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Remark (Comparison with the Path Category). Let $(Q, \leq )$ be a partially ordered set. We let $\operatorname{Path}[Q]$ denote the underlying category of the strict $2$-category $\operatorname{Path}_{(2)}[Q]$. The category $\operatorname{Path}[Q]$ can be described concretely as follows:

  • The objects of $\operatorname{Path}[Q]$ are the elements of $Q$.

  • If $x$ and $y$ are elements of $Q$, then a morphism from $x$ to $y$ in $\operatorname{Path}[Q]$ is given by a finite linearly ordered subset

    \[ S = \{ x = x_0 < x_1 < x_2 < \cdots < x_ n = y \} \subseteq Q \]

    having least element $x$ and largest element $y$.

Note that $\operatorname{Path}[Q]$ can also be realized as the path category of a directed graph $\mathrm{Gr}(Q)$ (as defined in Construction Here $\mathrm{Gr}(Q)$ denotes the underlying directed graph of the category $Q$, given concretely by

\[ \operatorname{Vert}( \mathrm{Gr}(Q) ) = Q \quad \quad \operatorname{Edge}( \mathrm{Gr}(Q) ) = \{ (x,y) \in Q: x < y \} \]

where we regard each ordered pair $(x,y) \in \operatorname{Edge}( \mathrm{Gr}(Q) )$ as an edge with source $s(x,y) = x$ and target $t(x,y) = y$.