Construction 1.1.3.9 (The Category of Simplices). Let $S S_{\bullet }$ be a simplicial set. We define a category $\operatorname{{\bf \Delta }}_{S}$ as follows:
The objects of $\operatorname{{\bf \Delta }}_{S}$ are pairs $([n], \sigma )$, where $[n]$ is an object of $\operatorname{{\bf \Delta }}$ and $\sigma $ is an $n$-simplex of $S$.
A morphism from $([n], \sigma )$ to $([n'], \sigma ')$ in the category $\operatorname{{\bf \Delta }}_{S}$ is a nondecreasing function $f: [n] \rightarrow [n']$ with the property that the induced map $S_{n'} \rightarrow S_{n}$ carries $\sigma '$ to $\sigma $.
We will refer to $\operatorname{{\bf \Delta }}_{S}$ as the category of simplices of $S$. If $k$ is an integer, we let $\operatorname{{\bf \Delta }}_{S, \leq k}$ denote the full subcategory of $\operatorname{{\bf \Delta }}_{S}$ spanned by those objects $([n], \sigma )$ satisfying $n \leq k$.