Corollary 1

Tag 03RS, permanent. Cite it as [Bourbaki, Tag 03RS].

Theory of Sets, chapter III, § 7. Inverse limits and direct limits

Let $(\mathrm{E}_\alpha, f_{\beta\alpha})$ and $(\mathrm{F}_\alpha, g_{\beta\alpha})$ be two direct systems of sets relative to the same index set I; let $\mathrm{E} = \varinjlim \mathrm{E}_\alpha$ , $\mathrm{F} = \varinjlim \mathrm{F}_\alpha$ , and for each $\alpha \in \mathrm{I}$ let $f_\alpha$ (resp. $g_\alpha$ ) be the canonical mapping of $\mathrm{E}_\alpha$ (resp. $\mathrm{F}_\alpha$ ) into E (resp. F). For each $\alpha \in \mathrm{I}$ let $u_\alpha$ be a mapping of $\mathrm{E}_\alpha$ into $\mathrm{F}_\alpha$ such that, whenever $\alpha \leqslant \beta$ , the diagram

\begin{array}{ccc} \mathrm{E}_\alpha & \overset{u_\alpha}{\longrightarrow} & \mathrm{F}_\alpha \\ {\scriptstyle f_{\beta\alpha}}\big\downarrow & & \big\downarrow{\scriptstyle g_{\beta\alpha}} \\ \mathrm{E}_\beta & \underset{u_\beta}{\longrightarrow} & \mathrm{F}_\beta \end{array}

is commutative. Then there exists a unique mapping $u : \mathrm{E} \rightarrow \mathrm{F}$ such that, for each $\alpha \in \mathrm{I}$ , the diagram

\begin{array}{ccc} \mathrm{E}_\alpha & \overset{u_\alpha}{\longrightarrow} & \mathrm{F}_\alpha \\ {\scriptstyle f_\alpha}\big\downarrow & & \big\downarrow{\scriptstyle g_\alpha} \\ \mathrm{E} & \underset{u}{\longrightarrow} & \mathrm{F} \end{array}

is commutative.

Put $v_\alpha = g_\alpha \circ u_\alpha$ . If $\alpha \leqslant \beta$ , then by (22) we have

v_\beta \circ f_{\beta\alpha} = g_\beta \circ u_\beta \circ f_{\beta\alpha} = g_\beta \circ g_{\beta\alpha} \circ u_\alpha = g_\alpha \circ u_\alpha = v_\alpha.

We may therefore apply Proposition 6 to the mappings $v_\alpha$ , whence the existence and uniqueness of a mapping $u : \mathrm{E} \to \mathrm{F}$ such that

u \circ f_\alpha = v_\alpha = g_\alpha \circ u_\alpha

for all $\alpha \in \mathrm{I}$ .

¶ A family of mappings $u_\alpha : \mathrm{E}_\alpha \to \mathrm{F}_\alpha$ which satisfies the conditions of Corollary 1 is called a direct system of mappings of $(\mathrm{E}_\alpha, f_{\beta\alpha})$ into $(\mathrm{F}_\alpha, g_{\beta\alpha})$ . The mapping defined in Corollary 1 is called the direct limit of the family $(u_\alpha)$ and is written $u = \varinjlim u_\alpha$ when there is no risk of ambiguity.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Proposition 6