Subsection 8.1. (0NSC) · ScalingStacks

8.1. Additive $\kappa$ -variant

Let

\psi\colon\Cat_{\infty}^{\perf}\longrightarrow\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\kappa};{\mathcal{S}}_{\infty})

be the functor obtained by first taking the Yoneda embedding and then restricting the presheaves to the $\infty$ -category $(\Cat_{\infty}^{\ex})^{\kappa}$ . Corollary 5.24 allow us to choose a fixed set $\underline{{\mathcal{E}}_{\mathrm{A}}^{\kappa}}$ of representatives of split-exact sequences in $(\Cat_{\infty}^{\perf})^{\kappa}$ . We denote by $\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}$ the localization of $\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\kappa};{\mathcal{S}}_{\infty})$ with respect to the set of maps

\Cone(\psi({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\psi({\mathcal{C}}))\longrightarrow\psi({\mathcal{B}})\,,

where ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is a split-exact sequence in $\underline{{\mathcal{E}}_{\mathrm{A}}^{\kappa}}$ . Let $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}$ be the following composite

\Cat_{\infty}^{\perf}\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\kappa};{\mathcal{S}}_{\infty})\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}\,,

where $\gamma$ is the localization functor.

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 $\underline{{\mathcal{E}}_{\mathrm{A}}^{\kappa}}$ ). ∎

Next, we localize $\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}$ with respect to the set of maps

(8.4)

\Cone\left(\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}})\longrightarrow\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{B}})\right)\longrightarrow\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{B}}/{\mathcal{A}})\,,

where ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}/{\mathcal{A}}$ is a strict-exact sequence in $\underline{{\mathcal{E}}_{\mathrm{wL}}^{\kappa}}$ (see section 5.5). Let $\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}$ be the following composite

\Cat_{\infty}^{\perf}\stackrel{{\scriptstyle\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}}}{{\longrightarrow}}\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}\,,

where $\gamma$ is the localization functor.

0NNX

Proposition 8.5. The functor $\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}$ preserves $\kappa$ -filtered colimits and sends strict-exact sequences to cofiber sequences in $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$

\displaystyle{\mathcal{A}}\longrightarrow{\mathcal{B}}\longrightarrow{\mathcal{B}}/{\mathcal{A}}

\displaystyle\mapsto

\displaystyle\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{A}})\longrightarrow\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{B}})\longrightarrow\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{B}}/{\mathcal{A}})\,.

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

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

0NNY

Proof. The result follows from propositions 8.3 and 5.30, and from the universal property of localization (see section 2.5). ∎

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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8.1. Additive κ\kappa-variant

8.1. Additive $\kappa$ -variant