Kerodon

Notation 10.2.2.13. Let $\operatorname{{\bf \Delta }}_{+}$ be the augmented simplex category (Definition 10.2.1.10). We let $\operatorname{{\bf \Delta }}_{\operatorname{inj},+}$ denote the (non-full) subcategory of $\operatorname{{\bf \Delta }}_{+}$ whose morphisms are strictly increasing functions $[m] \hookrightarrow [n]$. Note that $\operatorname{{\bf \Delta }}_{\operatorname{inj},+}$ can be obtained from the category $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ by adjoining an initial object $[-1]$ satisfying $\operatorname{Hom}_{ \operatorname{{\bf \Delta }}_{+, \operatorname{inj}} }( [n], [-1] ) = \emptyset $ for $n \geq 0$. Consequently, $\operatorname{{\bf \Delta }}_{\operatorname{inj},+}$ can be identified with the left cone $\operatorname{{\bf \Delta }}^{\triangleleft }_{\operatorname{inj}}$ (see Example 4.3.2.5).