Categories of rough sets and textures

It is known that the theories of rough sets and fuzzy sets have successful applications in computing. Textures, as a theoretical model, provide a new perspective for both rough sets and fuzzy sets. Indeed, recent papers have shown that there is a natural link between rough sets and textures while a texture is an alternative point-set based setting for fuzzy sets. Relations are representatives of information systems and induce approximation operators. Therefore, the first step for the categorical discussions on rough sets involves the category REL of sets and relations. In this context, we observe that power sets and pairs of rough set approximation operators form a category denoted by R-APR. In particular, we prove that R-APR is isomorphic to a full subcategory of the category cdrTex whose objects are complemented textures and morphisms are complemented direlations. Therefore, cdrTex may be regarded as a suitable abstract model of rough set theory. Here, we show that R-APR and cdrTex are new examples of dagger symmetric monoidal categories.