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Example To every simplicial set $X_{\bullet }$, we can associate a directed graph $\mathrm{Gr}( X_{\bullet } )$ as follows:

  • The vertex set $\operatorname{Vert}( \mathrm{Gr}(X_{\bullet } ) )$ is the set of $0$-simplices of the simplicial set $X_{\bullet }$.

  • The edge set $\operatorname{Edge}( \mathrm{Gr}(X_{\bullet } ) )$ is the set of nondegenerate $1$-simplices of the simplicial set $X_{\bullet }$.

  • For every edge $e \in \operatorname{Edge}( \mathrm{Gr}(X_{\bullet }) ) \subseteq X_1$, the source $s(e)$ is the vertex $d_1(e)$, and the target $t(e)$ is the vertex $d_0(e)$ (here $d_0, d_1: X_1 \rightarrow X_0$ are the face maps of Notation