Multicoreflective subcategories and coprime objects
type: Item
instance of: scholarly article
title: Multicoreflective subcategories and coprime objects
zbMATH Open document ID: 0688.18001
DOI: 10.1016/S0166-8641(89)80002-1
author: Reinhard Börger
published in: Topology and its Applications
publication date: 1989-01-01T00:00:00Z
review text: Connectedness notions like (usual or pathwise) connectedness in a category can often be described by component categories, as studied by Herrlich, Preuß, Salicrup, Strecker, Tholen, Tiller and the author of the present paper under different conditions. Independently, G. Salicrup and the author observed that component categories are multicoreflective in the sense of Diers. In the present paper, multicoreflectivity is studied in general without a smallness condition assumed by Diers. A multicoreflectivity criterion in the spirit of the special adjoint functor theorem is given. As a consequence, multicoreflective subcategories are always closed under connected colimits. In the category \({\mathcal C}omp\) of compact Hausdorff spaces, the latter condition is also sufficient, and the multicoreflection maps are always monic. \({\mathcal C}omp\) is not covered by the mentioned result on component categories. Moreover, the notion of multicoreflectivity in \({\mathcal C}omp\) is compared to different generalized notions of component categories, and counterexamples for the false implications are given. Another categorical description of connectedness is given by coprime objects, i.e. objects whose covariant set-valued hom-functor preserves coproducts. Those objects were introduced by R. E. Hoffmann under the name ``Z-objects''. In the present paper, a shorter proof is given for Hoffmann's unpublished result that the class of coprime objects is always closed under the formation of connected colimits, and characterizations of coprime objects are given in categories with universal coproducts or with a certain disjointness condition. Finally the results are applied to the dual of the category \({\mathcal R}ng\) of unital rings. It turns out that the category of powers of \({\mathbb{Z}}\) is reflective in \({\mathcal R}ng\) if and only if there are no uncountable measurable cardinals.
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 13A05
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 54D10
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 4127471
zbMATH Keywords: multiorthogonality
zbMATH Keywords: multicoreflective subcategories
zbMATH Keywords: connected colimits
zbMATH Keywords: compact Hausdorff spaces
zbMATH Keywords: component categories
zbMATH Keywords: connectedness
zbMATH Keywords: coprime objects
zbMATH Keywords: unital rings
zbMATH Keywords: category of powers of \({\mathbb{Z}}\)
zbMATH Keywords: measurable cardinals
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4127471