Dense Maps and Pullback Stability
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.
KeywordsEquivalence Relation Closure Operator Dense Subgroup Open Mapping Theorem Pullback Diagram
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- The notion of initial morphism (with respect to a closure operator) appears in Dikranjan  while the notion of openness is intrinsic to the notion of LawvereTierney topology (see Johnstone ) 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 . The construction of Barr’s reflector appears in Barr  and has been used by various authors; see, for example, Carboni and Mantovani . The notion of total density appears for the first time in Soundararajan  for abelian groups. The term weak total density was used in Dikranjan and Shakhmatov , although the notion appeared much earlier (under the name total density) in Dikranj an and Prodanov  (see Dikranjan, Prodanov and Stoyanov  for further information). Total closure operators operators were constructed by Tonolo [1995b] for applications to topological groups.Google Scholar