The Stacks project

Lemma 4.2.19. A functor is an equivalence of categories if and only if it is both fully faithful and essentially surjective.

Proof. Let $F : \mathcal{A} \to \mathcal{B}$ be essentially surjective and fully faithful. As by convention all categories are small and as $F$ is essentially surjective we can, using the axiom of choice, choose for every $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ an object $j(X)$ of $\mathcal{A}$ and an isomorphism $i_ X : X \to F(j(X))$. Then we apply Lemma 4.2.18 using that $F$ is fully faithful. $\square$

Comments (0)

There are also:

  • 7 comment(s) on Section 4.2: Definitions

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Your email address will not be published.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 02C3. Beware of the difference between the letter 'O' and the digit '0'.