ScalingStacks

Proof of Theorem 4.4. Corollary 4.6 identifies the functor $\Gamma_{\sf c}\colon\spaces_{B}^{\geq n}\to\mathbf{H}(\mfld_{n}^{B},\Space)$ as the composition $\Gamma_{\sf c}\colon\spaces_{B}^{\geq n}\xrightarrow{\Omega^{n}_{B}}\disk_{n}^{B}\xrightarrow{\int}\mathbf{H}(\mfld_{n}^{B},\Space)$. Theorem 3.24 gives that $\int$ is fully faithful, so it remains to argue that $\Omega_{B}^{n}$ is fully faithful with essential image the group-like $\disk_{n}^{B}$-algebras in spaces. This is immediate because, for instance, $\spaces_{/B}$ is an $\infty$-topos (Theorem 5.1.3.6 of [Lu2]).

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