ScalingStacks

Let \(\mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)^{\mathrm{iso}}\) denote the full subspace of \(\mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)\) on the invertible arrows in \(\mathcal C\). In particular, the composite \(\mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)^{\mathrm{iso}}\rightarrow\mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C) \xrightarrow{s}\mathcal C^{\simeq}\) is an equivalence. We now show that the composite map of spaces \[ \mathrm{LMod}(\mathcal C)^{\simeq} \times_{\mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)} \mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)^{\mathrm{iso}} \rightarrow\mathrm{LMod}(\mathcal C)^{\simeq} \rightarrow\mathrm{Alg}_{\mathbb E_1}(\mathcal C)^{\simeq}\] is an equivalence, which concludes the proof as \(\{ \mathrm{id}_X\} \in \mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)\) is an object of the full subspace \(\mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)^{\mathrm{iso}}\).