Remark 2.4.4.19. In the situation of Corollary 2.4.4.18, the simplicial subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{Path}[Q]_{\bullet }$ can be described more concretely:
The objects of $\operatorname{\mathcal{C}}$ are elements of the subset $Q_{-} \cup Q_{+} \subseteq Q$.
Let $a$ and $b$ be objects of $\operatorname{\mathcal{C}}$, and write $\operatorname{Hom}_{ \operatorname{Path}[Q]}(a,b)_{\bullet } = \operatorname{N}_{\bullet }(P_{a,b} )$, where $P_{a,b}$ is the collection finite linearly ordered $J \subseteq Q$ having smallest element $a$ and largest element $b$, ordered by reverse inclusion. Then $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(a,b)_{\bullet }$ can be identified with the nerve of the partially ordered subset $P'_{a,b} \subseteq P_{a,b}$ given by
\[ P'_{a,b} = \begin{cases} \{ J \in P_{a,b}: q \in J \} & \text{ if } a \leq q \leq b \\ P_{a,b} & \text{ otherwise. } \end{cases} \]
Stated more informally, $\operatorname{\mathcal{C}}$ is a simplicial subcategory of $\operatorname{Path}[Q]_{\bullet }$ whose morphisms are paths which, when possible, contain the element $q$.