Categorical Structure of Closure Operators pp 259-304 | Cite as

# Epimorphisms and Cowellpoweredness

## Abstract

Characterizing the epimorphisms of a concrete category and settling the question whether the category is cowellpowered can be a challenging problem and has been the theme of many research papers (see the Notes at the end of this chapter). In many cases, closure operators offer themselves as a natural tool to tackle the problem. We concentrate here on results for those categories of topology and algebra where this approach proves to be successful. These include criteria for epimorphisms in subcategories of modules and fields, recent or new results on cowellpowered and non-cowellpowered subcategories of topological spaces, and a rather direct proof of Uspenskij’s recent discovery of a non-dense epimorphism in the category of Hausdorff topological groups.

## Keywords

Closure Operator Full Subcategory Forgetful Functor Accessible Category Presentable Category## Preview

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## Notes

- The surjectivity of epimorphisms in
**Top**_{1}was noted by Burgess [1965] who also described the epimorphisms in**Haus**, while Baron [1968] characterized the epimorphisms in**Top**_{0}.Google Scholar - The closure operators ipro
^{𝒫}and iesp^{𝒫}were defined in Dikranjan and Giuli [1987], while esp^{𝒫}appears in Dikranjan and Giuli [1986] and in Giuli and Hušek [1986], in two particular cases.Google Scholar - The axioms
*S(n*) were introduced by Viglino [1969], and the axioms*S(α*), for infinite ordinals α, by Porter and Votaw [1973], while*S*_{n}appears in Arens [1978]. The axioms*S*(η) and*S*[*η*], depending on an arbitrary order type*η*(with*S(η*) =*S[η] = S(α*) when^{η}= α is an ordinal) were introduced by Dikranjan and Watson [1994].Google Scholar - The first example of a non-cowellpowered subcategory of
**Top**(see Example 8.2(2)) was given by Herrlich [1975]. Schröder [1983] proved that**Ury**and*S*_{n},*n*> 1, are not cowellpowered. Giuli and Hušek [1986] established non-cowellpoweredness of**Haus**(compact spaces). This was extended to the smaller subcategory**Haus(e**-compact spaces) by Giuli and Simon [1990]; the proof of Theorem 8.6 follows essentially the proof given there. Tozzi [1986] proved that the category**SUS**is cowellpowered. Non-cowellpoweredness of**Haus**(compact Hausdorff spaces), as well as cowellpoweredness of certain subcategories of**Top**(see Example 8.5), was established in Dikranjan and Giuli [1986]. Cowellpoweredness of*S(n*) and of**sUry**was shown by Dikranjan, Giuli and Tholen [1989]. The proof of Theorem 8.6*, isolated essentially from Dikranjan and Watson [1994], is substantially simpler than all its predecessors given for**Ury**and*S(n*). Cowellpoweredness of*S*[*η*] can be characterized in terms of properties of the order type*η*(see Dikranjan and Watson [1994] and Exercise 8.M). The epimorphisms in**QUnif**_{0}were described by Holgate [1992], while Theorem and Corollary 8.7 come from Dikranjan and Künzi [1995]. The proof of Theorem 8.8 is taken from Uspenskij [1994], where a compact connected manifold without boundary (either finite-dimensional or a Hilbert cube manifold) is considered instead of*A*.Google Scholar - The description of the epimorphisms of
**Fld**belongs to general categorical knowledge but seems hard to track down in the literature. Theorem 8.9 appears to be new, and so do the assertions of Exercises 8.W, X, Y.Google Scholar