Connected and disconnected maps
There exists a number of factorizations of continuous maps between topological spaces, in which the first (resp. second) factor is in a sense ``connected'' (resp. ``disconnected''). For example, S.~Eilenberg and G.~T.~Whyburn provided (monotone, light)-factorizations for continuous maps between compact metric spaces, where monotone (resp. light) means that the fibres of the respective factor are connected (resp. totally disconnected). E.~Michael extended the result for those maps, whose codomain is a \(T_1\)-space. Moreover, P.~J.~Collins constructed weaker (concordant, dissonant)-factorizations of arbitrary continuous maps, where concordant (resp. dissonant) means that the fibres of the corresponding factor are contained in quasi-components of its domain (resp. meet every quasi-component in at most one point), whereas \textit{G.~E.~Strecker} proposed (submonotone, superlight)-factorizations, which replace quasi-components with components in the above. Using the concept of factorization structure (called factorization system in the current paper) for morphisms (or, more generally, sources) in a category, several authors tried to obtain a category-theoretic generalization of the above-mentioned factorizations. For example, \textit{H.~Herrlich} has come out with (concentrated, dispersed)-factorizations, which could be defined in an abstract \((\mathcal{E},\mathcal{M})\)-category, whereas \textit{W.~Tholen} used the notion of \(\mathcal{E}\)-prereflection motivated by a concept of \textit{R.~Börger}. The current manuscript shows another categorical counterpart of the factorizations in question, motivated by a particular relation on the objects of a category, defined by \textit{H.~Herrlich} in terms of constant morphisms. The authors propose two analogues of this relation for the morphisms of a given category (both of them extending the original relation of \textit{H.~Herrlich} in the presence of a terminal object), thereby arriving at a factorization system in abstract categories with certain properties, whose examples include the above topological factorizations. In conclusion of the paper (the last section), the authors compare their approach to a similar one taken by \textit{M.~M.~Clementino} and \textit{W.~Tholen} on slice categories.
scientific article; zbMATH DE number 5871641
scientific article