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Remark 5.1.1.11. Let $q: X \rightarrow S$ be a morphism of simplicial sets and let $e$ be an edge of the simplicial set $X$. The following conditions are equivalent:

  • The edge $e$ is $q$-cartesian.

  • For every pullback diagram of simplicial sets

    \[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^-{q'} \ar [r]^-{f} & X \ar [d]^-{q} \\ S' \ar [r] & S, } \]

    and every edge $e'$ of $X'$ satisfying $f(e') = e$, the edge $e'$ is $q'$-cartesian.

  • For every pullback diagram of simplicial sets

    \[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^-{q'} \ar [r]^-{f} & X \ar [d]^-{q} \\ \Delta ^ n \ar [r] & S } \]

    and every edge $e'$ of $X'$ satisfying $f(e') = e$, the edge $e'$ is $q'$-cartesian.