ScalingStacks

Proof.

Consider the diagram of spaces To prove that the horizontal map is an equivalence, it suffices to show that for every \(A\in \mathrm{Alg}_{\mathbb E_1}(\mathcal C)\), the induced map between fibers \[\mathrm{Hom}_{\mathrm{Op}}(\mathbb E_2,\mathcal C) \times_{\mathrm{Hom}_{\mathrm{Op}}(\mathbb E_1,\mathcal C)} \{A\} \rightarrow\mathrm{Hom}_{\mathrm{Op}}(\mathbb A_2 \otimes \mathbb E_1,\mathcal C) \times_{\mathrm{Hom}_{\mathrm{Op}}(\mathbb E_1,\mathcal C)} \{A\}\] is an equivalence.

Since \(\mathcal C\) is presentably symmetric monoidal, it follows that centralizers and centers exist [Lur17, Cor. 5.3.1.15] and hence, by applying corollary 7.4.12, that the latter space is equivalent to \[\left(\mathrm{Alg}_{\mathbb E_1}(\mathcal C)_{/Z_1(A)}\right)^{\simeq} \times_{\left(\mathrm{Alg}_{\mathbb E_1}(\mathcal C)_{/A}\right)^{\simeq} } \{\mathrm{id}_A\}.\] Applying lemma 7.4.14 to the \(\infty\)-category \(\mathrm{Alg}_{\mathbb E_1}(\mathcal C)\), we find that the functor \[\mathrm{Alg}_{\mathbb E_2}(\mathcal C)_{/Z_1(A)} \times_{\mathrm{Alg}_{\mathbb E_1}(\mathcal C)/A} \{\mathrm{id}_A\}\rightarrow\mathrm{Alg}_{\mathbb E_2}(\mathcal C) \times_{\mathrm{Alg}_{\mathbb E_1}(\mathcal C)} \{A\}\] is an equivalence. It therefore suffices to show that the forgetful functor \[ \mathrm{Alg}_{\mathbb E_2}(\mathcal C)_{/Z_1(A)} \times_{\mathrm{Alg}_{\mathbb E_1}(\mathcal C)/A} \{\mathrm{id}_A\} \rightarrow\mathrm{Alg}_{\mathbb E_1}(\mathcal C)_{/Z_1(A)} \times_{\mathrm{Alg}_{\mathbb E_1}(\mathcal C)_{/A}} \{\mathrm{id}_A\}\] is an equivalence.

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.

But since \(\mathbb E_1 \rightarrow\mathbb E_2\) is \(0\)-surjective, i.e. essentially surjective on objects and \((-1)\)-connected on multi-hom spaces \(\mathbb E_1(n)\simeq S_n \rightarrow\mathbb E_2(n) = \mathrm{Conf}(n, \mathbb{R}^2)\), and since \(\mathrm{Ar}^{-1}(\mathcal C) \rightarrow\mathcal C\) is \(0\)-faithful³⁶, it follows from proposition 7.5.3 that the space of lifts ([00G6]) is contractible. ◻