ScalingStacks

Proof. Regarding $\tilde{F}_{R}^{\infty}$ as a simplicial category, $\End_{\tilde{F}_{R}^{\infty}}(R^{\infty})$ is (by construction) the $A_{\infty}$ ring space $\ell R$. Furthermore, we have that

$$\pi_{0}(\mathrm{Map}_{\Psi(\tilde{F}_{R}^{\infty})}(R^{\lor\infty},i_{!}i^{*}R^{\lor\infty}))\cong\pi_{0}(i_{!}i^{*}R^{\lor\infty})\cong m\pi_{0}R$$

and by construction

$$\pi_{0}(\End_{\Psi(\tilde{F}_{R}^{\infty})}(R^{\lor\infty}))\cong\ell\pi_{0}R.$$

Therefore, equation 9.49 implies that as groups there is an isomorphism

	$\displaystyle\pi_{0}(\End_{{\mathcal{C}}^{\prime}}(G^{\prime}))\cong\ell(\pi_{0}(R))/m(\pi_{0}(R))\cong\ell(\pi_{0}(\Omega^{\infty}R))/m(\pi_{0}(\Omega^{\infty}R))$
	$\displaystyle\cong\mu\pi_{0}(\Omega^{\infty}R)\cong\pi_{0}(\mu\Omega^{\infty}R),$

where the last isomorphism follows from [33, 5.1]. Finally, the universal property of the cofiber in spectra implies that there is a ring structure induced on $\pi_{0}(\End_{{\mathcal{C}}^{\prime}}(G^{\prime}))$ induced by the ring structure on $m\pi_{0}(R)$ quotiented by the two-sided ideal $\ell\pi_{0}(R)$. Inspection of $\pi_{0}$ shows that this multiplication coincides with the ring structure on $\pi_{0}(\End_{{\mathcal{C}}^{\prime}}(G^{\prime}))$ induced by composition. ∎