The Stacks project

Definition 4.33.6. Assume $p : \mathcal{S} \to \mathcal{C}$ is a fibred category.

  1. A choice of pullbacks1 for $p : \mathcal{S} \to \mathcal{C}$ is given by a choice of a strongly cartesian morphism $f^\ast x \to x$ lying over $f$ for any morphism $f: V \to U$ of $\mathcal{C}$ and any $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$.

  2. Given a choice of pullbacks, for any morphism $f : V \to U$ of $\mathcal{C}$ the functor $f^* : \mathcal{S}_ U \to \mathcal{S}_ V$ described above is called a pullback functor (associated to the choices $f^*x \to x$ made above).

[1] This is probably nonstandard terminology. In some texts this is called a “cleavage” but it conjures up the wrong image. Maybe a “cleaving” would be a better word. A related notion is that of a “splitting”, but in many texts a “splitting” means a choice of pullbacks such that $g^*f^* = (f \circ g)^*$ for any composable pair of morphisms. Compare also with Definition 4.36.2.

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Your email address will not be published.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