Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 4.1.5.7. Let $\operatorname{\mathcal{C}}$ be a category, let $[0]$ denote the category having a single object and a single morphism, and let $F: \operatorname{\mathcal{C}}\rightarrow [0]$ be the unique functor. Then $F$ automatically satisfies condition $(A)$ of Definition 4.1.5.1. Condition $(B)$ asserts that for every morphism $v: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$, the composition map

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \quad \quad u \mapsto v \circ u \]

is bijective for every object $X \in \operatorname{\mathcal{C}}$: that is, $v$ is an isomorphism. It follows that $F$ is a fibration in groupoids if and only if the category $\operatorname{\mathcal{C}}$ is a groupoid. Similarly, $F$ is an opfibration in groupoids if and only if $\operatorname{\mathcal{C}}$ is a groupoid.