Essential completeness in categories of completely regular frames.
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.)
scientific article; zbMATH DE number 6148660
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