Separated and connected maps
type: Item
instance of: scholarly article
title: Separated and connected maps
zbMATH Open document ID: 0935.18003
DOI: 10.1023/A:1008636615842
author: Maria Manuel Clementino
author: Walter Tholen
published in: Applied Categorical Structures
publication date: 2000-05-02T00:00:00Z
review text: 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{…
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 18A25
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 18A32
Mathematics Subject Classification ID: 18E40
Mathematics Subject Classification ID: 54B30
zbMATH DE Number: 1236101
zbMATH Keywords: separation
zbMATH Keywords: closure operator
zbMATH Keywords: connectedness
zbMATH Keywords: comma categories
reviewed by: Dikran Dikranjan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1236101