2.2.6 Direct Colimits
Let $\webleft (X_{\alpha },f_{\alpha \beta }\webright )_{\alpha ,\beta \in I}\colon \webleft (I,\preceq \webright )\to \mathsf{Top}$ be a direct system of sets.
The direct colimit of $\webleft (X_{\alpha },f_{\alpha \beta }\webright )_{\alpha ,\beta \in I}$ is the direct colimit of $\webleft (X_{\alpha },f_{\alpha \beta }\webright )_{\alpha ,\beta \in I}$ in $\mathsf{Sets}$ as in 
, .
The direct colimit of $\webleft (X_{\alpha },f_{\alpha \beta }\webright )_{\alpha ,\beta \in I}$ is the pair $\smash {\Big(\displaystyle \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright ),\webleft\{ \mathrm{inj}_{\alpha }\webright\} _{\alpha \in I}\Big)}$ consisting of:
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The Colimit. The set $\displaystyle \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$ defined by
\[ \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left.\webleft(\coprod _{\alpha \in I}X_{\alpha }\webright)\middle /\mathord {\sim }\right., \]
where $\mathord {\sim }$ is the equivalence relation on $\coprod _{\alpha \in I}X_{\alpha }$ generated by declaring $\webleft (\alpha ,x\webright )\sim \webleft (\beta ,y\webright )$ iff there exists some $\gamma \in I$ satisfying the following conditions:
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We have $\alpha \preceq \gamma $.
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We have $\beta \preceq \gamma $.
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We have $f_{\alpha \gamma }\webleft (x\webright )=f_{\beta \gamma }\webleft (y\webright )$.
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The Cocone.The collection
\[ \webleft\{ \mathrm{inj}_{\gamma }\colon X_{\gamma }\to \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )\webright\} _{\gamma \in I} \]
of maps of sets defined by
\[ \mathrm{inj}_{\gamma }\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (\gamma ,x\webright )\webright ] \]
for each $\gamma \in I$ and each $x\in X_{\gamma }$.
We will prove Construction 2.2.6.1.2 below in a bit, but first we need a lemma (which is interesting in its own right).
For each $\alpha ,\beta \in I$ and each $x\in X_{\alpha }$, if $\alpha \preceq \beta $, then we have
\[ \webleft (\alpha ,x\webright )\sim \webleft (\beta ,f_{\alpha \beta }\webleft (x\webright )\webright ) \]
in $\displaystyle \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$.
Taking $\gamma =\beta $, we have $f_{\alpha \gamma }=f_{\alpha \beta }$, we have $f_{\beta \gamma }=f_{\beta \beta }\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{X_{\beta }}$, and we have
\begin{align*} f_{\alpha \beta }\webleft (x\webright ) & = f_{\beta \beta }\webleft (f_{\alpha \beta }\webleft (x\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{X_{\beta }}\webleft (f_{\alpha \beta }\webleft (x\webright )\webright ),\\ & = f_{\alpha \beta }\webleft (x\webright ).\end{align*}
As a result, since $\alpha \preceq \beta $ and $\beta \preceq \beta $ as well, Item (a), Item (b), and Item (c) of Construction 2.2.6.1.2 are met. Thus we have $\webleft (\alpha ,x\webright )\sim \webleft (\beta ,f_{\alpha \beta }\webleft (x\webright )\webright )$.
We can now prove Construction 2.2.6.1.2:
We claim that $\operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$ is the colimit of the direct system of sets $\webleft (X_{\alpha },f_{\alpha \beta }\webright )_{\alpha ,\beta \in I}$.
Commutativity of the Colimit Diagram
First, we need to check that the colimit diagram defined by $\operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$ commutes, i.e. that we have for each $\alpha ,\beta \in I$ with $\alpha \preceq \beta $. Indeed, given $x\in X_{\alpha }$, we have
\begin{align*} \webleft [\mathrm{inj}_{\beta }\circ f_{\alpha \beta }\webright ]\webleft (x\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{inj}_{\beta }\webleft (f_{\alpha \beta }\webleft (x\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (\beta ,f_{\alpha \beta }\webleft (x\webright )\webright )\webright ]\\ & = \webleft [\webleft (\alpha ,x\webright )\webright ]\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{inj}_{\alpha }\webleft (x\webright ), \end{align*}
where we have used Lemma 2.2.6.1.3 for the third equality.
Proof of the Universal Property of the Colimit
Next, we prove that $\operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$ as constructed in Construction 2.2.6.1.2 satisfies the universal property of a direct colimit. Suppose that we have, for each $\alpha ,\beta \in I$ with $\alpha \preceq \beta $, a diagram of the form
in $\mathsf{Sets}$. We claim that there exists a unique map $\phi \colon \smash {\displaystyle \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )}\overset {\exists !}{\to }C$ making the diagram
commute. To this end, first consider the diagram
Lemma. If $\webleft (\alpha ,x\webright )\sim \webleft (\beta ,y\webright )$, then we have
\[ \webleft[\coprod _{\alpha \in I}i_{\alpha }\webright]\webleft (x\webright )=\webleft[\coprod _{\alpha \in I}i_{\alpha }\webright]\webleft (y\webright ). \]
Proof. Indeed, if $\webleft (\alpha ,x\webright )\sim \webleft (\beta ,y\webright )$, then there exists some $\gamma \in I$ satisfying the following conditions:
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We have $\alpha \preceq \gamma $.
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We have $\beta \preceq \gamma $.
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We have $f_{\alpha \gamma }\webleft (x\webright )=f_{\beta \gamma }\webleft (y\webright )$.
We then have
\begin{align*} \webleft[\coprod _{\alpha \in I}i_{\alpha }\webright]\webleft (x\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}i_{\alpha }\webleft (x\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [i_{\gamma }\circ f_{\alpha \gamma }\webright ]\webleft (x\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}i_{\gamma }\webleft (f_{\alpha \gamma }\webleft (x\webright )\webright )\\ & = i_{\gamma }\webleft (f_{\beta \gamma }\webleft (x\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [i_{\gamma }\circ f_{\beta \gamma }\webright ]\webleft (x\webright )\\ & = i_{\beta }\webleft (y\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft[\coprod _{\alpha \in I}i_{\alpha }\webright]\webleft (y\webright ). \end{align*}
This finishes the proof of the lemma. Continuing, by , of , there then exists a map $\phi \colon \smash {\displaystyle \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )}\overset {\exists !}{\to }C$ making the diagram
commute. In particular, this implies that the diagram
also commutes, and thus so does the diagram
This finishes the proof.1
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 .