Variant 10.2.1.9. Let $X^{\bullet }$ be a cosimplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$. We will say that an object $X \in \operatorname{\mathcal{C}}$ is a totalization of $X^{\bullet }$ if it is a limit of the diagram $X^{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}) \rightarrow \operatorname{\mathcal{C}}$. If this condition is satisfied, then $X$ is uniquely determined up to isomorphism. To emphasize this, we will often denote $X$ by $\operatorname{Tot}( X^{\bullet } )$ and refer to it as the totalization of $X^{\bullet }$.
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