Connected and disconnected maps
type: Item
instance of: scholarly article
title: Connected and disconnected maps
zbMATH Open document ID: 1228.18002
author: Juan-Miguel Gracia
published in: Applied Categorical Structures
publication date: 2011-03-30T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 18A32
Mathematics Subject Classification ID: 18A20
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54D05
zbMATH DE Number: 5871641
zbMATH Keywords: component
zbMATH Keywords: concordant map
zbMATH Keywords: constant morphism
zbMATH Keywords: dissonant map
zbMATH Keywords: factorization system
zbMATH Keywords: Galois connection
zbMATH Keywords: kernel pair
zbMATH Keywords: light map
zbMATH Keywords: monotone map
zbMATH Keywords: quasi-component
zbMATH Keywords: regular epimorphism
reviewed by: Sergejs Solovjovs
MaRDI profile type: Publication
full work available at URL: s10485-009-9206-3
OpenAlex ID: W2016415722
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DOI: 10.1007/S10485-009-9206-3
description: scientific article; zbMATH DE number 5871641