Here are some examples of direct colimits of sets.
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The Prüfer Group. The Prüfer group $\mathbb {Z}\webleft (p^{\infty }\webright )$ is defined as the direct colimit
\[ \mathbb {Z}\webleft (p^{\infty }\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{n\in \mathbb {N}}\webleft (\mathbb {Z}_{/p^{n}}\webright ); \]
see
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