Direct Limits and Limits of Nets

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?

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– Todd Trimble Aug 1 '11 at 10:19
Unless someone has an answer that goes beyond the above threads, Todd is right (at least about the first two; the third is a bit different). – David Corwin Aug 1 '11 at 10:35