Kerodon

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

Variant 8.4.0.27. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor. The following conditions are equivalent:

  • The functor $U$ is an opfibration in groupoids (Definition 4.2.2.1).

  • The functor $U$ is a cocartesian fibration and every morphism of $\operatorname{\mathcal{E}}$ is $U$-cocartesian.

  • The functor $U$ is a cocartesian fibration and, for every object $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a groupoid.