Categorical Structure of Closure Operators pp 305-329 | Cite as

# Dense Maps and Pullback Stability

## Abstract

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.

## Keywords

Equivalence Relation Closure Operator Dense Subgroup Open Mapping Theorem Pullback Diagram## Preview

Unable to display preview. Download preview PDF.

## Notes

- 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