Kerodon

Example 3.3.3.14. Let $Q$ be a partially ordered set, and let $\operatorname{N}_{\bullet }(Q)$ denote its nerve. By definition, the nondegenerate $n$-simplices of $\operatorname{N}_{\bullet }(Q)$ can be identified with the strictly increasing functions $\sigma : \{ 0 < 1 < \cdots < n \} \rightarrow Q$. The construction $([n], \sigma ) \mapsto \operatorname{im}(\sigma )$ determines an isomorphism from the category of nondegenerate simplices $\operatorname{{\bf \Delta }}_{\operatorname{N}_{\bullet }(Q)}^{\mathrm{nd}}$ to the partially ordered set $\operatorname{Chain}[Q]$ of Notation 3.3.2.1.