Lexicographic sums and fibre-faithful maps
The notion of lexicographic sums is defined and investigated in categories \({\mathcal X}\) with fibres, a terminal object 1, small hom-sets \(| Y|\colon = X(1,Y)\), \(Y\) an object of \({\mathcal X}\). Starting with lexicographic or \(l\)-sums in the category of posets and \(T_ 1\)-spaces in section 1 and taking these as paradigmatic cases, an \(l\)-sum of objects \(B_ y\in X\), \(y\in | Y|\), over \(Y\) is defined (section 2) as an \(L\in{\mathcal X}\) together with \(m\colon L\to Y\), \(j_ y : B_ y\to L\), \(y\in | Y|\), such that (1) \((j_ y\mid y\in | Y|)\) is the family of fibres of \(m\), (2) for every morphism \(f\colon X\to Y\) and its family of fibres \((i_ y\mid y\in | Y|)\) and any family of morphisms \(h_ y\colon f^{-1}(y)\to B_ y\), \(y\in | Y|\), there is a unique \(h\colon X\to L\) with \(m\circ h=f\) and \(h\circ i_ y= j_ y\circ h_ y\), \(y\in | Y|\). Several general results on \(l\)-sums are proved. In section 3 connections between \(l\)-sums and exponentiable points are investigated and in proposition 3.3 a necessary and sufficient condition for the existence of \(l\)-sums is given in terms of exponentiable points, multiple pullbacks and a disjointness condition. This leads in section 4 to one of the main theorems (4.2) on the existence of \(l\)-sums. In section 5 the connections between \(l\)-sums and coproducts are investigated, whereas in section 6 an interesting associative law for \(l\)-sums is proved. \(l\)-sums (as above) give rise to a factorization \(f=m\circ e\) for any morphism \(f\colon X\to Y\), with \(m\colon L\to Y\) the projection of the \(l\)-sum. This factorization is called the fibre-factorization of \(f\) and gives rise to the definition of fibre-faithful and fibre-trivial maps, which are investigated in sections 7 to 9. In section 10 resp. 11 the paradigmatic cases of ordered sets and \(T_ 1\)-spaces are considered; section 12 contains several interesting examples. It seems to the reviewer that such a systematic treatment of lexicographic sums has long been due.
scientific article; zbMATH DE number 458989