Having defined \(\mathrm{BSbim}\) together with its inclusion \(\mathrm{BSbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\), we will now define \(\mathrm{Sbim}\) as the homwise completion of \(\mathrm{BSbim}\) under \(\mathbb{Z}\)-shifts, direct sums and splitting of idempotents inside of \(\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\).
To formally implement this, we need to construct a factorization system on the presentably symmetric monoidal \(\infty\)-category \(\mathrm{add}_{k}^{B\mathbb{Z}}\) of additive \(k\)-linear \(\infty\)-categories with a \(\mathbb{Z}\)-action, equipped with the Day convolution monoidal structure, constructed in subsection 4.3.
We will use the following terminology:
A functor \(F\colon \mathcal C\rightarrow\mathcal D\) between idempotent-complete \(\infty\)-categories is called dominant if every object in \(\mathcal D\) is a retract of an object in the image of \(F\).
A morphism in \(\mathrm{add}_{k}^{B\mathbb{Z}}\) is dominant, resp. fully faithful, if its underlying functor is.
The (dominant, fully faithful)-functors define a factorization system on \(\mathrm{add}_{k}^{B\mathbb{Z}}\) which is of small generation and compatible with the symmetric monoidal structure.
Proof.
Consider the sequence of morphisms in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) \[\mathrm{Cat}_{\infty}\rightarrow\mathrm{Cat}_{\infty}^{\sqcup} \rightarrow\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Mod}_{\mathrm{CProj}_k}(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}) \simeq \mathrm{add}_k\] left adjoint to the respective forgetful functors (see proposition 3.1.11 for the first two functors, and observation 4.2.4 for the latter one). We will successively lift the (surjective-on-objects, fully faithful)-factorization system on \(\mathrm{Cat}_{\infty}\) to \(\mathrm{add}_k\).
Step 1: We first show that the (surjective-on-objects, fully faithful)-functors define a factorization system on \(\mathrm{Cat}_{\infty}^{\sqcup}\) which is of small generation and compatible with the symmetric monoidal structure. Given a morphism \(F\colon \mathcal C\rightarrow\mathcal D\) in \(\mathrm{Cat}_{\infty}^{\sqcup}\), i.e. a functor between \(\infty\)-categories with finite coproducts that preserves finite coproducts, we consider its (surjective-on-objects, fully faithful) factorization in \(\mathrm{Cat}_{\infty}\) 
where \(\widetilde{F}\) is surjective on objects and \(\iota\) is fully faithful. Using observation B.2.1 and lemma B.2.2 we can deduce that the factorization system on \(\mathrm{Cat}_{\infty}\) restricts to one on \(\mathrm{Cat}_{\infty}^{\sqcup}\) which is of small generation and compatible with the symmetric monoidal structure, provided we can show that \(\mathrm{Im}(F)\) admits finite coproducts and that \(\widetilde{F}\) and \(\iota\) preserve them. In fact, since \(\iota\) is fully faithful, it is enough to prove that \(\mathrm{Im}(F)\) is closed under finite coproducts in \(\mathcal D\). To this end, let \(S\) be a finite set and \(I\colon S \rightarrow\mathrm{Im}(F)\) a diagram. Since \(\widetilde{F}\colon \mathcal C\rightarrow\mathrm{Im}(F)\) is surjective on objects, we can lift \(I\) to a functor \(\widetilde{I} \colon S \rightarrow\mathcal C\) which has a colimit \(\mathrm{colim}~\widetilde{I}\in \mathcal C\) by assumption. Since \(F\) preserves coproducts and using the factorization, the coproduct \(\mathrm{colim}~\iota \circ I\) agrees with \(F(\mathrm{colim}~\widetilde{I})\) and hence is in the image of \(F\), as required.
Step 2: To lift from \(\mathrm{Cat}_{\infty}^{\sqcup}\) to \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\), observe that the symmetric monoidal left adjoint \((-)^{\mathrm{idem}}:\mathrm{Cat}_{\infty}^{\sqcup} \rightarrow\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) is a reflective localization, i.e. that the right adjoint is fully faithful. We observe the following:
A morphism in \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) is of the form \((F)^{\mathrm{idem}}\) for a fully faithful morphism \(F\) in \(\mathrm{Cat}_{\infty}^{\sqcup}\) if and only if it is fully faithful.
