Kerodon

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

Corollary 4.1.5.11. Let $f: X \rightarrow S$ be a morphism of simplicial sets. Then $f$ is an inner covering if and only if, for every simplex $\sigma : \Delta ^ n \rightarrow S$, the fiber product $\Delta ^{n} \times _{S} X$ is isomorphic to the nerve of a category.

Proof. Suppose $f$ is an inner covering. For every simplex $\sigma : \Delta ^ n \rightarrow S$, it follows from Remark 4.1.5.7 that the projection map $\Delta ^ n \times _{S} X \rightarrow \Delta ^ n$ is also an inner covering map, so that $\Delta ^ n \times _{S} X$ is isomorphic to the nerve of a category by virtue of Proposition 4.1.5.10. Conversely, to show that $f$ is an inner covering map, it will suffice to show that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r] \ar@ {^{(}->}[d] & X \ar [d]^{f} \\ \Delta ^ n \ar [r] & S } \]

has a unique solution for $0 < i < n$. If the fiber product $\Delta ^{n} \times _{S} X$ is the nerve of a category, then the existence and uniqueness of the desired solution follow from (and uniqueness) of the desired solution follow from Proposition 1.2.3.1. $\square$