# Colimit of $\frac{1}{n} \mathbb{Z}$

We should have $\displaystyle\mathbb{Q} = \lim_{\rightarrow} \frac 1n \mathbb{Z}$ but a few things are confusing me. Since the index category is a set, we should get the coproduct: $\bigsqcup \frac 1n \mathbb{Z}$. But in this coproduct we have $1/2 \neq 2/4$ which clearly is wrong. This make me suspect that the index category is not an index set. What we probably should get is a union (and not a coproduct) in the category of rings (in which $1/2 = 2/4$).

Question: Help me understand what the index category is (eg, what are its maps) and am I correct when believing we should get an ordinary union?

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Just a clarification: what is $\frac{1}{n}\mathbb{Z}$ ? – magma Mar 26 '13 at 0:30

Index category is the category of positive integers with a morphism from n to m iff n|m. The limit is not coproduct but union of $\frac{1}{n}\mathbb{Z}$ with natural embedings (which is the same as union of $\frac{1}{n}\mathbb{Z}$ inside $\mathbb{Q}$ — hence the equality).