ScalingStacks

Notation 7.14. Given a small stable $\infty$-category ${\mathcal{A}}$, we denote by $K^{w}_{{\mathcal{A}}}$ the object

$${\mathcal{B}}\mapsto|(S^{\infty}_{\bullet}(\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}}))))_{\mathrm{iso}}|$$

in $\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\omega})_{*}$ and by $K_{{\mathcal{A}}}$ the object

$${\mathcal{B}}\mapsto K(\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}})))$$

in $\mathrm{Pre}_{{\mathcal{S}}_{\infty}}((\Cat_{\infty}^{\perf})^{\omega})$. Note that the value of $K_{{\mathcal{A}}}$ at ${\mathcal{S}}_{\infty}^{\omega}$ is precisely the $K$-theory spectrum $K({\mathcal{A}})$ of ${\mathcal{A}}$, similarly and that $K^{w}_{{\mathcal{A}}}$ is the delooping of the $K$-theory space.