Categorical Structure of Closure Operators pp 177-224 | Cite as

# Regular Closure Operators

## Abstract

Regular closure operators provide the key instrument for attacking the epimorphism problem in a subcategory A of the given (and, in general, better behaved) category *χ*. Depending on A one defines the A-regular closure operator of *χ* in such a way that its dense morphisms in A are exactly the epimorphisms of A. Now everything depends on being able to “compute” the A-regular closure effectively. The strong modification of a closure operator as introduced in 6.6 turns out to play a major role in this, as we shall see in the following two chapters. It arises quite naturally after we have provided two powerful criteria for closedness with respect to the A-regular closure (6.4, 6.5) . At least for additive categories this leads to a complete characterization of regular closure operators as maximal closure operators. The rest of the chapter is devoted to the (quite particular) case of weakly hereditary closure operators which, roughly, characterize torsionfree classes in algebraic contexts and give a general notion of disconnectedness in topological contexts.

## Keywords

Closure Operator Full Subcategory Left Adjoint Unique Morphism Pullback Diagram## Preview

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## Notes

- The regular closure and epi-closure have been defined for categories of algebras by Isbell [1966] and for categories of topological spaces by Salbany [1976] and Cagliari and Cicchese [1983]. The modification formula 6.3 was given by Dikranjan and Giuli [1984] in topological contexts and by Dikranjan, Giuli and Tholen [1989] for general categories. Frolík communicated a proof of Theorem 6.5 in the context of subcategories of
**Top**to Dikranjan and Giuli in 1983 (see Dikranjan and Giuli [1983]). A categorical version (but more restrictive than the one given in 6.5) was given by Dikranjan, Giuli and Tholen [1989]. Strong modifications were defined by Dikranjan [1992], and the characterization of weakly hereditary regular closure operators (see 6.8) was given by Clementino [1992], [1993]. The correspondences of 6.10 concerning pointed topological spaces are new.Google Scholar