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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 Setbased topological categories. Although very closely related, neither the natural notion of cfinality (designed in total analogy to cinitiality) nor the notion of cquotient (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 kfinal maps cannot be described as the class of cquotient maps for any closure operator c, and the class of kquotients cannot be described as the class of cfinal maps for any c. We also discuss the behaviour of cfinal 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) 139151
