ScalingStacks

proposition 7.7.5 implies that the map \(Z_1(A) \rightarrow A\) is \(0\)-truncated as a morphism in \(\mathrm{Alg}_{\mathbb E_1}(\mathcal C)\). Hence, any section \(A\rightarrow Z_1(A)\) is \((-1)\)-truncated. Therefore, the map  ([00G4]) is equivalent to the map \[\mathrm{Alg}_{\mathbb E_2}(\mathcal C)_{/^{-1}Z_1(A)} \times_{\mathrm{Alg}_{\mathbb E_1}(\mathcal C)/A} \{\mathrm{id}_A\} \rightarrow\mathrm{Alg}_{\mathbb E_1}(\mathcal C)_{/^{-1}Z_1(A)} \times_{\mathrm{Alg}_{\mathbb E_1}(\mathcal C)_{/A}} \{\mathrm{id}_A\}\] where \(- /^{-1} Z_1(A)\) denote full subcategories of \((-1)\)-truncated \(\mathbb E_1\)-maps (see notation 5.5.1). To prove this is an equivalence, it suffices to show that \[\mathrm{Alg}_{\mathbb E_2}(\mathcal C)_{/^{-1}Z_1(A)} \rightarrow\mathrm{Alg}_{\mathbb E_1}(\mathcal C)_{/^{-1} Z_1(A)}\] is an equivalence. But given any \((X\hookrightarrow Z_1(A))\) in \(\mathrm{Alg}_{\mathbb E_1}(\mathcal C)_{/^{-1}Z_1(A)}\), the fiber of ([00G5]) is equivalent to the space of dashed lifts in \(\mathrm{Op}\) where \(\mathrm{Ar}^{-1}(\mathcal C)\) denotes the full symmetric monoidal subcategory of the arrow category \(\mathrm{Ar}(\mathcal C) \coloneqq \mathrm{Fun}([1], \mathcal C)\) on the \((-1)\)-truncated morphisms.