Separated and connected maps
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 the closure operator is intended in the sense of \textit{E. Giuli} and the reviewer [Topology Appl. 27, 129-143 (1987; Zbl 0634.54008)] where these notions where introduced. Recently the authors offered a very detailed study of these concepts [Topology Appl. 75, No. 2, 143-181 (1997; Zbl 0906.18003)] and gave a characterization of the pair's disconnectedness/connectedness in the sense of Arkhangel'skij and Wiegandt in this context via the subcategory \(\nabla(C)\) of \(C\)-connected objects and the subcategory \(\Delta(C)\) of \(C\)-separated objects. This paper is an impressive continuation of this project concentrated on the case of comma categories. The counterpart of \(\Delta(C)\) (resp. \(\nabla(C)\)) for morphisms is the subcategory \(\text{Sep}(C)\) (resp. \(\text{Conn}(C)\)) of \(C\)-separated (resp. \(C\)-connected) morphisms. After a thourogh study of the permanence properties of these classes, the authors introduce also dissonant and concordant morphisms and explain their relation to separated and connected morphisms. The authors establish precise conditions that ensure the existence of \((\text{Conn}^*(C), \text{Sep}(C))\)-factorizations, where \(\text{Conn}^*(C)\) denotes the class of strong epimorphisms in \(\text{Conn}(C)\). This rich paper culminates with a solution of the following longstanding problem (see Problem 7.2 in the monograph of \textit{W. Tholen} and the reviewer [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 Diagonal Theorem [cf. \textit{E. Giuli} and \textit{M. Husek}, Ann. Mat. Pura Appl., IV. Ser. 145, 337-346 (1986; Zbl 0617.54006); \textit{E. Giuli, S. Mantovani} and \textit{W. Tholen}, J. Pure Appl. Algebra 51, No. 1/2, 129-140 (1988; Zbl 0651.18002)] says that in every decent category \({\mathcal X}\) any strongly epireflective subcategory of \({\mathcal X}\) must be of the form \(\Delta(C)\) of some closure operator \(C\). The authors provide a very natural counter-example where the Diagonal Theorem fails.
scientific article; zbMATH DE number 1236101
scientific article