The Stacks project

Lemma 8.2.5. Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category, see Categories, Section 4.33. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and let $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$. Denote $x, y : \mathcal{C}/U \to \mathcal{S}$ also the corresponding $1$-morphisms, see Categories, Lemma 4.41.2. Then

  1. the $2$-fibre product $\mathcal{S} \times _{\mathcal{S} \times \mathcal{S}, (x, y)} \mathcal{C}/U$ is fibred in setoids over $\mathcal{C}/U$, and

  2. $\mathit{Isom}(x, y)$ is the presheaf of sets corresponding to this category fibred in setoids, see Categories, Lemma 4.39.6.

Proof. Omitted. Hint: Objects of the $2$-fibre product are $(a : V \to U, z, (\alpha , \beta ))$ where $\alpha : z \to a^*x$ and $\beta : z \to a^*y$ are isomorphisms in $\mathcal{S}_ V$. Thus the relationship with $\mathit{Isom}(x, y)$ comes by assigning to such an object the isomorphism $\beta \circ \alpha ^{-1}$. $\square$

Comments (2)

Comment #7555 by Jiangfan on

There are two in the object of the 2-fibre product, one of them should be redundant?

There are also:

  • 2 comment(s) on Section 8.2: Presheaves of morphisms associated to fibred categories

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Your email address will not be published.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



View Lemma 8.2.5 as pdf