ScalingStacks

Carrying out the same analysis as above, we see that the quotient

$${\mathcal{C}}^{\prime}=\Psi(\tilde{F}_{R}^{\infty})/\Psi(F_{R})$$

can be described as modules over $\End_{{\mathcal{C}}^{\prime}}(G^{\prime})$, where $G^{\prime}$ is the cofiber of the map

$$i_{!}i^{*}R^{\lor\infty}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}R^{\lor\infty}$$

(here $R^{\lor\infty}$ is regarded as an object of $\tilde{F}_{R}^{\infty}$) and hence as a spectrum $\End_{{\mathcal{C}}^{\prime}}(G^{\prime})$ is equivalent to the cofiber in spectra of the map

(9.49)

$$\mathrm{Map}_{\Psi(\tilde{F}_{R}^{\infty})}(R^{\lor\infty},i_{!}i^{*}R^{\lor\infty})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\End_{\Psi(\tilde{F}_{R}^{\infty})}(R^{\lor\infty}).$$