We call the monoidal \(k\)-linear stable \((\infty,2)\)-category with local shifts \[{\mathbf K}^b_{\mathrm{loc}}\left(\mathrm{Sbim}\right) \in \mathrm{Alg}_{\mathbb E_1} \left( \mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\right)\] the chain complex Soergel \((\infty, 2)\)-category.
6.4 The monoidal \((\infty,2)\)-category of chain complexes of Soergel bimodules[00D3]
Recall from Sections 3.4.2 and 3.4.3 that the forgetful functor \(\mathrm{st}\rightarrow\mathrm{add}\) from the presentably symmetric monoidal \(\infty\)-category of small stable \(\infty\)-categories to that of small additive \(\infty\)-categories has a symmetric monoidal left adjoint \[{\mathbf K}^b\colon \mathrm{add}\rightarrow\mathrm{st},\] which sends an ordinary additive \(1\)-category \(\mathcal A\) to the \(\infty\)-category of bounded chain complexes in \(\mathcal A\) with chain maps and (higher) chain homotopies between them. By proposition 4.3.2, this is compatible with linearity and \(\mathbb{Z}\)-action and induces a symmetric monoidal left adjoint \[{\mathbf K}^b\colon \mathrm{add}_{k}^{B\mathbb{Z}}\rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\] of the forgetful functor \(\mathrm{st}^{B\mathbb{Z}}_{k}\rightarrow\mathrm{add}_{k}^{B\mathbb{Z}}\). Applying \({\mathbf K}^b\) homwise, it follows from subsection A.10 and subsection A.8.11 that we obtain symmetric monoidal left-adjoints of the respective forgetful functors: \[{\mathbf K}^b_{\mathrm{loc}}:= \mathrm{Cat}[{\mathbf K}^b] \colon \mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}] \rightarrow\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\] \[ {\mathbf K}^b_{\mathrm{loc}}\colon \mathrm{Alg}_{\mathbb E_1}\left(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]\right) \rightarrow\mathrm{Alg}_{\mathbb E_1}\left(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}] \right).\]
The unit of the adjunction ([00D4]) is a monoidal \(\mathrm{add}_{k}^{B\mathbb{Z}}\)-enriched \((\infty,2)\)-functor \[ \mathrm{Sbim}\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}).\]
Since the functor \({\mathbf K}^b_{\mathrm{loc}}\) is defined by applying \({\mathbf K}^b\) homwise, we immediately obtain the following explicit description of \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\), which completes the construction of \({\mathbf K}^b_{\mathrm{loc}}\) as described in theorem A:
The objects of \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) agree with those of \(\mathrm{Sbim}\), while the stable \(k\)-linear hom-categories are given by \[\underline{\mathrm{Hom}}_{{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})}(n,m) = \left\{ \begin{array}{lr} 0 & n \neq m \\ {\mathbf K}^b(\mathrm{Sbim}_n) & n = m \end{array}\right. ,\] with \(\mathbb{Z}\)-action given by internal (i.e. non-homological!) grading shift.
The functor \(\mathrm{Sbim}\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) from ([00D6]) sends objects to themselves and is on hom-categories given by the additive \(k\)-linear \(\mathbb{Z}\)-equivariant functor \(\mathrm{Sbim}_n \hookrightarrow {\mathbf K}^b(\mathrm{Sbim}_n)\) including Soergel bimodules as chain complexes concentrated in degree zero. In particular, it is faithful as an \((\infty,2)\)-functor.
Note that while \(\mathrm{Sbim}\) is a \((2,2)\)-category, the category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) is a true \((\infty,2)\)-category with non-trivial higher cells.
The following makes the connection to section 2 and justifies the notation \(h_1\mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) from there.
The homotopy \(1\)-category \(h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) of the monoidal \((\infty,2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) agrees with the monoidal \(1\)-category \(h_1\mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) from definition 2.3.2.
Moreover, after taking homotopy \(1\)-categories, the monoidal \((\infty,2)\)-functor \(\mathrm{BSbim}\rightarrow\mathrm{Sbim}\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) becomes the ordinary monoidal \(1\)-functor \[h_1 \mathrm{BSbim}\rightarrow h_1 \mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\] from ([001D]).
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2