Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 1.2.6.2. Fix an integer $n \geq 0$. Let $G$ be the directed graph with vertex set $\operatorname{Vert}(G) = \{ v_0, v_1, \ldots , v_ n \} $, and edge set $\operatorname{Edge}(G) = \{ e_1, \ldots , e_ n \} $, where each edge $e_{i}$ has source $s(e_ i) = v_{i-1}$ and target $t(e_ i) = v_ i$; we can represent $G$ graphically by the diagram

\[ \xymatrix { v_0 \ar [r]^{ e_1 } & v_1 \ar [r]^{e_2} & \cdots \ar [r]^{e_{n-1}} & v_{n-1} \ar [r]^{e_ n} & v_ n. } \]

Let $v_ i$ and $v_ j$ be a pair of vertices of $G$. Then:

  • If $i \leq j$, there is a unique path from $v_ i$ to $v_ j$, given by the sequence of edges $(e_{j}, e_{j-1}, \ldots , e_{i+1})$.

  • If $i > j$, then there are no paths from $v_ i$ to $v_ j$.

It follows that the path category $\operatorname{Path}[G]$ is isomorphic to the linearly ordered set $[n] = \{ 0 < 1 < 2 < \cdots < n \} $ (regarded as a category).