ScalingStacks

1.4.6 From prebraidings to braidings[000F]

In Subsection 1.4.4 we outlined the construction of a prebraiding on the monoidal functor \(h_1\mathrm{BSbim}\hookrightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) between ordinary monoidal \(1\)-categories, which is compatible with the fiber functor \(h_1 H_{\mathrm{loc}}: h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow h_1 \mathrm{st}^{B\mathbb{Z}}_{k}\). To complete the proof of Theorem B, it remains to show that this admits a unique lift to a braiding on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) such that the fiber functor is braided.

We prove this in steps, combining the machinery described above, as follows, see section 8.

Applying a version of the techniques described in Subsection 1.4.2, it follows that our prebraiding on \(h_1\mathrm{BSbim}\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) over \(h_1\mathrm{st}^{B\mathbb{Z}}_{k}\) lifts uniquely to a prebraiding on the \((\infty,2)\)-functor \(\mathrm{BSbim}\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) over \(\mathrm{st}^{B\mathbb{Z}}_{k}\).

In order to proceed, we note a crucial property of prebraidings, see corollary 8.3.7: given an adjunction \(F \colon \mathcal C\rightleftarrows \mathcal D\colon G\) in which the left adjoint is symmetric monoidal, the data of a prebraiding on an \(\mathbb E_1\)-algebra morphism \(c \rightarrow G(d)\) is equivalent to the data of a prebraiding on its adjunct \(F(c) \rightarrow d\). We use this to extend our prebraiding over \(\mathrm{st}^{B\mathbb{Z}}_{k}\) from one on \(\mathrm{BSbim}\hookrightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) to one on \(\mathrm{Sbim}\hookrightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\), and then we use it again to extend the latter to one on the defining equivalence \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \xrightarrow{\sim} {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\).

So, we have obtained a prebraiding, i.e. a \(\mathbb T_2\)-structure on the identity morphism of \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \in \mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathrm{st}^{B\mathbb{Z}}_{k}})\). Because the identity morphism is invertible, this is equivalent to an \(\mathbb A_2 \otimes \mathbb E_1\)-structure on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \in \mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathrm{st}^{B\mathbb{Z}}_{k}}\).

A final task is to lift this \(\mathbb A_2 \otimes \mathbb E_1\)-structure to an \(\mathbb E_2 \simeq \mathbb E_1 \otimes \mathbb E_1\) structure. Recall that prebraidings on identity functors between ordinary monoidal \(1\)-categories, i.e. \(\mathbb A_2\otimes \mathbb E_1\)-structures in \(\mathrm{Cat}_1\), are precisely the same as braidings, i.e. \(\mathbb E_2\)-structures. More generally, we show in subsection 7.7 that \(\mathbb A_2 \otimes \mathbb E_1\)- and \(\mathbb E_2\)-algebras agree in general \((2,1)\)-operads.

This observation applies to our situation due to a crucial truncatedness result regarding the endomorphism \(\infty\)-operad of the object \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \in \mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathrm{st}^{B\mathbb{Z}}_{k}}\). Namely, while we expect it to be quite complicated in general, we prove in corollary 8.4.4 that the maximal sub-\(\infty\)-operad in the image of its \(\mathbb E_1\)-structure is in fact just a \((2,1)\)-operad. This suffices for our purposes since the map of \(\infty\)-operads \(\mathbb E_1 \rightarrow\mathbb E_2\) is surjective on path components of mapping spaces.

Thus, our prebraiding on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) over \(\mathrm{st}^{B\mathbb{Z}}_{k}\) extends uniquely to an \(\mathbb E_2\)-algebra structure establishing our main goal.

After now having introduced the key ideas and concepts, we finish this introduction by giving an outline of the organization of the paper.