ScalingStacks

The ind-completion \(\operatorname{Ind}(\mathcal C)\) of a small \(\infty\)-category \(\mathcal C\) is defined to be the smallest full subcategory of \(\mathcal P(\mathcal C)\) which contains the image of the Yoneda embedding and is closed under filtered colimits, [Lur09, Rem. 5.3.5.2, Prop. 5.3.5.3]. Then, \(\operatorname{Ind}(\mathcal C)\) has filtered colimits and the inclusion \(\mathcal C\rightarrow\operatorname{Ind}(\mathcal C)\) is characterized by the universal property that for any \(\infty\)-category \(\mathcal D\) with filtered colimits, it induces an equivalence \[ \mathrm{Fun}^{\omega}(\operatorname{Ind}(\mathcal C),\mathcal D) \rightarrow\mathrm{Fun}(\mathcal C,\mathcal D),\] where \(\mathrm{Fun}^{\omega}\) denotes the full subcategory of functors which preserve filtered colimits.