ScalingStacks

Proof. We begin by handling the unstable case. Theorem 4.23 implies that we can model ${\mathcal{A}}$ by a small spectral category (which we still denote by ${\mathcal{A}}$). Following [56, 3.3], we consider the following sequence of simplicial spectral categories

$${\mathcal{A}}_{\bullet}\stackrel{{\scriptstyle I}}{{\longrightarrow}}PS_{\bullet}{\mathcal{A}}\stackrel{{\scriptstyle Q}}{{\longrightarrow}}S_{\bullet}{\mathcal{A}}\,,$$

where ${\mathcal{A}}_{\bullet}$ is a constant simplicial object and $PS_{\bullet}{\mathcal{A}}$ is the simplicial path object of $S_{\bullet}{\mathcal{A}}$. By applying the functor ${\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}$ to this sequence, we obtain an induced morphism

$$\Theta\colon{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}(PS^{\infty}_{\bullet}{\mathcal{A}})/{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}({\mathcal{A}}_{\bullet})\longrightarrow{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}(S^{\infty}_{\bullet}{\mathcal{A}})$$

of simplicial objects in ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$. We now show that each component $\Theta_{n}$ of $\Theta$ is an equivalence. For each $n\geq 0$, we have a split-exact sequence

in which

	$\displaystyle I_{n}$	$\displaystyle(A)=(\ast\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A\stackrel{{\scriptstyle\mathrm{Id}}}{{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}}A\stackrel{{\scriptstyle\mathrm{Id}}}{{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}}\cdots\stackrel{{\scriptstyle\mathrm{Id}}}{{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}}A),$
	$\displaystyle Q_{n}$	$\displaystyle(\ast\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{0}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\cdots\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{n})=(A_{1}/A_{0}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\cdots\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{n}/A_{0}),$
	$\displaystyle S_{n}$	$\displaystyle(\ast\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{0}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\cdots\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{n-1})=(\ast\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\ast\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{0}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\cdots\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{n-1}),$
	$\displaystyle R_{n}$	$\displaystyle(\ast\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{0}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\cdots\longrightarrow A_{n-1})=A_{0}.$

By the construction of ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$ (and of ${\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}$), we conclude that the induced morphisms

$$\Theta_{n}\colon{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}(PS^{\infty}_{n}{\mathcal{A}})/{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}({\mathcal{A}})\longrightarrow{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}(S^{\infty}_{n}{\mathcal{A}})\qquad n\geq 0\,,$$

are equivalences in ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$. This allow us to obtain the following cocartesian square

and so a natural equivalence

$$\Sigma({\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}({\mathcal{A}}))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}|{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}(S^{\infty}_{\bullet}{\mathcal{A}})|$$

in ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$. By combining this equivalence with the equivalences

(7.18)		$\displaystyle{\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}(S^{\infty}_{\bullet}{\mathcal{A}})$	$\displaystyle=$	$\displaystyle|(\mathrm{Fun}^{\ex}(-,\Idem(S^{\infty}_{\bullet}{\mathcal{A}})))_{iso}|$
			$\displaystyle\simeq$	$\displaystyle|(S^{\infty}_{\bullet}\mathrm{Fun}^{\ex}(-,\Idem({\mathcal{A}})))_{iso}|$
			$\displaystyle=$	$\displaystyle K^{w}({\mathcal{A}})\,,$

where (7.18) follows from lemma 7.16, we conclude that $\Sigma({\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}({\mathcal{A}}))\simeq K^{w}_{\mathcal{A}}$ in ${\mathcal{M}}_{\mathrm{add}}^{\mathrm{un}}$. The identification in the stable setting follows from the unstable considerations and the usual passage from results on the $K$-theory space to the $K$-theory spectrum. ∎