A morphism in \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) is of the form \((F)^{\mathrm{idem}}\) for a surjective-on-objects morphism \(F\) in \(\mathrm{Cat}_{\infty}^{\sqcup}\) if and only if it is dominant.
Since the class of dominant functors is stable under retracts in \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\), it therefore follows from lemma B.2.5 that the (surjective-on-objects, fully faithful)-factorization system on \(\mathrm{Cat}_{\infty}^{\sqcup}\) induces the (dominant, fully faithful)-factorization system on \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\), and that this factorization system is of small generation and compatible with the monoidal structure on \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\).
Step 3: Since 
is a monadic adjunction, whose underlying monad \(\mathrm{CProj}_k \otimes - \colon \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\rightarrow\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) preserves colimits (and in particular geometric realizations), and preserves dominant functors (since \(\otimes\) is compatible with the (dominant, fully faithful)-factorization system on \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\), as follows from Step 2), it follows fromlemma B.2.7 that the (dominant, fully faithful) functors form a factorization system on \(\mathrm{add}_k\) which is of small generation and compatible with its symmetric monoidal structure.
Step 4: Lastly, by lemma B.2.3, the (dominant, fully faithful)-factorization system induces one on \(\mathrm{Fun}(B\mathbb{Z}, \mathrm{add}_k)\) which is of small generation and compatible with the Day convolution symmetric monoidal structure. ◻
We will use the following terminology:
A morphism \(F\colon \mathcal C\rightarrow\mathcal D\) in \(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]\) is called faithful if its underlying \((\infty,2)\)-functor is (see notation 5.3.3). It is called surjective-on-objects-and-dominant-on-1-morphisms if it is surjective on objects and if for each \(c, c'\in \mathcal C\), the induced additive functor \(\underline{\mathrm{Hom}}_{\mathcal C}(c,c') \rightarrow\underline{\mathrm{Hom}}_{\mathcal C}(Fc,Fc')\) is dominant.
A morphism in \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}])\) is called faithful or surjective-on-objects-and-dominant-on-1-morphisms if the underlying morphism in \(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]\) is.
Applying the factorization system from proposition 6.2.2 homwise leads to the following corollary.
The (surjective-on-objects-and-dominant-on-1-morphisms, faithful)-functors define factorization systems on \(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]\) and \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}])\), respectively, which are of small generation and compatible with the symmetric monoidal structure.
Proof.
This follows immediately from applying theorem B.4.1 and then theorem B.3.1 to the factorization system of proposition 6.2.2. ◻
The forgetful functor \(\mathrm{add}_k \rightarrow\mathrm{Cat}_{\infty}\) has a symmetric monoidal left adjoint ‘linearization functor’ \[\mathrm{Lin}_k\colon \mathrm{Cat}_{\infty}\rightarrow\mathrm{add}_k,\] which is the composite of symmetric monoidal left adjoints \[ \mathrm{Cat}_{\infty}\xrightarrow{(-)^{\sqcup, \mathrm{idem}}}\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\xrightarrow{\mathrm{CProj}_k \otimes -} \mathrm{Mod}_{\mathrm{CProj}_k}(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}) \simeq \mathrm{add}_k,\] where \((-)^{\sqcup, \mathrm{idem}}\) freely adjoints finite coproducts and splittings of idempotents (see proposition 3.1.11) and \(\mathrm{CProj}_k\otimes -\) constructs free \(\mathrm{CProj}_k\)-modules (see proposition 3.1.8). This induces a symmetric monoidal left adjoint of the forgetful functor \(\mathrm{add}_{k}^{B\mathbb{Z}}\rightarrow\mathrm{Cat}_{\infty}^{B\mathbb{Z}}\) \[\mathrm{Fun}(B\mathbb{Z}, \mathrm{Lin}_k(-))\colon \mathrm{Cat}_{\infty}^{B\mathbb{Z}}\rightarrow\mathrm{add}_{k}^{B\mathbb{Z}},\] which we will also denote by \(\mathrm{Lin}_k(-) \colon \mathrm{Cat}_{\infty}^{B\mathbb{Z}}\rightarrow\mathrm{add}_{k}^{B\mathbb{Z}}\). Unpacking observation 3.5.13, the forgetful functor \(\mathrm{Cat}_{\infty}^{B \mathbb{Z}} \rightarrow\mathrm{Cat}_{\infty}\) (i.e. the functor \(\mathrm{ev}_*\colon \mathrm{Fun}(B\mathbb{Z}, \mathrm{Cat}_{\infty}) \rightarrow\mathrm{Cat}_{\infty}\)) has a left adjoint \[- \times \mathbb{Z}: \mathrm{Cat}_{\infty}\rightarrow\mathrm{Cat}_{\infty}^{B\mathbb{Z}}\] which sends an \(\infty\)-category \(\mathcal C\) to the \(\infty\)-category \(\mathcal C\times \mathbb{Z}\) with free \(\mathbb{Z}\)-action, and which is symmetric monoidal with respect to the Day convolution structure on \(\mathrm{Cat}_{\infty}^{B\mathbb{Z}}\).
