Interior operators and their applications

Abstract

Categorical closure operators were introduced by Dikranjan and Giuli in [DG87] and then developed by these authors and Tholen in [DGT89]. These operators have played an important role in the development of Categorical Topology by introducing topological concepts, such as connectedness, separatedness and compactness, in an arbitrary category and they provide a uni ed approach to various mathematical notions. Motivated by the theory of these operators, the categorical notion of interior operators was introduced by Vorster in [Vor00]. While there is a notational symmetry between categorical closure and interior operators, a detailed analysis shows that the two operators are not categorically dual to each other, that is: it is not true in general that whatever one does with respect to closure operators may be done relative to interior operators. Indeed, the continuity condition of categorical closure operators can be expressed in terms of images or equivalently, preimages, in the same way as the usual topological closure describes continuity in terms of images or preimages along continuous maps. However, unlike the case of categorical closure operators, the continuity condition of categorical interior operators can not be described in terms of images. Consequently, the general theory of categorical interior operators is not equivalent to the one of closure operators. Moreover, the categorical dual closure operator introduced in [DT15] does not lead to interior operators. As a consequence, the study of categorical interior operators in their own right is interesting.