Kerodon

Notation 9.2.1.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. It follows from Proposition 8.4.5.3 that there exists an $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ and a functor $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}$-completion of $\operatorname{\mathcal{C}}$. In this case, the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is uniquely determined up to equivalence and depends functorially on $\operatorname{\mathcal{C}}$. To emphasize this dependence, we will often denote $\widehat{\operatorname{\mathcal{C}}}$ by $\operatorname{Ind}(\operatorname{\mathcal{C}})$ and refer to it as the $\operatorname{Ind}$-completion of $\operatorname{\mathcal{C}}$. In §9.2.4, we will give a more explicit description of the $\infty $-category $\operatorname{Ind}(\operatorname{\mathcal{C}})$ in the case where $\operatorname{\mathcal{C}}$ is essentially small (Proposition 9.2.4.5).