Kerodon

Construction 1.4.5.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma : \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$ be an $n$-simplex of $\operatorname{\mathcal{C}}$. For $0 \leq i \leq n$, let $C_{i}$ denote the object of $\operatorname{\mathcal{C}}$ given by the image of the $i$th vertex of $\Delta ^ n$. For $0 \leq i \leq j \leq n$, let $f_{ij}: C_ i \rightarrow C_ j$ denote the image under $\sigma $ of the edge of $\Delta ^ n$ joining the $i$th vertex to the $j$th vertex, and let $[f_{ij}] \in \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( C_ i, C_ j )$ denote the homotopy class of $f_{ij}$. Then we can regard $( \{ C_ i \} _{0 \leq i \leq n}, \{ [f_{ij} ] \} _{0 \leq i \leq j \leq n} )$ as a functor from the linearly ordered set $[n]$ to the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Let $u(\sigma )$ denote the corresponding $n$-simplex of $\operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$. Then the construction $\sigma \mapsto u(\sigma )$ determines a map of simplicial sets