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Separation versus connectedness. (English) Zbl 0906.18003

For a category \({\mathcal X}\) with finite products and a closure operator \(C\) an object \(X\) is \(C\)-separated (\(C\)-connected) if the diagonal \(\Delta_X\) in \(X\times X\) is \(C\)-closed (\(C\)-dense, respectively). Here closure operator is intended in the sense of Giuli and the reviewer [cf. E. Giuli and D. Dikranyan, Topology Appl. 27, 129-143 (1987; Zbl 0634.54008)] where these notions where introduced. While the subcategory \(\Delta(C)\) of \(C\)-separated objects is better understood and described via the diagonal theorem [cf. E. Giuli and H. Hušek, Ann. Mat. Pura Appl., IV. Ser. 145, 337-346 (1986; Zbl 0617.54006) and E. Giuli, S. Mantovani and W. Tholen, J. Pure Appl. Algebra 51, No. 1/2, 129-140 (1988; Zbl 0651.18002)] the subcategory \(\nabla(C)\) of \(C\)-connected objects was studied only recently in the monograph [D. Dikranyan and W. Tholen, “Categorical structure of closure operators. With applications to topology, algebra and discrete mathematics”, Mathematics and its Applications 346, Kluwer Academic Publishers, Dordrecht (1995; Zbl 0853.18002)].
The present authors obtain a stronger form of the diagonal theorem and introduce the important notion of coregular closure operator that permits to describe similarly the subcategory \(\nabla(C)\) of \(C\)-connected objects. The pairs disconnectedness/connectedness are considered often as a pair of left constant and right constant categories. The authors contribute essentially also in this direction by proving that the category \(\nabla(C)\) is left constant iff \(C\) is regular, while the category \(\Delta(C)\) is right constant iff \(C\) is coregular. The authors give a very elegant diagram (Union Jack) of four commutative triangles of Galois correspondences involving closure operators of a category \({\mathcal X}\), subcategories of \({\mathcal X}\) and morphisms of \({\mathcal X}\). Only one of these triangles was known previously: namely the factorization of the Pumplün-Röhrl connection obtained by G. Castellini, J. Koslowski and G. E. Strecker [Topology Appl. 44, No. 1-3, 69-76 (1992; Zbl 0774.18002)] for idempotent closure operators and by Tholen and the reviewer [loc. cit.] in the general case. The paper contains also many nice examples.

MSC:

18B30 Categories of topological spaces and continuous mappings (MSC2010)
54B30 Categorical methods in general topology
54D05 Connected and locally connected spaces (general aspects)
18E40 Torsion theories, radicals
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