Connected and Disconnected Maps |
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| Authors: | Gareth Boxall David Holgate |
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| Affiliation: | 1.Department of Pure Mathematics,University of Leeds,Leeds,UK;2.Department of Mathematical Sciences,University of Stellenbosch,Stellenbosch,South Africa |
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| Abstract: | 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 (F,G)({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 |
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