A Categorical Approach to Convergence

Abstract

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 is then investigated
whose objects are the pairs consisting of objects of the original category and convergence classes on them.
We study full subcategories of this category which are obtained by imposing some natural convergence
axioms. In particular, we find sucient conditions for the subcategories to be cartesian closed. We also
investigate behavior of the closure operators associated with the convergence in a natural way.