Epimorphisms and Cowellpoweredness
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.
KeywordsClosure Operator Full Subcategory Forgetful Functor Accessible Category Presentable Category
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- The surjectivity of epimorphisms in Top 1 was noted by Burgess  who also described the epimorphisms in Haus, while Baron  characterized the epimorphisms in Top 0.Google Scholar
- The closure operators ipro𝒫 and iesp𝒫 were defined in Dikranjan and Giuli , while esp𝒫 appears in Dikranjan and Giuli  and in Giuli and Hušek , in two particular cases.Google Scholar
- The axioms S(n) were introduced by Viglino , and the axioms S(α), for infinite ordinals α, by Porter and Votaw , while S n appears in Arens . 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 .Google Scholar
- The first example of a non-cowellpowered subcategory of Top (see Example 8.2(2)) was given by Herrlich . Schröder  proved that Ury and S n, n > 1, are not cowellpowered. Giuli and Hušek  established non-cowellpoweredness of Haus(compact spaces). This was extended to the smaller subcategory Haus(e-compact spaces) by Giuli and Simon ; the proof of Theorem 8.6 follows essentially the proof given there. Tozzi  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 . Cowellpoweredness of S(n) and of sUry was shown by Dikranjan, Giuli and Tholen . The proof of Theorem 8.6*, isolated essentially from Dikranjan and Watson , 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  and Exercise 8.M). The epimorphisms in QUnif 0 were described by Holgate , while Theorem and Corollary 8.7 come from Dikranjan and Künzi . The proof of Theorem 8.8 is taken from Uspenskij , 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