Remark 9.2.0.7 (063G)

Remark 9.2.0.7. Let $\operatorname{\mathcal{C}}_0$ be an $\infty $-category and let us abuse notation by identifying $\operatorname{\mathcal{C}}_0$ with a full subcategory of its $\operatorname{Ind}$-completion $\operatorname{\mathcal{C}}= \operatorname{Ind}(\operatorname{\mathcal{C}}_0)$. Using the results of §9.2.4, we will see that every object $C \in \operatorname{\mathcal{C}}$ can be realized the colimit of a small filtered diagram $\{ C_ i \} _{i \in \operatorname{\mathcal{I}}}$ in the $\infty $-category $\operatorname{\mathcal{C}}_0$ (Corollary 9.2.4.21). If $D$ is the colimit of a small filtered diagram $\{ D_ j \} _{j \in \operatorname{\mathcal{J}}}$ in $\operatorname{\mathcal{C}}_0$, then we have canonical homotopy equivalences

\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( C, D) & = & \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( \varinjlim _{i \in \operatorname{\mathcal{I}}} C_ i, D ) \\ & \simeq & \varprojlim _{i \in \operatorname{\mathcal{I}}} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_ i, D) \\ & = & \varprojlim _{i \in \operatorname{\mathcal{I}}} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_ i, \varinjlim _{j} D_ j ) \\ & \simeq & \varprojlim _{i \in \operatorname{\mathcal{I}}} \varinjlim _{j \in \operatorname{\mathcal{J}}} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_ i, D_ j) \\ & = & \varprojlim _{i \in \operatorname{\mathcal{I}}} \varinjlim _{j \in \operatorname{\mathcal{J}}} \operatorname{Hom}_{\operatorname{\mathcal{C}}_0}( C_ i, D_ j). \end{eqnarray*}

Here the limits and colimits are computed in the $\infty $-category of spaces $\operatorname{\mathcal{S}}$, and the equivalence on the fourth line comes from the observation that each $C_ i$ is compact as an object of $\operatorname{\mathcal{C}}$. We therefore obtain the following more informal description of the $\operatorname{Ind}$-completion:

Objects of $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ are represented by small filtered diagrams $\{ C_ i \} _{i \in \operatorname{\mathcal{I}}}$ in the $\infty $-category $\operatorname{\mathcal{C}}_0$.
Morphism spaces in $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ are given by the formula

\[ \operatorname{Hom}_{ \operatorname{Ind}(\operatorname{\mathcal{C}}_0) }( \{ C_ i \} _{i \in \operatorname{\mathcal{I}}}, \{ D_ j \} _{j \in \operatorname{\mathcal{J}}} ) = \varprojlim _{i \in \operatorname{\mathcal{I}}} \varinjlim _{j \in \operatorname{\mathcal{J}}} \operatorname{Hom}_{\operatorname{\mathcal{C}}_0}( C_ i, D_ j). \]