The Stacks project

Remark 90.6.4. Let $p : \mathcal{F} \to \mathcal{C}_\Lambda$ be a category cofibered in groupoids, and let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$. We denote by $\mathcal{F}_ x$ the category of objects over $x$. An object of $\mathcal{F}_ x$ is an arrow $y \to x$. A morphism $(y \to x) \to (z \to x)$ in $\mathcal{F}_ x$ is a commutative diagram

There is a forgetful functor $\mathcal{F}_ x \to \mathcal{F}$. We define the functor $p_ x : \mathcal{F}_ x \to \mathcal{C}_\Lambda$ as the composition $\mathcal{F}_ x \to \mathcal{F} \xrightarrow {p} \mathcal{C}_\Lambda$. Then $p_ x : \mathcal{F}_ x \to \mathcal{C}_\Lambda$ is a predeformation category (proof omitted). In this way we can pass from an arbitrary category cofibered in groupoids over $\mathcal{C}_\Lambda$ to a predeformation category at any $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$.