Basic Properties of Closure Operators
Categorical closure operators as defined in this chapter for any category with a suitable subobject structure provide simultaneously a coherent closure operation for the subobjects of each object of the category. The notions of closedness and denseness associated with a closure operator are discussed from a factorization point of view. This leads to a symmetric presentation of the fundamental properties of idempotency vis-a-vis weak hereditariness and of hereditariness vis-a-vis minimality. Further important properties are given by additivity and productivity which are briefly discussed at the end of the chapter.
KeywordsDirect Product Commutative Diagram Closure Operator Closure Operation Dosure Operator
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- The categorical notion of closure operator as introduced in 2.2 (with the key ingredient given by the continuity condition) goes back to Dikranjan and Giuli [1987a] and includes both, the lattice-theoretic closure operations (see Exercise 2.E) and the Lawvere-Tierney topologies or universal closure operations of Topos- and Sheaf Theory (see Chapter 9), as special instances. Principal properties like idempotency, (weak) hereditariness, and additivity are already discussed in the Dikranjan-Giuli paper, although the “symmetric” approach to idempotency / weak hereditariness and to hereditariness / minimality as given in 2.4 and 2.5 is not yet apparent in that paper. Earlier papers in Categorical Topology are mostly concerned with particular instances of closure operators, predominantly in the category of topological spaces (see Chapters 6–8). To be mentioned particularly is the paper by Cagliari and Cicchese  which introduces for epireflective subcategories of Top a stronger notion of closure operator, with the continuity condition stated explicitly as one of the axioms.Google Scholar