Separation versus connectedness
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. \textit{E. Giuli} and \textit{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. \textit{E. Giuli} and \textit{H. Hušek}, Ann. Mat. Pura Appl., IV. Ser. 145, 337-346 (1986; Zbl 0617.54006) and \textit{E. Giuli, S. Mantovani} and \textit{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 [\textit{D. Dikranyan} and \textit{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 \textit{G. Castellini, J. Koslowski} and \textit{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.
four commutative triangles of Galois correspondences
scientific article; zbMATH DE number 988060
scientific article