Lemma 2.2.6.1.3 (01FG): Identification of $x$ with $f_{\alpha \beta }\webleft (x\webright )$ in Direct Colimits

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

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 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:

We have $\alpha \preceq \gamma $.
We have $\beta \preceq \gamma $.
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

¹Incidentally, the conditions

\[ \webleft\{ i_{\alpha }=\phi \circ \mathrm{inj}_{\alpha }\webright\} _{\alpha \in I} \]

show that $\phi $ must be given by

\[ \phi \webleft (\webleft [\webleft (\alpha ,x\webright )\webright ]\webright )=\webleft (i_{\alpha }\webleft (x\webright )\webright )_{\alpha \in I} \]

for each $\webleft [\webleft (\alpha ,x\webright )\webright ]\in \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$, although we would need to show that this assignment is well-defined were we to prove Construction 2.2.6.1.2 in this way. Instead, invoking , of gave us a way to avoid having to prove this, leading to a cleaner alternative proof.