Remark 4.25.2. The lemma above is often used to construct the free something on something. For example the free abelian group on a set, the free group on a set, etc. The idea, say in the case of the free group on a set $E$ is to consider the functor

$F : \textit{Groups} \to \textit{Sets},\quad G \longmapsto \text{Map}(E, G)$

This functor commutes with limits. As our family of objects we can take a family $E \to G_ i$ consisting of groups $G_ i$ of cardinality at most $\max (\aleph _0, |E|)$ and set maps $E \to G_ i$ such that every isomorphism class of such a structure occurs at least once. Namely, if $E \to G$ is a map from $E$ to a group $G$, then the subgroup $G'$ generated by the image has cardinality at most $\max (\aleph _0, |E|)$. The lemma tells us the functor is representable, hence there exists a group $F_ E$ such that $\mathop{\mathrm{Mor}}\nolimits _{\textit{Groups}}(F_ E, G) = \text{Map}(E, G)$. In particular, the identity morphism of $F_ E$ corresponds to a map $E \to F_ E$ and one can show that $F_ E$ is generated by the image without imposing any relations.

Another typical application is that we can use the lemma to construct colimits once it is known that limits exist. We illustrate it using the category of topological spaces which has limits by Topology, Lemma 5.14.1. Namely, suppose that $\mathcal{I} \to \textit{Top}$, $i \mapsto X_ i$ is a functor. Then we can consider

$F : \textit{Top} \longrightarrow \textit{Sets},\quad Y \longmapsto \mathop{\mathrm{lim}}\nolimits _\mathcal {I} \mathop{\mathrm{Mor}}\nolimits _{\textit{Top}}(X_ i, Y)$

This functor commutes with limits. Moreover, given any topological space $Y$ and an element $(\varphi _ i : X_ i \to Y)$ of $F(Y)$, there is a subspace $Y' \subset Y$ of cardinality at most $|\coprod X_ i|$ such that the morphisms $\varphi _ i$ map into $Y'$. Namely, we can take the induced topology on the union of the images of the $\varphi _ i$. Thus it is clear that the hypotheses of the lemma are satisfied and we find a topological space $X$ representing the functor $F$, which precisely means that $X$ is the colimit of the diagram $i \mapsto X_ i$.

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