Remark 2.4.5.4 (00LM)

Remark 2.4.5.4. Let $n \geq 0$ be a nonnegative integer. For every pair of integers $0 \leq i < j \leq n$, we can identify morphisms from $i$ to $j$ in the path category $\operatorname{Path}[n]$ with subsets $S \subseteq [n]$ having least element $i$ and largest element $j$. The construction $S \mapsto ( \{ i, i+1, \ldots , j-1, j \} \setminus S )$ then induces a bijection $\operatorname{Hom}_{ \operatorname{Path}[n]}(i,j) \simeq P( \{ i+1, i+2, \ldots , j-2, j-1 \} )$, which extends to uniquely to an isomorphism of simplicial sets

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Path}[n] }(i, j )_{\bullet } & \simeq & \operatorname{N}_{\bullet }( P( \{ i+1, i+2, \ldots , j-2, j-1\} ) ) \\ & \simeq & \operatorname{\raise {0.1ex}{\square }}^{ \{ i+1, i+2, \ldots , j-2, j-1 \} } \\ & \simeq & \operatorname{\raise {0.1ex}{\square }}^{j-i-1}. \end{eqnarray*}

In particular, we have a canonical isomorphism of simplicial sets $\operatorname{Hom}_{ \operatorname{Path}[n] }(0,n)_{\bullet } \simeq \operatorname{\raise {0.1ex}{\square }}^{n-1}$.

Under these isomorphisms, the composition law on $\operatorname{Path}[n]_{\bullet }$ is given for $i < j < k$ by the construction

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Path}[n] }(j, k )_{\bullet } \times \operatorname{Hom}_{\operatorname{Path}[n] }(i, j )_{\bullet } & \simeq & \operatorname{\raise {0.1ex}{\square }}^{ \{ j+1, \ldots , k-1 \} } \times \operatorname{\raise {0.1ex}{\square }}^{ \{ i+1, \ldots , j-1 \} } \\ & \simeq & \operatorname{\raise {0.1ex}{\square }}^{ \{ j+1, \ldots , k-1 \} } \times \{ 0\} \times \operatorname{\raise {0.1ex}{\square }}^{ \{ i+1, \ldots , j-1 \} } \\ & \hookrightarrow & \operatorname{\raise {0.1ex}{\square }}^{ \{ j+1, \ldots , k-1 \} } \times \Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{ \{ i+1, \ldots , j-1 \} } \\ & \simeq & \operatorname{\raise {0.1ex}{\square }}^{ \{ i+1, \ldots , k-1 \} } \\ & \simeq & \operatorname{Hom}_{\operatorname{Path}[n] }(i, k)_{\bullet }. \end{eqnarray*}