1.3 Braiding theorem[0005]
To motivate our main theorem, recall that braided categories of quantum group representations do not exist in isolation: Rather, they come equipped with forgetful fiber functors to the category of vector spaces, carrying representations to their underlying vector spaces. Likewise, our monoidal \((\infty,2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) from theorem A comes equipped with a monoidal ‘fiber functor’, see ([00DH]), more precisely with a morphism in \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}])\) of the form \[H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}.\] In particular, here \(\mathrm{st}_{k}\) plays the role of a \(2\)-categorical version of the category of vector spaces. Hence \(\mathrm{st}^{B\mathbb{Z}}_{k}= \mathrm{Fun}(B \mathbb{Z}, \mathrm{st}_k)\) may be thought of as \(2\)-vector spaces equipped with a \(\mathbb{Z}\)-action. The Day convolution symmetric monoidal structure on \(\mathrm{st}^{B\mathbb{Z}}_{k}\) may be interpreted as being the natural convolution structure induced by thinking of these as \(2\)-vector spaces graded by the abelian Lie group \(U(1) \simeq B\mathbb{Z}\).
Homwise, the functor \(H_{\mathrm{loc}}\) is induced by the functors \({\mathbf K}^b(\mathrm{Sbim}_n) \rightarrow\mathbf D({}_{R_n} \mathrm{grbmod}_{R_n})\) which send chain complexes of Soergel bimodules to their corresponding objects in the derived \(\infty\)-category of the abelian category of graded \(R_n\)-bimodules. They then act as morphisms in \(\mathrm{st}^{B\mathbb{Z}}_{k}\) between certain stable \(\infty\)-categories of graded \(R_n\)-modules.
[0006]
Theorem B. (corollary 8.2.2).
There exists a unique braided monoidal (i.e., \(\mathbb E_2\)-algebra) structure on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \in \mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\) that enhances its monoidal structure and satisfies the following conditions.
The fiber functor \(H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\) is braided monoidal.
The braiding \(1 \otimes 1 \xrightarrow{\sim} 1 \otimes 1\) in \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) admits an equivalence with the Rouquier complex \(F(\sigma) \in \mathrm{End}_{{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})}(2) \coloneqq {\mathbf K}^b(\mathrm{Sbim}_2)\) corresponding to the braid group generator \(\sigma \in \operatorname{Br}_2\).
We emphasize that the uniqueness statement of Theorem B must be interpreted in the \(\infty\)-categorical sense, where the notion of “unique up to unique isomorphism” from classical category theory is generalized to the notion of being parametrized by a contractible \(\infty\)-groupoid.
We prove Theorem B in obstruction-theoretic terms, as explained further in Subsection 1.4. In fact, Theorem B is a special case of a more general result, namely Theorem 8.2.1, that applies to a general homwise additive 2-category (in place of \(\mathrm{Sbim}\)) and a general \(\mathrm{st}^{B\mathbb{Z}}_{k}\)-enriched \(\infty\)-category (in place of \(\mathrm{st}^{B\mathbb{Z}}_{k}\)).
The braided monoidal \(2\)-category \(\mathcal H\) is recovered in remark 8.2.3 from the braided monoidal \((\infty,2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) by taking its homotopy \(2\)-category \(h_2({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}))\) obtained by quotienting out all \(3\)-morphisms, see subsection 5.4, i.e. by applying \(H_0\) to its \(2\)-hom-objects (which are objects of the derived \(\infty\)-category of \(k\)). This is the sense in which theorem A and theorem B enhance the \(2\)-categorical statement from subsection 1.1.