Proposition 9.25 (0NPS) · ScalingStacks

Proposition 9.25. The two functors

\mathrm{Loc},\gamma^{\ast}\colon{\mathcal{M}}_{\mathrm{loc}}^{\kappa}\longrightarrow\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}

are canonically equivalent, where $\gamma^{\ast}$ is the right adjoint of the localization functor.

0NPT

Proof. Let us denote by ${\bf L}$ the endofunctor $\mathrm{Loc}\circ\gamma$ of $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ . Note that we have a natural transformation $\Id\Rightarrow{\bf L}$ . Making use of the definition of $V(-)$ and of the fact that colimits in $\infty$ -categories commute, we observe that ${\bf L}$ is a localization functor on $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ [52, 5.2.7.4]. Therefore, it suffices to show that a map in $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ becomes an equivalence in ${\mathcal{M}}_{\mathrm{loc}}^{\kappa}$ if and only if it becomes an equivalence after application of ${\bf L}$ . This follows from the fact that for every small stable $\infty$ -category ${\mathcal{A}}$ , we have an equivalence $\gamma(V({\mathcal{A}}))\simeq{\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}})$ : note that we have cofiber sequences in ${\mathcal{M}}_{\mathrm{loc}}^{\kappa}$

{\mathcal{U}}_{\mathrm{loc}}^{\kappa}(\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\longrightarrow{\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{F}}_{\kappa}\Sigma_{\kappa}^{(n)}({\mathcal{A}}))\longrightarrow{\mathcal{U}}_{\mathrm{loc}}^{\kappa}(\Sigma_{\kappa}^{(n+1)}({\mathcal{A}}))\,.

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