Closure Operators, Functors, Factorization Systems

  • D. Dikranjan
  • W. Tholen
Part of the Mathematics and Its Applications book series (MAIA, volume 346)


The functorial presentation of closure operators and their well-behavedness “along functors” are the dominating themes of this chapter. Briefly, closure operators are equivalently described by (generalized functorial) factorization systems. The interplay between closure operators and preradicals which we have seen for R-modules in 3.4 extends to arbitrary categories; it is described by adjunctions which are (largely) compatible with the compositional structure. With the notion of continuity for functors between categories that come equipped with closure operators we can define — like in topology — initial and final structures. This permits us to “transport” closure operators along functors. For adjoint functors and for M-fibrations, these “transported” closure operators can be computed effectively.


Topological Group Factorization System Closure Operator Natural Transformation Full Subcategory 
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  1. A comprehensive study of pointed endofunctors and their applications was given by Kelly [1980]. Prereflections were introduced by Börger [1981] and studied by Tholen [1987] and by Rosický and Tholen [1988]. The functorial definition of closure operator appears already in the paper by Dikranjan and Giuli [1987a], with the case of no mono-assumption and the connection with generalized factorization systems being presented by Dikranjan, Giuli and Tholen [1989]. The functorial views of factorization systems can be traced back (at least) to Linton [1969] and culminates in their presentation as Eilenberg-Moore algebras (see Korostenski and Tholen [1993]). While maximal closure operators defined by preradicals of modules have been present at least implicitly in the literature on torsion theories for some time, they appeared in the formal setting of closure operators not before Dikranjan and Giuli [1987a], and their study in the context of arbitrary categories is certainly new. Likewise, the concept of continuity of functors between categories equipped with closure operators, and the notion of mixed continuity of morphisms with respect to two given closure operators were developed in the course of writing this book, in order to interpret abstractly the various constructions for transporting closure operators along functors. Of these, only the special, but fundamental case of a modification along a reflexion appears in the literature (see the Notes of Chapter 6). External closure operators were studied by Castellini [1986a]. The characterization of density classes with respect to a closure operator was given and communicated to the authors by Tonolo in 1992 and is due to appear in Tonolo [1995].Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • D. Dikranjan
    • 1
  • W. Tholen
    • 2
  1. 1.Department of MathematicsUniversity of UdineUdineItaly
  2. 2.Department of Mathematics and StatisticsYork UniversityNorth YorkCanada

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