Kerodon

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Proposition 2.4.4.13. Let $S_{\bullet }$ be a simplicial set and let $G$ be its underlying directed graph (Example 1.1.5.4), so that $G_{\bullet }$ can be identified with the $1$-skeleton of $S_{\bullet }$. Let $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Path}[ S])$ denote the unit map. Then:

  • The restriction $u|_{ G_{\bullet } }$ factors uniquely as a composition

    \[ G_{\bullet } \xrightarrow {u_0} \operatorname{N}_{\bullet }( \operatorname{Path}[S ]_0 ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Path}[ S] ). \]
  • The map $u_0$ induces an isomorphism of categories $\operatorname{Path}[G] \xrightarrow {\sim } \operatorname{Path}[S]_0$.

Proof. The first assertion follows immediately from Example 2.4.3.9, since $G_{\bullet }$ is a simplicial set of dimension $\leq 1$. To prove the second assertion, we note that Theorem 2.4.4.10 guarantees that $\operatorname{Path}[S]_0$ is a free category, whose objects can be identified with the vertices of $S_{\bullet }$ and whose indecomposable morphisms can be identified with elements of the set $E(S,0)$ of Notation 2.4.4.9. By definition, $E(S,m)$ consists of pairs $(\sigma , \overrightarrow {I} )$, where $\sigma $ is a nondegenerate $n$-simplex of $S_{\bullet }$ for $n > 0$ and $\overrightarrow {I} = (I_0 \supseteq \cdots \supseteq I_ m)$ is a chain of subsets of $[n]$ satisfying $I_0 = [n]$ and $I_ m = \{ 0, n \} $. In the case $m =0$, the equality $I_0 = I_ m$ forces $n = 1$, so that $E(S,0)$ can be identified (via the morphism $u_0$) with the collection of nondegenerate $1$-simplices of $S_{\bullet }$: that is, with the collection of edges of the graph $G$. The freeness of $\operatorname{Path}[ S ]_0$ now guarantees that the induced map $\operatorname{Path}[G] \simeq \operatorname{Path}[S]_0$ is an isomorphism of categories (see Proposition 1.2.6.11). $\square$