Lemma 19.2.3. Suppose that, in (19.2.0.1), $\mathcal{C}$ is the category of sets and $A$ is a finite set, then the map is a bijection.

Proof. Let $f : A \to \mathop{\mathrm{colim}}\nolimits B_\beta$. The range of $f$ is finite, containing say elements $c_1, \ldots , c_ r \in \mathop{\mathrm{colim}}\nolimits B_\beta$. These all come from some elements in $B_\beta$ for $\beta \in \alpha$ large by definition of the colimit. Thus we can define $\widetilde{f} : A \to B_\beta$ lifting $f$ at a finite stage. This proves that (19.2.0.1) is surjective. Next, suppose two maps $f : A \to B_\gamma , f' : A \to B_{\gamma '}$ define the same map $A \to \mathop{\mathrm{colim}}\nolimits B_\beta$. Then each of the finitely many elements of $A$ gets sent to the same point in the colimit. By definition of the colimit for sets, there is $\beta \geq \gamma , \gamma '$ such that the finitely many elements of $A$ get sent to the same points in $B_\beta$ under $f$ and $f'$. This proves that (19.2.0.1) is injective. $\square$

There are also:

• 2 comment(s) on Section 19.2: Baer's argument for modules

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).