Lemma 4.34.13. Let $\mathcal{A} \to \mathcal{B} \to \mathcal{C}$ be functors between categories. If $\mathcal{A}$ is fibred in groupoids over $\mathcal{B}$ and $\mathcal{B}$ is fibred in groupoids over $\mathcal{C}$, then $\mathcal{A}$ is fibred in groupoids over $\mathcal{C}$.

**Proof.**
One can prove this directly from the definition. However, we will argue using the criterion of Lemma 4.34.2. By Lemma 4.32.12 we see that $\mathcal{A}$ is fibred over $\mathcal{C}$. To finish the proof we show that the fibre category $\mathcal{A}_ U$ is a groupoid for $U$ in $\mathcal{C}$. Namely, if $x \to y$ is a morphism of $\mathcal{A}_ U$, then its image in $\mathcal{B}$ is an isomorphism as $\mathcal{B}_ U$ is a groupoid. But then $x \to y$ is an isomorphism, for example by Lemma 4.32.2 and the fact that every morphism of $\mathcal{A}$ is strongly $\mathcal{B}$-cartesian (see Lemma 4.34.2).
$\square$

## 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Your email address will not be published.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).

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.

## Comments (0)

There are also: