Proposition 8.3 (0NNV) · ScalingStacks

0NNV

Proposition 8.3. The functor $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}$ preserves $\kappa$ -filtered colimits and sends split-exact sequences

in $\Cat_{\infty}^{\perf}$ to (split) cofiber sequences in $\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}$ . Moreover, $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}$ is universal with respect to these two properties, i.e., given any stable presentable $\infty$ -category ${\mathcal{D}}$ , we have an equivalence of $\infty$ -categories

(\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}})^{\ast}:\mathrm{Fun}^{\mathrm{L}}(\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}},{\mathcal{D}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Fun}_{\underline{\mathrm{add}}}^{\kappa}(\Cat_{\infty}^{\perf},{\mathcal{D}})\,,

where the right-hand side denotes the full subcategory of $\mathrm{Fun}(\Cat_{\infty}^{\ex},{\mathcal{D}})$ of morphisms of $\infty$ -categories which satisfy the above two conditions.

0NNW

Proof. The result follows from the analogue of the argument for lemma 6.4 in the context of $\kappa$ -compact objects and $\Ind_{\kappa}$ , and from the universal property of Bousfield localization (see section 2.5 the functor $\psi$ preserves $\kappa$ -filtered colimits and proposition 5.27 shows that any split-exact sequence can be approximated by a $\kappa$ -filtered colimit of split-exact sequences in $\underline{{\mathcal{E}}_{\mathrm{A}}^{\kappa}}$ ). ∎