The Stacks project

Lemma 4.39.8. Let $\mathcal{C}$ be a category. The construction of Lemma 4.39.6 which associates to a category fibred in setoids a presheaf is compatible with products, in the sense that the presheaf associated to a $2$-fibre product $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ is the fibre product of the presheaves associated to $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$.

Proof. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. The lemma just says that

\[ \mathop{\mathrm{Ob}}\nolimits ((\mathcal{X} \times _\mathcal {Y} \mathcal{Z})_ U)/\! \cong \quad \text{equals} \quad \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ U)/\! \cong \ \times _{\mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ U)/\! \cong } \ \mathop{\mathrm{Ob}}\nolimits (\mathcal{Z}_ U)/\! \cong \]

the proof of which we omit. (But note that this would not be true in general if the category $\mathcal{Y}_ U$ is not a setoid.) $\square$

Comments (0)

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Your email address will not be published.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 04SD. Beware of the difference between the letter 'O' and the digit '0'.