Remark 4.6.8.14 (01LM)

Remark 4.6.8.14. Let $K$ be a simplicial set. Note that, for $n > 0$, every nondegenerate simplex $\sigma : \Delta ^ n \rightarrow \Sigma (K)$ satisfies $\sigma (0) = x$ and $\sigma (n) = y$. Using Theorem 2.4.4.10, we see that for each $m \geq 0$, $\operatorname{Path}[ \Sigma (K) ]_{m}$ can be identified with the path category of a directed graph $G_ m$ with vertex set $\operatorname{Vert}(G_ m) = \{ x,y\} $, where each edge of $G_ m$ has source $x$ and target $y$. These path categories are easy to describe: they satisfy

\[ \operatorname{Hom}_{ \operatorname{Path}[G_ m] }( x, x) = \{ \operatorname{id}_{x} \} \quad \quad \operatorname{Hom}_{\operatorname{Path}[G_ m] }( y,y) = \{ \operatorname{id}_{y} \} \]

\[ \operatorname{Hom}_{ \operatorname{Path}[G_ m] }(x,y) = \operatorname{Edge}(G_ m) \quad \quad \operatorname{Hom}_{ \operatorname{Path}[G_ m] }( y,x) = \emptyset . \]

Allowing $m$ to vary, we conclude that the simplicial category $\operatorname{Path}[ \Sigma (K) ]_{\bullet }$ satisfies

\[ \operatorname{Hom}_{ \operatorname{Path}[\Sigma (K)] }(x,x)_{\bullet } = \{ \operatorname{id}_{x} \} \quad \operatorname{Hom}_{ \operatorname{Path}[\Sigma (K)] }(y,x)_{\bullet } = \emptyset \quad \operatorname{Hom}_{ \operatorname{Path}[\Sigma (K)] }(y,y)_{\bullet } = \{ \operatorname{id}_{y} \} . \]

That is, $\operatorname{Path}[\Sigma (K)]_{\bullet }$ can be identified with the simplicial category $\operatorname{\mathcal{E}}[ \Phi (X) ]$ of Notation 4.6.8.1. Moreover, we can identify $m$-simplices of $\Phi (X)$ with elements of the set $E( \Sigma (K), m)$ defined in Notation 2.4.4.9.