Abstract
A Hausdorff space X is characterized by the property that its diagonal Δ X ⊆ X × X is (Kuratowski-) closed. In this way, every closure operator C of a category χ defines the Delta-subcategory Δ(C) of objects with C-closed diagonal, and sub categories appearing as Delta-subcategories are in any “good” category χ characterized as the strongly epireflective ones. What then is the regular closure operator induced by Δ (C)? Under quite “topological” conditions on χ, we show for additive C that this closure can be computed as the idempotent hull of the strong modification of C, at least where it matters: for subobjects in Δ(C) (see Theorem 7.4). This leads to a complete characterization of additive regular closure operators in the given context.
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Notes
The Salbany Correspondence of Theorem 7.1 appears in Tholen [1988], while the Generating Diagonal Theorem was proved by Giuli and Hušek [1986] for the category Top and by Giuli, Mantovani and Tholen [1988] in categorical generality. The notion of essential equivalence for closure operators was introduced (under a different name) by Dikranjan [1992] who also proved the crucial Theorems 7.4 and 7.5 in the context of topological categories over Set. The PR-Correspondence appears in the paper by Pumplün and Röhrl [1985], with its factorization through the conglomerate of idempotent closure operators being discussed by Castellini, Koslowski and Strecker [1992a]. Hoffmann[1982] introduced the maximal epi-preserving extension of Section 7.7, with more general categorical studies appearing in Giuli, Mantovani and Tholen [1988]; its description via a closure operator has its origins in the paper [1987b] by Dikranjan and Giuli. Nabla subcategories were defined but hardly studied in the Dikranjan-Giuli paper [1987a], while the “companions” 7.8 of Δ(C) appear for the first time in Dikranjan [1992].
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© 1995 Springer Science+Business Media Dordrecht
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Dikranjan, D., Tholen, W. (1995). Subcategories Defined by Closure Operators. In: Categorical Structure of Closure Operators. Mathematics and Its Applications, vol 346. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8400-5_7
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DOI: https://doi.org/10.1007/978-94-015-8400-5_7
Publisher Name: Springer, Dordrecht
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