Definition 4.2.2.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. We say that $U$ is a fibration in groupoids if the following conditions are satisfied:
- $(A)$
For every object $Y \in \operatorname{\mathcal{E}}$ and every morphism $\overline{f}: \overline{X} \rightarrow U(Y)$ in $\operatorname{\mathcal{C}}$, there exists a morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{E}}$ with $\overline{X} = U(X)$ and $\overline{f} = U(f)$.
- $(B)$
For every morphism $g: Y \rightarrow Z$ in $\operatorname{\mathcal{E}}$ and every object $X \in \operatorname{\mathcal{E}}$, the diagram of sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y) \ar [r]^-{g \circ } \ar [d]^{U} & \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z) \ar [d]^{U} \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Y) ) \ar [r]^-{ U(g) \circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Z) ) } \]is a pullback square.
In this case, we will also say that $\operatorname{\mathcal{E}}$ is fibered in groupoids over $\operatorname{\mathcal{C}}$.