What is a Quotient Map with Respect to a Closure Operator?

ISSN： 0927-2852

Source： Applied Categorical Structures, Vol.9, Iss.2, 2001-03, pp. : 139-151

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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.