Kerodon

Construction 1.3.1.1. For every integer $n \geq 0$, let us view the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n-1 < n \} $ as a category (where there is a unique morphism from $i$ to $j$ when $i \leq j$). For any category $\operatorname{\mathcal{C}}$, we let $\operatorname{N}_{n}( \operatorname{\mathcal{C}})$ denote the set of all functors from $[n]$ to $\operatorname{\mathcal{C}}$. Note that for any nondecreasing map $\alpha : [m] \rightarrow [n]$, precomposition with $\alpha $ determines a map of sets $\operatorname{N}_{n}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{m}(\operatorname{\mathcal{C}})$. We can therefore view the construction $[n] \mapsto \operatorname{N}_{n}(\operatorname{\mathcal{C}})$ as a simplicial set. We will denote this simplicial set by $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ and refer to it as the nerve of $\operatorname{\mathcal{C}}$.