#abelian groups

Allen Hatcher

the stable homotopy category by Cary Malkiewich

Fundamental Theorem of Abelian Groups

Compute the colimit (in the category of abelian groups) of the diagram \[\mathbb{Z} \overset{2!}{\to} \mathbb{Z} \overset{3!}{\to} \mathbb{Z} \dots\] where the map \(\mathbb{Z} \overset{n}{\to} \mathbb{Z}\) factors into \(\mathbb{Z} \to n \mathbb{Z} \hookrightarrow \mathbb{Z}\). The answer's \(\mathbb{Q}\), it takes a little work, in general you just need your collection of transition maps to intersect every nontrivial subgroup of \(\mathbb{Z}\), because then you can specify where each \(\frac{1}{n}\) in \(\mathbb{Q}\) gets sent to to define the unique mediating morphism.