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

Remark Let $I$ be a set equipped with a partial ordering $\leq _ I$. Then we can regard $I$ as a category whose objects are the elements of $I$, with morphisms given by

\[ \operatorname{Hom}_{I}(i,j) = \begin{cases} \ast & \text{ if } i \leq _ I j \\ \emptyset & \text{otherwise.} \end{cases} \]

We will denote the nerve of this category by $\operatorname{N}_{\bullet }(I)$, and refer to it as the nerve of the partially ordered set $I$. For each $n \geq 0$, we can identify $n$-simplices of $\operatorname{N}_{\bullet }(I)$ with monotone functions $[n] \rightarrow I$: that is, with nondecreasing sequences $(i_0 \leq _ I i_1 \leq _ I \cdots \leq _ I i_ n)$ of elements of $I$.