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
\[ F : \textit{TopGroup} \longrightarrow \textit{Sets},\quad H \longmapsto \mathop{\mathrm{lim}}\nolimits _\mathcal {I} \mathop{\mathrm{Mor}}\nolimits _{\textit{TopGroup}}(G_ i, H) \]
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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