Kerodon

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Proposition 5.1.1.13. Let $q: X \rightarrow S$ be a morphism simplicial sets and let $e: x \rightarrow y$ be an edge of $X$. Then:

  • The edge $e$ is $q$-cartesian if and only if the natural map

    \[ X_{/e} \rightarrow X_{/y} \times _{ S_{/q(y)} } S_{ /q(e) } \]

    is a trivial Kan fibration of simplicial sets.

  • The edge $e$ is $q$-cocartesian if and only if the natural map

    \[ X_{e/} \rightarrow X_{x/} \times _{ S_{q(x)/} } S_{ q(e) / } \]

    is a trivial Kan fibration of simplicial sets.

Proof. We will prove the first assertion; the proof of the second is similar. By definition, the natural map $X_{/e} \rightarrow X_{/y} \times _{ S_{/q(y)} } S_{ / q(e) }$ is a trivial Kan fibration if and only if, for every integer $n \geq 0$, every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & X_{/e} \ar [d] \\ \Delta ^{n} \ar@ {-->}[ur] \ar [r] & X_{/y} \times _{ S_{/q(y)} } S_{ /q(e) } } \]

admits a solution. By virtue of Lemma 4.3.6.14, this is equivalent to the datum of a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+2}_{n+2} \ar [r]^-{\sigma _0} \ar [d] & X \ar [d] \\ \Delta ^{n+2} \ar@ {-->}[ur] \ar [r] & S, } \]

where $\sigma _0$ carries the final edge $\operatorname{N}_{\bullet }( \{ n+1 < n+2 \} ) \subseteq \Lambda ^{n+2}_{n+2}$ to $e$. $\square$