caa a22 4500 445378670 CHVBK 20180317143011.0 cr unu---uuuuu 170323e20110201xx s 000 0 eng 10.1007/s10485-009-9206-3 doi (NATIONALLICENCE)springer-10.1007/s10485-009-9206-3 Connected and Disconnected Maps [Elektronische Daten] [Gareth Boxall, David Holgate] A new relation between morphisms in a category is introduced—roughly speaking (accurately in the categories Set and Top), f ∥ g iff morphisms w:dom(f)→dom(g) never map subobjects of fibres of f non-constantly to fibres of g. (In the algebraic setting replace fibre with kernel.) This relation and a slight weakening of it are used to define "connectedness” versus "disconnectedness” for morphisms. This parallels and generalises the classical treatment of connectedness versus disconnectedness for objects in a category (in terms of constant morphisms). The central items of study are pairs $({\mathcal F},{\mathcal G})$ of classes of morphisms which are corresponding fixed points of the polarity induced by the ∥-relation. Properties of such pairs are examined and in particular their relation to (pre)factorisation systems is analysed. The main theorems characterise: (a)factorisation systems which factor morphisms through a regular epimorphic "connected” morphism followed by a "disconnected” morphism, and(b)pairs $({\mathcal F},{\mathcal G})$ consisting of "connected” versus "disconnected” morphisms which induce a (regular) factorisation system.This suggests a generalisation of the pair (Concordant, Dissonant) of classes of continuous maps which was shown by Collins to yield the factorisation system (Concordant quotient, Dissonant) on Top. Springer Science+Business Media B.V., 2009 Concordant nationallicence Dissonant nationallicence Connected nationallicence Disconnected nationallicence Torsion nationallicence Torsionfree nationallicence Factorisation system nationallicence Boxall Gareth Department of Pure Mathematics, University of Leeds, LS2 9JT, Leeds, UK aut Holgate David Department of Mathematical Sciences, University of Stellenbosch, 7600, Stellenbosch, South Africa aut Applied Categorical Structures Springer Netherlands 19/1(2011-02-01), 301-319 0927-2852 19:1<301 2011 19 10485 https://doi.org/10.1007/s10485-009-9206-3 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10485-009-9206-3 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Boxall Gareth Department of Pure Mathematics, University of Leeds, LS2 9JT, Leeds, UK aut NATIONALLICENCE 700 1- Holgate David Department of Mathematical Sciences, University of Stellenbosch, 7600, Stellenbosch, South Africa aut NATIONALLICENCE 773 0- Applied Categorical Structures Springer Netherlands 19/1(2011-02-01), 301-319 0927-2852 19:1<301 2011 19 10485 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer