Kerodon

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

Remark 2.4.3.4 (Relationship with Ordinary Path Categories). Let $Q$ be a partially ordered set and let $\mathrm{Gr}(Q)$ denote the associated directed graph, given concretely by

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

Then the path category $\operatorname{Path}[ \mathrm{Gr}(Q) ]$ of Construction 1.2.6.1 is the underlying category of the simplicial category $\operatorname{Path}[Q]_{\bullet }$ of Notation 2.4.3.1 (see Remark 2.3.6.2). In other words, we can regard $\operatorname{Path}[Q]_{\bullet }$ as a simplicially enriched version of $\operatorname{Path}[ \mathrm{Gr}(Q) ]$. Beware that the simplicial enrichment is nontrivial in general: that is, the simplicial mapping sets $\operatorname{Hom}_{\operatorname{Path}[Q]}( x, y)_{\bullet }$ are usually not constant.