Lemma 5.30.6. The category of topological groups has colimits and colimits commute with the forgetful functor to the category of groups.

**Proof.**
We will use the argument of Categories, Remark 4.25.2 to prove existence of colimits. Namely, suppose that $\mathcal{I} \to \textit{Top}$, $i \mapsto G_ i$ is a functor into the category $\textit{TopGroup}$ of topological groups. Then we can consider

This functor commutes with limits. Moreover, given any topological group $H$ and an element $(\varphi _ i : G_ i \to H)$ of $F(H)$, there is a subgroup $H' \subset H$ of cardinality at most $|\coprod G_ i|$ (coproduct in the category of groups, i.e., the free product on the $G_ i$) such that the morphisms $\varphi _ i$ map into $H'$. Namely, we can take the induced topology on the subgroup generated by the images of the $\varphi _ i$. Thus it is clear that the hypotheses of Categories, Lemma 4.25.1 are satisfied and we find a topological group $G$ representing the functor $F$, which precisely means that $G$ is the colimit of the diagram $i \mapsto G_ i$.

To see the statement on commutation with the forgetful functor to groups we will use Categories, Lemma 4.24.5. Indeed, the forgetful functor has a right adjoint, namely the functor which assigns to a group the corresponding chaotic (or indiscrete) topological group. $\square$

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