Kerodon

Remark 10.2.2.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Every augmented semisimplicial object of $\operatorname{\mathcal{C}}$ determines a semisimplicial object of $\operatorname{\mathcal{C}}$, by restriction along the inclusion of full subcategories $\operatorname{{\bf \Delta }}^{\operatorname{op}}_{\operatorname{inj}} \hookrightarrow \operatorname{{\bf \Delta }}^{\operatorname{op}}_{\operatorname{inj},+}$. For this reason, we will sometimes use the notation $\overline{X}_{\bullet }$ to indicate an augmented semisimplicial object of $\operatorname{\mathcal{C}}$, to distinguish it from the underlying simplicial object $X_{\bullet } = \overline{X}_{\bullet }|_{ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} )}$.