Kerodon

Notation 3.3.3.12. Let $X$ be a simplicial set and let $\operatorname{{\bf \Delta }}_{X}$ be the category of simplices of $X$ (Construction 1.1.3.9). By definition, the objects of $\operatorname{{\bf \Delta }}_{X}$ are given by pairs $([n], \sigma )$, where $n$ is a nonnegative integer and $\sigma $ is an $n$-simplex of $X$. We let $\operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}$ denote the full subcategory of $\operatorname{{\bf \Delta }}_{X}$ spanned by those pairs $([n], \sigma )$ where $\sigma $ is a nondegenerate $n$-simplex of $X$. We will refer to $\operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}$ as the category of nondegenerate simplices of $X$.