# Kerodon

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

Definition 5.1.1.1. Let $q: X \rightarrow S$ be a morphism of simplicial sets, and let $e$ be an edge of $X$. We say that $e$ is $q$-cartesian if every lifting problem

$\xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{n} \ar [r]^-{\sigma _0} \ar [d] & X \ar [d]^-{q} \\ \Delta ^ n \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur] & S }$

admits a solution, provided that $n \geq 2$ and the composite map

$\Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ n-1 < n \} ) \hookrightarrow \Lambda ^ n_{n} \xrightarrow {\sigma _0} X$

corresponds to the edge $e$.

We say that $e$ is $q$-cocartesian if every lifting problem

$\xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{0} \ar [r]^-{\sigma _0} \ar [d] & X \ar [d]^-{q} \\ \Delta ^ n \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur] & S }$

admits a solution, provided that $n \geq 2$ and the composite map

$\Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \hookrightarrow \Lambda ^ n_{0} \xrightarrow {\sigma _0} X$

corresponds to the edge $e$.