The Stacks project

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. $\square$


Comments (0)

There are also:

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

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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05NR. Beware of the difference between the letter 'O' and the digit '0'.