Connected and Disconnected Maps

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) 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 (F ,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.