I'd like to clear up something that came up in the comments.  There are **two** natural ways to fit the finite cyclic groups together in a <a href="http://en.wikipedia.org/wiki/Diagram_%28category_theory%29">diagram</a>.  One is to take the morphisms $\mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/m\mathbb{Z}, m | n$ given by sending $1$ to $1$.  This gives a diagram (inverse system) whose <a href="http://en.wikipedia.org/wiki/Limit_%28category_theory%29">limit</a> (inverse limit) is the <a href="http://en.wikipedia.org/wiki/Profinite_group#Profinite_completion">profinite completion</a> of $\mathbb{Z}$.  This diagram also makes sense in the category of unital rings, since they also respect the ring structure, giving the profinite integers the structure of a commutative ring.  

This is **not** the diagram relevant to understanding the circle group.  Instead, one needs to take the morphisms $\mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/m\mathbb{Z}, n | m$ given by sending $1$ to $\frac{m}{n}$.  This is the diagram relevant to understanding the cyclic groups as subgroups of their colimit (direct limit), which is, as I have said, $\mathbb{Q}/\mathbb{Z}$.  And this group, in turn, "compactifies" or "completes" to the circle group in whichever way you prefer.