Categorification of a Set Relation Geometry Induced by Rough Operators

In the present work we investigate Pawlak's rough set theory from a categorical point of view, by introducing specific categories of {\em lower} and {\em upper operators} in order to analyze in a generalized setting the usual approximant operators of rough set theory. We determine several embeddings and isomorphisms between these categories and suitable categories of finitary matroids, set partitions and equivalence relations, some of which already investigated in recent papers. Using the aforementioned isomorphic categories, we exhibit several categorical properties of lower and upper operators. In addition, as one of the main applications of rough set theory concerns Pawlak's information systems and Granular Computing, in the last part of the paper we translate in categorical terms the occurrence of rough sets in Granular Computing and, to this end, we need to work with a category ${\bf PR}$ of {\em pairings} (i.e. generalizations of Pawlak's information systems) and {\em pairing homomorphisms}. More specifically, we exhibit several categorical properties of pairings, such as balancedness, completeness, exactness, $(\textnormal{RegEpi},\textrm{Mono}$-$\textrm{Source})$-factorizability and prove that ${\bf PR}$ is Heyting but, in general, it does not admit coproducts.