ScalingStacks

1.4.4 The prebraiding[000D]

Another key ingredient of our proof of Theorem B is the following: \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) is generated, in an appropriate sense, by a monoidal sub-2-category, restricted to which the fiber functor to \(\mathrm{st}^{B\mathbb{Z}}_{k}\) is faithful.

We begin by considering the sub-2-category⁸ \(\mathrm{Sbim}\subset {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) described as follows: it contains all the same objects, but on hom-objects we pass to the subcategories \(\mathrm{Sbim}_n \subseteq {\mathbf K}^b(\mathrm{Sbim}_n)\) of Soergel bimodules (seen as complexes concentrated in degree 0). This is a monoidal sub-2-category: Soergel bimodules are closed under parabolic induction.

Consider now the restricted fiber functor, i.e. the composite \(\mathrm{Sbim}\hookrightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\) of the inclusion followed by the fiber functor. Here, we arrive at an interesting tension. While the fiber functor \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\) is not faithful but (will be) braided, this composite is faihtful but not braided: The braiding of \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) does not restrict to one on \(\mathrm{Sbim}\). Indeed, Rouquier complexes are typically genuine complexes, not concentrated in degree 0.

However, we now make a key observation: the \((\infty,2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) is obtained by applying the left adjoint \({\mathbf K}^b\) to the hom-objects of the \((2,2)\)-category \(\mathrm{Sbim}\). In this sense, \(\mathrm{Sbim}\) generates \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\), which suggests that the braiding of \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) might be uniquely determined by its values on \(\mathrm{Sbim}\). In fact, for each \(m\), there is a full additive monoidal subcategory \(\mathrm{BSbim}_m \subseteq \mathrm{Sbim}_m\) of Bott–Samelson bimodules ([000M]) which generates \(\mathrm{Sbim}_m\) under sums, retracts, and grading shifts. These Bott–Samelson bimodules assemble into a sub-2-category \(\mathrm{BSbim}\subset \mathrm{Sbim}\), which suggests that the braiding of \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) might even be uniquely determined by its values on \(\mathrm{BSbim}\).

In order to explore this idea further, let us imagine constructing the braiding on the monoidal \((\infty,2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) of Theorem B by hand. This certainly requires, for every pair of objects \(m,n \in \mathbb{N}_0\), a braiding isomorphism \(m \otimes n \xrightarrow{\sim} n \otimes m\). Up to equivalence, this 1-morphism is already determined by condition ([0008]) of Theorem B: it must be given by the Rouquier complex for the positive \((m,n)\)-shuffle braid in \(\operatorname{Br}_{m+n}\) (see also figure [0016]). Of course, the braiding has to be natural: any complex \(C \in {\mathbf K}^b(\mathrm{Sbim}_m)\), seen as an endomorphism of \(m \in {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\), should “slide” along the \(m\) parallel strands through the braiding by means of a specified homotopy equivalence (see figure [0021]), and likewise for any \(C' \in {\mathbf K}^b(\mathrm{Sbim}_n)\). In fact, the discussion in the previous paragraph suggests that it suffices to specify these “slide” homotopy equivalences merely for objects \(C \in \mathrm{BSbim}_m \subseteq {\mathbf K}^b(\mathrm{Sbim}_m)\).

We formalize such a specification through the key notion of a prebraiding on the monoidal functor \(\mathrm{BSbim}\hookrightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\). We first describe such data in the simple setting of ordinary categories, i.e. in the \((2,1)\)-category \(\mathrm{Cat}\): a prebraiding on a monoidal functor \(F \colon \mathcal A\rightarrow\mathcal B\) between ordinary categories consists of equivalences \(F(x) \otimes F(y) \xrightarrow{\sim} F(y) \otimes F(x)\) that are natural in \(x,y \in \mathcal A\) and satisfy two appropriate analogs of the hexagon axioms for braidings; see definition 2.4.1. In particular, a prebraiding on the identity functor \(\mathrm{id}_\mathcal A\) is equivalent to a braiding on \(\mathcal A\). On the other hand, as we will see in subsection 1.4.5, in the \(\infty\)-categorical context, a prebraiding is a much sparser structure. Still, a prebraiding on a monoidal functor between \((\infty,2)\)-categories involves a substantial amount of coherence data.

Inspired by the results of Subsection 1.4.2, as a core input to our proof of Theorem B we construct a prebraiding on the ordinary monoidal functor \(h_1\mathrm{BSbim}\hookrightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) in subsection 2.5 whose prebraiding equivalence is given by the homotopy equivalence classes of the Rouquier complexes of positive \((m,n)\)-shuffle braids. This is based on explicit computations using the diagrammatic formulation of Soergel bimodules, [EW16, EK10]. Furthermore, this prebraiding is compatible with the fiber functor \(h_1H_{\mathrm{loc}}\colon h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\).

Finally, having constructed such a prebraiding at the level of homotopy categories, it remains to show that it lifts uniquely to a fully homotopy coherent braiding on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) and on its fiber functor \(H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\).