A Note on Categories about Rough Sets
This work provides foundational insights for researchers in rough set theory and category theory, but it is incremental as it builds on existing categorical frameworks without introducing new paradigms.
The paper tackles the problem of understanding the categorical structure of rough set theory by defining and relating categories of approximation spaces, rough closure/interior spaces, and information systems, proving an intrinsic property and establishing relationships between these categories.
Using the concepts of category and functor, we provide some insights and prove an intrinsic property of the category ${\bf AprS}$ of approximation spaces and relation-preserving functions, the category ${\bf RCls}$ of rough closure spaces and continuous functions, and the category ${\bf RInt}$ of rough interior spaces and continuous functions. Furthermore, we define the category ${\bf IS}$ of information systems and O-A-D homomorphisms, and establish the relationship between the category ${\bf IS}$ and the category ${\bf AprS}$ by considering a subcategory ${\bf NeIS}$ of ${\bf IS}$ whose objects are information systems and whose arrows are non-expensive O-A-D homomorphisms with surjective attribute mappings.