Definition 4.32.2. Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a category over $\mathcal{C}$.

The

*fibre category*over an object $U\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ is the category $\mathcal{S}_ U$ with objects\[ \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U) = \{ x\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}) : p(x) = U\} \]and morphisms

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{S}_ U}(x, y) = \{ \phi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {S}(x, y) : p(\phi ) = \text{id}_ U\} . \]A

*lift*of an object $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ is an object $x\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$ such that $p(x) = U$, i.e., $x\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$. We will also sometime say that*$x$ lies over $U$*.Similarly, a

*lift*of a morphism $f : V \to U$ in $\mathcal{C}$ is a morphism $\phi : y \to x$ in $\mathcal{S}$ such that $p(\phi ) = f$. We sometimes say that*$\phi $ lies over $f$*.

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