Combining these left adjoints, we obtain a symmetric monoidal left adjoint \[ \mathrm{Lin}_k(- \times \mathbb{Z}): \mathrm{Cat}_{\infty}\rightarrow\mathrm{add}_{k}^{B\mathbb{Z}}.\]
Applying \(\mathrm{Lin}_k(-\times \mathbb{Z})\) homwise, this induces by subsection A.10 and subsection A.8.11 symmetric monoidal left adjoints \[ \mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}}\coloneqq \mathrm{Cat}[\mathrm{Lin}_k(-\times \mathbb{Z})]\colon \mathrm{Cat}_{(\infty, {2})} = \mathrm{Cat}[\mathrm{Cat}_{\infty}] \rightarrow\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]\] and (abusing notation) \[\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} \coloneqq \mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{Lin}_k(-\times \mathbb{Z})])\colon \mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}_{(\infty, {2})}) \rightarrow\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}])\] of the respective forgetful functors.
The functor \(\mathrm{BSbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) of monoidal \((\infty,2)\)-categories from corollary 6.1.3 factors through a monoidal \(\mathrm{add}_{k}^{B\mathbb{Z}}\)-enriched functor \[\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathrm{BSbim}) \rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}}).\]
Proof.
Since \(\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) is an object in \(\mathrm{Alg}_{\mathbb E_1} (\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}])\), the statement immediately follows from the adjunction ([00CQ]). ◻
Explicitly, \(\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathrm{BSbim})\) has objects natural numbers and additive \(k\)-linear hom-categories with \(\mathbb{Z}\)-action given by the \(k\)-linearization \(\mathrm{Lin}_k(\mathrm{BSbim}_n \times \mathbb{Z})\) of the 1-category \(\mathrm{BSbim}_n \times \mathbb{Z}\) with free \(\mathbb{Z}\)-action.
Define the monoidal \(k\)-linear (2,2)-category with local shifts \(\mathrm{Sbim}\in \mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}])\), the Soergel \((2,2)\)-category, as the unique factorization 
of \(\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathrm{BSbim}) \rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) with respect to the (surjective-on-objects-and-dominant-on-morphisms, faithful)-factorization system on \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}])\).
We denote the composite monoidal shift-preserving functor \(\mathrm{BSbim}\rightarrow\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathrm{BSbim}) \rightarrow\mathrm{Sbim}\) by \[ \iota\colon \mathrm{BSbim}\rightarrow\mathrm{Sbim}.\]
Composing with the monoidal \((\infty,2)\)-functor \(\mathrm{BSbim}\rightarrow\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathrm{BSbim})\) (i.e. the unit of the adjunction ([00CQ])), we can summarize the categories and functors defined so far thus:
The above defined functors assemble into a commuting diagram of monoidal \((2,2)\)-functors 
where the top and right diagonal functors are faithful and the right diagonal functor is shift-preserving and \(k\)-linear.
Intuitively, \(\mathrm{Sbim}\) is the smallest locally \(k\)-linear additive and idempotent-complete ‘sub’-\((\infty,2)\)-category (mind warning 6.0.4) of \(\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) that is closed under the homwise \(\mathbb{Z}\)-action and contains \(\mathrm{BSbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\). Indeed, it follows from the definition that \(\mathrm{Sbim}\) together with its faithful monoidal \(\mathrm{add}_{k}^{B\mathbb{Z}}\)-enriched functor \(\mathrm{Sbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) is initial amongst factorizations of the monoidal \((\infty,2)\)-functor \(\mathrm{BSbim}\rightarrow\mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\) through faithful monoidal \(\mathrm{add}_{k}^{B\mathbb{Z}}\)-enriched functors. More formally, it is the initial object of the pullback of the following span of \(\infty\)-categories: 
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2