Subcategories Defined by Closure Operators
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.
KeywordsClosure Operator Full Subcategory Finite Product Left Adjoint Topological Category
Unable to display preview. Download preview PDF.
- The Salbany Correspondence of Theorem 7.1 appears in Tholen , while the Generating Diagonal Theorem was proved by Giuli and Hušek  for the category Top and by Giuli, Mantovani and Tholen  in categorical generality. The notion of essential equivalence for closure operators was introduced (under a different name) by Dikranjan  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 , with its factorization through the conglomerate of idempotent closure operators being discussed by Castellini, Koslowski and Strecker [1992a]. Hoffmann introduced the maximal epi-preserving extension of Section 7.7, with more general categorical studies appearing in Giuli, Mantovani and Tholen ; 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 .Google Scholar