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.

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