Multicoreflective subcategories and coprime objects
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 In the present paper, a shorter proof is given for Hoffmann's unpublished result that the class of coprime objects is always closed under the formation of connected colimits, and characterizations of coprime objects are given in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.
scientific article; zbMATH DE number 4127471
scientific article