Coreflectively modified duality
This survey paper is concerned with, among other things, the question of the stability of various types of quotient maps under products. The author introduces a categorical technique, coreflectively modified duality, and shows how it can be used in the category CONV of convergence spaces to give a unified treatment of this question for various kinds of quotient maps among which are: quotient, hereditarily quotient, countably biquotient, and biquotient. Moreover, the technique can be used to simultaneously treat problems on the preservation under product of certain properties such as: sequentiality, Fréchetness, strong Fréchetness, \(k\)-ness, quasi-\(k\)-ness, countably bi-\(k\)-ness among others. The technique is also extended to the category CAP of convergence-approach spaces. By this method, then, classical results are unified and new results obtained. Many of the results are conveniently displayed in tables showing that if \(g\) is a certain type of quotient map then \(f\times g\) is a (perhaps different) type of quotient map if and only if \(f\) is a particular type of quotient map. The technique is based on a modification of the observation that if \(L\) is a bireflective subcategory (with reflector \(R\)) of a Cartesian closed topological category \(C\), then for each object \(X\) of \(C\) the coarsest \(C\)-object \(W\) with the same underlying set as \(X\) and for which \(W\times Y\) is finer than \(R(X\times Y)\), for all objects \(Y\) in \(C\), defines a reflection of \(X\) into the Cartesian-closed topological hull of \(L\).
scientific article; zbMATH DE number 2104660
scientific article