Definition 10.2.1.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$, let $\overline{X}_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$ satisfying $\overline{X}_{-1} = X$, and let $X_{\bullet } = \underline{X}_{\bullet } |_{ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) }$ denote its underlying simplicial object. We will say that $\overline{X}_{\bullet }$ exhibits $X$ as a geometric realization of $X_{\bullet }$ if it is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$, in the sense of Variant 7.1.3.5.
Similarly, if $\overline{X}^{\bullet }$ is an augmented cosimplicial object of $\operatorname{\mathcal{C}}$ satisfying $\overline{X}^{-1} = X$ and $X^{\bullet } = \overline{X}^{\bullet }|_{ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}) }$ is the underlying cosimplicial object, we say that $\overline{X}^{\bullet }$ exhibits $X$ as a totalization of $X^{\bullet }$ if it is a limit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$, in the sense of Definition 7.1.3.4.