ScalingStacks

Proof.

The space of lifts ([00EZ]) is by definition the fiber of \[\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, Z_0(f)) \xrightarrow{\mathrm{ev}_{1_A} \circ -} \mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, B)\] at \(f\). By the universal property of the centralizer and since \(I\) is initial in \(\mathrm{Alg}_{\mathbb E_0}(\mathcal C)\), the space \(\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, Z_0(f))\) is equivalent to the space \(\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A\otimes A, B)\times_{\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, B)}\{f\}\) of lifts in \(\mathrm{Alg}_{\mathbb E_0}(\mathcal C)\), where \(1_A \colon I \rightarrow A\) denotes the unique morphism in \(\mathrm{Alg}_{\mathbb E_0}(\mathcal C)\). Under this identification, the map \(\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, Z_0(f)) \xrightarrow{\mathrm{ev}_{1_A} \circ -}\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, B)\) unpacks to the composite \[\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A\otimes A, B)\times_{\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, B)}\{f\} \rightarrow\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A\otimes A, B) \rightarrow\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, B)\] of the projection and the precomposition with \(A\simeq I \otimes A \xrightarrow{1_A \otimes \mathrm{id}_A}A\otimes A\). Hence, the fiber of \(\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, Z_0(f)) \xrightarrow{\mathrm{ev}_{1_A} \circ -}\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, B)\) at \(f\) is equivalent to the space \[\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A\otimes A, B)\times_{\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal C)}(A, B)^{\times 2}} \{(f,f)\} =: \mathbb T_2(f). \qedhere\] ◻

Proof.

Apply proposition 7.4.11 for \(f=\mathrm{id}_A\) using that \(Z_0(A) = Z_0(\mathrm{id}_A)\). ◻

Proof.

Since \(\mathrm{Alg}_{\mathbb E_1}(\mathcal C) \rightarrow\mathcal C\) is conservative, the \(\infty\)-categories in ([00F6]) are \(\infty\)-groupoids, and equivalently given by the fibers of the respective maps of maximal subgroupoids.

For an \(\mathbb E_1\)-algebra \(A\) in \(\mathcal C\), it follows from the free-forgetful adjunction and initiality of the unit of \(\mathcal C\) that the free \(A\)-module \(_{A}A\) is initial in \(\mathrm{LMod}_A(\mathcal C)\). This induces a functor \(\mathrm{LMod}_A(\mathcal C) \simeq \mathrm{LMod}_{A}(\mathcal C)_{{}_{A}A/} \rightarrow\mathcal C_{A/}\) which sends a left \(A\)-module \(M\) to the morphism \[\mathrm{act}_1 \colon A\simeq A \otimes I \xrightarrow{\mathrm{id}_A \otimes !} A \otimes M \xrightarrow{\mathrm{act}} M.\]

Applying this functor \(\mathrm{act}_1 \colon \mathrm{LMod}_A(\mathcal C) \rightarrow\mathcal C_{A/}\) fiberwise induces a commuting diagram of spaces

By the universal property of the center, the \(\infty\)-category \(\mathrm{Alg}_{\mathbb E_1}(\mathcal C)_{/\mathfrak{Z}(X)}\) is equivalent to the \(\infty\)-category \(\mathrm{LMod}(\mathcal C) \times_{\mathcal C} \{X\}\), and hence the space \(\underline{\mathrm{Alg}}_{\mathbb E_1}(\mathcal C)_{/\mathfrak{Z}(X)} \times_{\mathcal C_{/X}} \{ \mathrm{id}_X\}\) is equivalent to the fiber of the top horizontal map in ([00F8]) at \(\{ \mathrm{id}_X\} \in \mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)\). The functor ([00F6]) is the induced map from the fiber of the top horizontal map of ([00F8]) at \(\mathrm{id}_X\) to the fiber of the bottom horizontal map of ([00F8]) at \(X\).

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 \(\mathrm{Hom}_{\mathrm{Cat}_{\infty}}([1], \mathcal C)^{\mathrm{iso}}\).

It suffices to verify that all fibers of ([00F9]) are contractible. By definition, for \(A\in \mathrm{Alg}_{\mathbb E_1}(\mathcal C)\) the fiber is the full subspace of \(\mathrm{LMod}_A(\mathcal C)^{\simeq} \simeq \left(\mathrm{LMod}_A(\mathcal C)_{{}_{A}A/}\right)^{\simeq}\) on those modules \(M\) for which the induced map \(\mathrm{act}_1 \colon A \rightarrow M\) is an equivalence. But since \(\mathrm{LMod}_A(\mathcal C) \rightarrow\mathcal C\) is conservative, this is the full subcategory \(\mathrm{LMod}_A(\mathcal C)_{{}_{A}A/^{\mathrm{iso}}}\) on the invertible module functors \({}_{A}A \rightarrow_{A}M\) and hence contractible. ◻

Proof.

By proposition 7.4.11, a \(\mathbb T_2 \otimes \mathbb E_1\)-structure on \(F\) is a lift in \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}_{({1}, {1})})\): Explicitly, the space of \(\mathbb T_2\otimes \mathbb E_1\)-structures on \(F\) is therefore the \(1\)-groupoid of weak lifts from theorem 2.6.4.([002J]). By theorem 2.6.4, this is equivalent to the set of prebraidings on \(F\) in the sense of definition 2.4.8. This completes the proof of part ([00FB]). Part ([00FC]) follows by applying statement ([00FB]) to the case \(F=\mathrm{id}_{\mathcal A}\). ◻