Kerodon

Remark 10.2.1.11. The augmented simplex category $\operatorname{{\bf \Delta }}_{+}$ of Definition 10.2.1.10 contains the simplex category $\operatorname{{\bf \Delta }}$ of Definition 1.1.0.2 as a full subcategory (spanned by the objects $[n]$ for $n \geq 0$). Moreover, $\operatorname{{\bf \Delta }}_{+}$ can be obtained from $\operatorname{{\bf \Delta }}$ by adjoining a single object $[-1]$, which is an initial object satisfying $\operatorname{Hom}_{ \operatorname{{\bf \Delta }}_{+} }( [n], [-1] ) = \emptyset $ for $n \geq 0$. In other words, $\operatorname{{\bf \Delta }}_{+}$ can be identified with the left cone $\operatorname{{\bf \Delta }}^{\triangleleft }$ (see Example 4.3.2.5).