Proposition 6

Let I be a directed set, let $(E_\alpha, f_{\beta\alpha})$ be a direct system of sets relative to I, let $E = \varinjlim E_\alpha$ be the direct limit, and for each $\alpha \in I$ let

be the canonical mapping. For each $\alpha \in I$, let $u_\alpha$ be a mapping of $E_\alpha$ into a set F such that

(23)

Then:

(a) There exists a unique mapping $u$ of E into F such that

(24)

(b) $u$ is surjective if and only if F is the union of the sets $u_\alpha(E_\alpha)$.

(c) $u$ is injective if and only if for each $\alpha \in I$ the relations $x \in E_\alpha$, $y \in E_\alpha$, $u_\alpha(x) = u_\alpha(y)$ imply that there exists $\beta \geqslant \alpha$ such that $f_{\beta\alpha}(x) = f_{\beta\alpha}(y)$.

(a) With the notation of no. 5, let $v$ be the mapping of G into F which agrees with $u_\alpha$ on $E_\alpha$ for each $\alpha \in I$ (Chapter II, § 4, no. 7, Proposition 8). The hypothesis implies that $v$ is compatible with the equivalence relation R (Chapter II, § 6, no. 5); hence there exists a unique mapping $u$ of $E = G/R$ into F such that $v = u \circ f$ (loc. cit.).

(b) Since E is the union of the $f_\alpha(E_\alpha)$, the relation $F = \bigcup_{\alpha \in I} u_\alpha(E_\alpha)$ is clearly necessary and sufficient for $u$ to be surjective.

(c) By Lemma 1 of no. 5 any two elements of $\mathbf{E}$ can always be written in the form $f_\alpha(x)$ and $f_\alpha(y)$, where $x \in \mathrm{E}_\alpha$ and $y \in \mathrm{E}_\alpha$, for a suitable choice of $\alpha \in \mathrm{I}$. It follows also from the lemma that the relation $f_\alpha(x) = f_\alpha(y)$ is equivalent to the existence of $\beta \geqslant \alpha$ such that $f_{\beta\alpha}(x) = f_{\beta\alpha}(y)$. Since $u_\alpha(x) = u(f_\alpha(x))$ and $u_\alpha(y) = u(f_\alpha(y))$, this completes the proof.

¶ If the mapping $u$ is bijective, it is sometimes said, by abuse of language, that F is the direct limit of the family $(\mathrm{E}_\alpha)$.