Dual closure operators and their applications

Abstract

Departing from a suitable categorical description of closure operators, this paper dualizes this notion and introduces some basic properties of dual closure operators. Usually these operators act on quotients rather than subobjects, and much attention is being paid here to their key examples in algebra and topology, which include the formation of monotone quotients (Eilenberg-Whyburn) and concordant quotients (Coffins). In fair categorical generality, these constructions are shown to be factors of the fundamental correspondence that relates connectecinesses and disconnectednesses in topology, as well as torsion classes and torsion-free classes in algebra. Depending on a given cogenerator, the paper also establishes a non-trivial correspondence between closure operators and dual closure operators in the category of R-modules. Dual closure operators must be carefully distinguished from interior operators that have been studied by other author

Epireflective subcategories and formal closure operators

“…Viewing M as the full subcategory of the arrow-category C 2 , closure operators on the codomain functor cod : M → C are precisely the Dikranjan-Giuli closure operators. Almost the same is true for Dikranjan-Tholen closure operators, as defined in [14], which generalize Dikranjan-Giuli closure operators by simply relaxing conditions on the class M (see also [12], [13] and [28] for intermediate generalizations). For Dikranjan-Tholen closure operators, the class M is an arbitrary class of morphisms containing isomorphisms and being closed under composition with them; the closure operators are then required to satisfy an additional assumption that each component of the natural transformation c is given by a morphism from the class M -our definition does not capture this additional requirement.…”

“…For Dikranjan-Tholen closure operators, the class M is an arbitrary class of morphisms containing isomorphisms and being closed under composition with them; the closure operators are then required to satisfy an additional assumption that each component of the natural transformation c is given by a morphism from the class M -our definition does not capture this additional requirement. M is a class of not necessarily monomorphisms already in the definition of a categorical closure operator given in [12]; however, instead of the additional condition on a closure operator as in [14], there is an additional "left-cancellation condition" on M as in [28] (although there M is a class of monomorphisms) -our definition of a closure operator for such M becomes the definition of a closure operator given in [12].…”

“…The motivation for the study of this type of closure operators comes from algebra, as explained in the Introduction (see the last section for some representative examples). Let us remark that these closure operators are not the same as dual closure operators studied in [14] (which are almost the same as coclosure operators in the sense of [12]). In the latter case, the functor to consider is the dual of the domain functor dom op : E op → C op .…”

Codenseness and Openness with Respect to an Interior Operator

“…It is shown in [24] that the category of groups does not have a categorical transformation, hence the two notions are not necessarily equivalent. Moreover, in any category for which all subobjects are normal, in particular, in all abelian categories (such as the category of modules over a ring, the category of all abelian groups), while there is an abundance of closure operators there is a unique interior operator, which is the discrete one (see [14]).…”

“…Since categorical interior operators are not compatible with taking images, they cannot be seen as endofunctors on a suitable class M of embeddings, hence the preservation property of interior operators fails; see [5][6][7]. Furthermore, the dual closure operator introduced in [14] is a categorical dual to closure operator and does not lead to interior operators.…”