We define the concept of a convergence class on an object of a given category by using certain generalized nets for expressing the convergence. The resulting topological category, whose objects are the pairs consisting of objects of the original category and convergence classes on them, is then investigated. We study the full subcategories of this category which are obtained by imposing on it some natural convergence axioms. In particular, we find sufficient conditions for the subcategories to be cartesian closed. We also investigate the behavior of the closure operator associated with the convergence in a natural way.