Lexicographic sums and fibre-faithful maps

The notion of lexicographic sum is introduced in general categories. Existence criteria are derived, particularly for locally cartesuan closed categories and for categories with suitable coproducts. Lexicographic sums satisfy a generalised associative law. More importantly, every morphism can be factored through the lexicographic sum of its fibres. This factorisation and the two types of maps arising from it, fibre-trivial and fibre-faithful, are studied partcularly for partially ordered sets and for T1-spaces