Definition 10.2.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For each object $C \in \operatorname{\mathcal{C}}$, we let $\underline{C}$ denote the simplicial object of $\operatorname{\mathcal{C}}$ given by the constant functor
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \{ C\} \hookrightarrow \operatorname{\mathcal{C}}. \]
We say that a simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ is constant if it is equal to $\underline{C}$, for some object $C \in \operatorname{\mathcal{C}}$ (in this case, we must have $C = X_{0}$). We say that $X_{\bullet }$ is essentially constant it is isomorphic to a constant simplicial object $\underline{C}$, for some $C \in \operatorname{\mathcal{C}}$.