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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: