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On the pullback stability of a quotient map with respect to a closure
operator

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Lurdes Sousa

There are well-known characterizations of the hereditary quotient maps in
the category of topological spaces, (that is, of quotient maps stable
under pullback along embeddings), as well as of universal quotient maps
(that is, of quotient maps stable under pullback). These are precisely the
so-called pseudo-open maps, as shown by Arhangel'skii, and the bi-quotient
maps of Michael, as shown by Day and Kelly, respectively. In this paper
hereditary and stable quotient maps are characterized in the broader
context given by a category equipped with a closure operator. To this end,
we derive explicit formulae and conditions for the closure in the codomain
of such a quotient map in terms of the closure in its domain.

Keywords: closure operator, quotient, pullback, closed morphism, open morphism, final morphism.

2000 MSC: 18A32, 18A30, 18A20, 54C10, 54B30.

*Theory and Applications of Categories*, Vol. 8, 2001, No. 6, pp 100-113.

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