Dense Maps and Pullback Stability

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


In this chapter we briefly discuss a particular type of closure operator, called Lawvere-Tierney topology, which generalizes the notion of Grothendieck topology and is a fundamental tool in Sheaf- and Topos Theory: Lawvere-Tierney topologies are simply idempotent and weakly hereditary closure operators (with respect to the class of monomorphisms) such that dense subobjects are stable under pullback. Localizations (=reflective subcategories with finite-limit preserving reflector) give rise to such closure operators. A Lawvere-Tierney topology allows for an effective construction of the reflector into its Delta-subcategory, which we describe in detail.


Equivalence Relation Closure Operator Dense Subgroup Open Mapping Theorem Pullback Diagram 
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  1. The notion of initial morphism (with respect to a closure operator) appears in Dikranjan [1992] while the notion of openness is intrinsic to the notion of LawvereTierney topology (see Johnstone [1977]) which assumes every morphism to be open. Modal closure operators were investigated by Castellini, Koslowski and Strecker [1992b]; Theorem 9.3 is very much related to the work of Cassidy, Hébert and Kelly [1985]. The construction of Barr’s reflector appears in Barr [1988] and has been used by various authors; see, for example, Carboni and Mantovani [1994]. The notion of total density appears for the first time in Soundararajan [1968] for abelian groups. The term weak total density was used in Dikranjan and Shakhmatov [1992], although the notion appeared much earlier (under the name total density) in Dikranj an and Prodanov [1974] (see Dikranjan, Prodanov and Stoyanov [1989] for further information). Total closure operators operators were constructed by Tonolo [1995b] for applications to topological groups.Google Scholar

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© 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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