ScalingStacks

Let ${\mathcal{A}}$ be a small stable $\infty$-category. We denote by $V({\mathcal{A}})$ the object

$$V({\mathcal{A}})=\colim_{n}\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{A}}))$$

in $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$ whose indexing maps are induced from the above diagram (9.14). Note that $V({\mathcal{A}})$ is functorial in ${\mathcal{A}}$ and that we have a natural map $\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}V({\mathcal{A}})$. We obtain then a well-defined functor $V$ along with a natural transformation:

(9.16)

$\displaystyle V(-):\Cat_{\infty}^{\ex}\longrightarrow\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$

$\displaystyle\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}\Rightarrow V(-)\,.$