Kerodon

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Corollary 5.1.1.15. Let $q: X \rightarrow S$ be a morphism of simplicial sets and let $e: x \rightarrow y$ be an edge of $X$. The following conditions are equivalent:

$(1)$

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

$(2)$

Let $f: B \rightarrow X$ be a morphism of simplicial sets, let $A$ be a simplicial subset of $B$, and let $Y$ denote the fiber product $X_{f|_{A} / } \times _{ S_{ (q \circ f|_{A})/} } S_{ (q \circ f)/ }$, so that the restriction map $X_{f/} \rightarrow X$ factors as a composition $X_{f/} \xrightarrow {\theta } Y \xrightarrow {\rho } X$. Then every lifting problem

$\xymatrix@R =50pt@C=50pt{ \{ 1\} \ar [r] \ar [d] & X_{f/} \ar [d]^-{\theta } \\ \Delta ^1 \ar [r]^-{e'} \ar@ {-->}[ur] & Y }$

admits a solution, provided that $\rho (e') = e$.

Proof. For a fixed simplicial set $B$ with a simplicial subset $A \subseteq B$, condition $(2)$ is equivalent to the requirement that every lifting problem

$\xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & X_{/e} \ar [d] \\ B \ar [r] \ar@ {-->}[ur] & X_{/y} \times _{ S_{/q(y)} } S_{ /q(e) } }$

admits a solution. This condition is satisfied for every inclusion of simplicial sets $A \subseteq B$ if and only if the map $X_{/e} \rightarrow X_{/y} \times _{ S_{/q(y)} } S_{ /q(e) }$ is a trivial Kan fibration: that is, if and only if $e$ is $q$-cartesian (Proposition 5.1.1.13). $\square$