. They then show that any strongly monocoreflective subcategory of \(\mathbf{CRFrm}\) has unique essential completions. For each \(L\in\mathbf{S}\) they are given by \(k_L\colon L\to S\mathfrak BL\), where \(\mathfrak BL\) denotes the Booleanization of \(L\), and \(k_L\) is the unique homomorphism such that \(s_{\mathfrak BL}\cdot k_L=\beta_L\), with \(\beta_L\colon L\to\mathfrak BL\) the Booleanization map. To do this they first establish that the essential monomorphisms in \(\mathbf{CRFrm}\) are precisely the maps whose ``co-lifts'' to Booleanizations are isomorphisms. Equivalently they are the maps \(h\) such that \(h_*(a)=0\) if and only if \(a=0\), that is, the dense and \(*\)-dense homomorphisms. Concrete examples of strongly monocoreflective subcategories of \(\mathbf{CRFrm}\) are given, and they include the subcategories of (i) realcomplete, (ii) compact, and (iii) paracompact frames, amongst others. In view of the significant role played by skeletal maps in this study, the authors end the paper with a result which shows that in the category \(\mathbf S_{\text{sk}}\), obtained from any monocoreflective subcategory by considering only skeletal maps, the epicomplete objects are exactly the essentially complete ones in \(\mathbf S\). Furthermore, they form an epi-monoreflective subcategory of \(\mathbf S_{\text{sk}}\) with reflection maps \(k_L\colon L\to S\mathfrak BL\). (Also submitted to MR.)
reviewed by: Themba Dube
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 54B30
zbMATH DE Number: 6148660
zbMATH Keywords: completely regular frames
zbMATH Keywords: essential monomorphisms
zbMATH Keywords: essential completions
zbMATH Keywords: strongly monocoreflective subcategories
zbMATH Keywords: Booleanizations of frames
MaRDI profile type: Publication
full work available at URL: s10485-011-9262-3
OpenAlex ID: W2077204451
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DOI: 10.1007/S10485-011-9262-3
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description: scientific article; zbMATH DE number 6148660