What is a quotient map with respect to a closure operator?
Abstract: 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.
