Proposition 2.18 (0NJS) · ScalingStacks

Proposition 2.18. Let ${\mathcal{C}}$ be a small $\infty$ -category and $\kappa$ an infinite regular cardinal.

•

The $\infty$ -category $\Ind_{\kappa}({\mathcal{C}})$ admits all $\kappa$ -small colimits that exist in ${\mathcal{C}}$ [52, 5.3.5.14, 5.5.1.1].
•

The functor ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind_{\kappa}({\mathcal{C}})$ preserves $\kappa$ -filtered colimits [52, 5.3.5.2, 5.3.5.3].
•

$\Ind_{\kappa}({\mathcal{C}})$ is a stable $\infty$ -category [53, 1.1.3.6].
•

The image of ${\mathcal{C}}$ in $\Ind_{\kappa}({\mathcal{C}})$ provides a set of compact objects which generates $\Ind({\mathcal{C}})$ under $\kappa$ -filtered colimits [52, 5.3.5.5,5.3.5.11].

•

The category $\Ind_{\kappa}$ is characterized by the property that it has $\kappa$ -small filtered colimits, admits a functor ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind_{\kappa}({\mathcal{C}})$ , and this functor induces an equivalence

\mathrm{Fun}_{\kappa}(\Ind({\mathcal{C}}),{\mathcal{D}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}({\mathcal{C}},{\mathcal{D}}),

for any ${\mathcal{D}}$ which admits $\kappa$ -filtered colimits (here $\mathrm{Fun}_{\kappa}(-,-)$ denotes the $\infty$ -category of functors that preserve $\kappa$ -small filtered colimits) [52, 5.3.5.10].