A net is a function from a directed set into a topological space, and it is said to converge to a point if certain conditions are satisfied. Similarly, a direct system is a function from a directed set into a category (e.g., abelian groups or topological spaces) that satisfies the additional property of having a mapping if one element is $\le$ another. Similarly, you can find the (direct) limit of a directed system.

I was wondering, is there any sense in which these two concepts have a common generalization? Both are different kinds of limits of functions of directed sets into another set. Could you put something like a topology on the category of abelian groups so that, in some sense, a directed system with a colimit is a net that converges?