Proposition 7.19 (0NNR) · ScalingStacks

0NNR

Proposition 7.19. Let ${\mathcal{A}}$ be a small stable $\infty$ -category. Then, the presheaves $K^{w}_{{\mathcal{A}}}$ and $K_{{\mathcal{A}}}$ (see notation 7.14) are local, i.e., given any split-exact sequnce ${\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ in ${\mathcal{E}}$ , the induced maps of spectra (see (6.5) and (6.9))

\map(\phi({\mathcal{D}}),K^{w}_{{\mathcal{A}}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Map}(\phi({\mathcal{C}})/\phi({\mathcal{A}}),K^{w}_{{\mathcal{A}}})

\map(\psi({\mathcal{D}}),K_{{\mathcal{A}}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Map}(\psi({\mathcal{C}})/\psi({\mathcal{A}}),K_{{\mathcal{A}}})

are equivalences.

0NNS

Proof. The argument is exactly the same in both cases. Therefore, we discuss only the stable $K_{{\mathcal{A}}}$ . Since ${\mathcal{B}}$ , ${\mathcal{C}}$ and ${\mathcal{D}}$ belong to $(\Cat_{\infty}^{\perf})^{\omega}$ , the spectral Yoneda lemma shows us that we need to prove that the induced sequence of spectra

is a cofiber sequence. Using corollary 4.27 it suffices to consider the split-exact sequence of small spectral categories

Note that, again by corollary 4.27, all of these spectral categories carry a natural Waldhausen structure inherited from the usual model structure on spectral modules. We will apply Waldhausen’s fibration theorem [84, 1.6.4]. We have the Waldhausen category $v\mathrm{rep}({\mathcal{C}},{\mathcal{A}})$ , whose weak equivalences are the morphisms $f$ such that $\Cone(f)$ is contractible, as well as the Waldhausen category $w\mathrm{rep}({\mathcal{C}},{\mathcal{A}})$ , with the same cofibrations as $v\mathrm{rep}({\mathcal{C}},{\mathcal{A}})$ but whose weak equivalences are those $f$ such that $\Cone(f)$ belongs to $\mathrm{rep}({\mathcal{D}},{\mathcal{A}})$ . Moreover, we have a natural inclusion $v\mathrm{rep}({\mathcal{C}},{\mathcal{A}})\subset w\mathrm{rep}({\mathcal{C}},{\mathcal{A}})$ and an equivalence $\mathrm{rep}({\mathcal{C}},{\mathcal{A}})^{w}\simeq\mathrm{rep}({\mathcal{C}},{\mathcal{A}})$ ; see [84, § 1.6]. The conditions of [84, 1.6.4] are satisfied, so we obtain a cofiber sequence of spectra

K(\mathrm{rep}({\mathcal{D}},{\mathcal{A}}))\longrightarrow K(\mathrm{rep}({\mathcal{C}},{\mathcal{A}}))\longrightarrow K(\mathrm{rep}({\mathcal{B}},{\mathcal{A}})).

∎