We let \(h_1\mathrm{BSbim}\) denote the \(1\)-category, whose set of object is \(\mathbb{N}_0\) and whose morphism sets between objects \(n, m\) are \[\mathrm{Hom}_{h_1\mathrm{BSbim}}(n, m) = \left\{ \begin{array}{ll} h_0\mathrm{BSbim}_n & \quad n = m \\ \{0\} & \quad n \neq m \end{array} \right.\] where \(h_0 \mathrm{BSbim}_n\) denotes the set of isomorphism classes of objects in \(\mathrm{BSbim}_n\), and whose composition of morphisms \(n \rightarrow n\) is induced by the monoidal structure \(\otimes_{R_n}\) of \(\mathrm{BSbim}_n\). The category \(h_1\mathrm{BSbim}\) admits a monoidal structure with monoidal product \(\boxtimes\colon h_1\mathrm{BSbim}\times h_1\mathrm{BSbim}\rightarrow h_1\mathrm{BSbim}\) defined on objects by \(n \boxtimes m = n+m\) and on morphisms using parabolic induction: \[h_0 \boxtimes \colon h_0\mathrm{BSbim}_n \times h_0\mathrm{BSbim}_m \rightarrow h_0\mathrm{BSbim}_{n+m}.\]
2.3 Isomorphism classes of Soergel bimodules[0019]
We have already seen concrete hints that Soergel bimodules for symmetric groups form a monoidal bicategory and in subsection 2.5 we will see elements of a braiding on the homotopy category. The rigorous construction of the braiding will be carried out in an \(\infty\)-categorical setting later, but it requires a coupling to the classical setting here to enable a few critically important computations. The optimal handover point between these two worlds turns out to be one categorical dimension lower than we have been working in so far. In this section, we prepare the descent to this common ground from the classical side.
The currently ad-hoc notation \(h_1\) will be justified later in corollary 6.1.3 when we pass to \(\infty\)-categories.
Let \(h_1\mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) be the monoidal \(1\)-category whose objects are \(n \in \mathbb{N}_0\) and the morphisms between objects \(n, m\) are \[ \mathrm{Hom}_{h_1\mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})}(n, m) = \left\{ \begin{array}{ll} h_0\mathrm{K}^b(\mathrm{Sbim}_n) & \quad n = m \\ \{0\} & \quad n \neq m \end{array} \right.\] The monoidal product is induced by parabolic induction on chain complexes of Soergel bimodules and again we have \(\mathbb{Z}\)-actions on the morphism sets, inherited from grading shifts of bimodules. Since the inclusion \(\mathrm{BSbim}_n \rightarrow\mathrm{K}^b(\mathrm{Sbim}_n)\) of Bott–Samelson bimodules as chain complexes concentrated in homological degree zero is compatible with parabolic induction, this defines a monoidal functor \[ h_1 K_{\mathrm{loc}}\colon h_1\mathrm{BSbim}\rightarrow h_1\mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim}).\] In fact, this intertwines the \(\mathbb{Z}\)-actions on morphism sets.
In corollary 6.4.3, we match ([001D]) with its \(\infty\)-categorical version.
We leave it to the reader to check that the involved categories are monoidal as claimed. The conceptual reason behind this is that these categories are the shadow of monoidal bicategories in the sense of [Bén67] (the objects are the same, but morphism categories are given for instance by \(\mathrm{K}^b(\mathrm{Sbim}_n)\) instead of the sets \(h_0 \mathrm{K}^b(\mathrm{Sbim}_n)\) in ([001C])). Although these monoidal bicategories play an important conceptual role in represention theory and quantum topology, see e.g. [EW16, HRW21, HRW21], the construction of the monoidal structure has not yet appeared in full detail in the literature. In the world of \(\infty\)-categories we will obtain an analogous construction in section 6.
Given a graded algebra \(A\), we call an object of the derived category \(\mathrm{D}(\mathrm{grmod}_A)\) of graded \(A\)-modules graded-perfect if it is quasi-isomorphic to a finite chain complex of finitely generated graded-projective \(A\)-modules (see reference [000S]).
Let \(h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\) be the symmetric monoidal \(1\)-category whose objects are the graded algebras \(R_n= k[x_1, \ldots, x_n]\) for \(n \in \mathbb{N}_0\) and whose morphism sets between algebras \(R_n\) and \(R_m\) are given by the set \[h_0 \mathrm{D}\left( {}_{R_n} \mathrm{grbmod}_{R_m}\right)^{\mathrm{gr-perf}}\] of isomorphism classes of objects in the derived category of graded \(R_n\)–\(R_m\) bimodules which are graded-perfect as right (i.e. \(R_m\)-)modules; composition is the derived graded tensor product over the respective polynomial algebras. Similar to definition 2.1.5, the monoidal structure is given by the derived graded tensor product \(\otimes_k^L\) over the ground ring \(k\), under the identification \(R_n \otimes^{L}_k R_m \simeq R_n \otimes_k R_m \simeq R_{n+m}\).
We define the monoidal functor \[ h_1 H_{\mathrm{loc}}\colon h_1 \mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}}),\] which takes an object \(n\) to the polynomial algebra \(R_n\) and a chain homotopy equivalence class of a chain complex of Soergel bimodules to the corresponding quasi-isomorphism class of complexes of graded bimodules.
The notation \(h_1 H_{\mathrm{loc}}\) will be justified in corollary 6.5.4, the notation \(H_{\mathrm{loc}}\) indicates ‘taking homology at \(1\)-morphism level’.
Our first main result is the construction of a prebraiding structure on ([001D]).
In the \(\infty\)-categorical setting, we will replace \(h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\) with a more natural target category with less restrictions on objects. Namely, the category \(h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\) is a full symmetric monoidal subcategory of the category \(h_1\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{\mathbb{Z}})\) whose objects are arbitrary flat graded algebras, and morphims are isomorphism classes of right-graded-perfect derived bimodules between them. By passing to module categories over these algebras, this can in turn be realized as a full subcategory of the category \(h_1 \mathrm{st}^{B\mathbb{Z}}_{k}\) of stable \(k\)-linear categories with a \(\mathbb{Z}\)-action and equivalence classes of \(k\)-linear exact \(\mathbb{Z}\)-equivariant functors between them. In the next sections, we will lift the composite functor \(h_1 \mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\rightarrow h_1 \mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\rightarrow h_1\mathrm{st}^{B\mathbb{Z}}_{k}\) to a functor of \((\infty,2)\)-categories.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2