Kerodon

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

Example 5.1.3.6. Let $q: X \rightarrow S$ be a morphism of simplicial sets and let $e$ be an edge of $X$ such that $q(e) = \operatorname{id}_{s}$ is a degenerate edge of $S$. Suppose that the fiber $X_{s} = \{ s\} \times _{S} X$ is an $\infty$-category (this condition is satisfied, for example, if $q$ is an inner fibration). The following conditions are equivalent:

• The edge $e$ is locally $q$-cartesian.

• The edge $e$ is locally $q$-cocartesian.

• The edge $e$ is an isomorphism in the $\infty$-category $X_{s}$.

To prove this, we can use Remark 5.1.3.5 to reduce to the situation where $S = \{ s\}$ consists of a single vertex. In this case, the edge $e$ is locally $q$-cartesian if and only if it is $q$-cartesian, and locally $q$-cocartesian if and only if it is $q$-cocartesian (Remark 5.1.3.4). The desired result now follows from Example 5.1.1.4.