In this work, we define a category Cap of covering approximation spaces whose morphisms are functions satisfying a refinement property. We give the relations among Cap, and the category Top of topological spaces and continuous functions, and the category Rere of reflexive approximation spaces and the relation preserving functions. Further, we discuss the textural versions diCap, dfDitop and diRere of these categories. Then we study the definability in Cap with respect to five covering-based approximation operators. In particular, it is observed that via the morphisms of Cap, we may get more information about the subsets of the universe. (C) 2018 Elsevier Inc. All rights reserved.