Kerodon

Remark 2.4.3.2. Let $Q$ be a partially ordered set. The simplicial category $\operatorname{Path}[Q]_{\bullet }$ can be regarded as a “thickened version” of $Q$. For every pair of elements $x,y \in Q$, the simplicial set $\operatorname{Hom}_{\operatorname{Path}[Q]}(x,y)_{\bullet }$ is empty if $x \nleq y$, and weakly contractible (see Definition 3.2.4.16) if $x \leq y$ (since it is the nerve of a partially ordered set with a largest element $\{ x,y\} $). In particular, there is a unique simplicial functor $\pi : \operatorname{Path}[Q]_{\bullet } \rightarrow Q$ which is the identity on objects (where we abuse notation by identifying $Q$ with the associated constant simplicial category of Example 2.4.2.4). The simplicial functor $\pi $ is a prototypical example of a weak equivalence in the setting of simplicial categories (see Definition 4.6.8.7).