The Stacks project

Definition 8.2.2. Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category, see Categories, Section 4.33. Given an object $U$ of $\mathcal{C}$ and objects $x$, $y$ of the fibre category, the presheaf of morphisms from $x$ to $y$ is the presheaf

\[ (f : V \to U) \longmapsto \mathop{\mathrm{Mor}}\nolimits _{\mathcal{S}_ V}(f^*x, f^*y) \]

described above. It is denoted $\mathit{Mor}(x, y)$. The subpresheaf $\mathit{Isom}(x, y)$ whose value over $V$ is the set of isomorphisms $f^*x \to f^*y$ in the fibre category $\mathcal{S}_ V$ is called the presheaf of isomorphisms from $x$ to $y$.

Comments (2)

Comment #8651 by Yu on

Typo: "The subpresheaf Isom(x,y) whose values over V is the set of isomorphisms..." should be "The subpresheaf Isom(x,y) whose value over V is the set of isomorphisms...".

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Your email address will not be published.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



View Definition 8.2.2 as pdf