Eraldo Giuli; Maria Manuel Clementino; Walter Tholen;
What is a quotient map with respect to a closure operator?
It is shown that there is no good answer to the question of the title, even if we restrict our attention to Set-based topological categories. Although very closely related, neither the natural notion of c-finality (designed in total analogy to c-initiality) nor the notion of c-quotient (modelled after the behaviour of topological quotient maps) provide universally satisfactory concepts. More dramatically, in the category Top with its natural Kuratowski closure operator k, the class of k-final maps cannot be described as the class of c-quotient maps for any closure operator c, and the class of k-quotients cannot be described as the class of c-final maps for any c. We also discuss the behaviour of c-final maps under crossing with an identity map, as in Whitehead's Theorem. In Top, this gives a new stability theorem for hereditary quotient maps.
Appl. Categ. Structures