ScalingStacks

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

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

∎