Algebra, Logic and Topology

 

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Author(s)
Eraldo Giuli; Maria Manuel Clementino; Walter Tholen;

Title
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.

Journal
Appl. Categ. Structures

Volume
9

Year
2001

Issue
2

Page(s)
139-151

 
     
 

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