ScalingStacks

Abstract

The Hecke algebras for all symmetric groups taken together form a braided monoidal category that controls all quantum link invariants of type A and, by extension, the standard canon of topological quantum field theories in dimension 3 and 4. Here we provide the first categorification of this Hecke braided monoidal category, which takes the form of an \(\mathbb E_2\)-monoidal \((\infty,2)\)-category whose hom-\((\infty,1)\)-categories are \(k\)-linear, stable, idempotent-complete, and equipped with \(\mathbb{Z}\)-actions. This categorification is designed to control homotopy-coherent link homology theories and to-be-constructed topological quantum field theories in dimension 4 and 5.

Our construction is based on chain complexes of Soergel bimodules, with monoidal structure given by parabolic induction and braiding implemented by Rouquier complexes, all modelled homotopy-coherently. This is part of a framework which allows to transfer the toolkit of the categorification literature into the realm of \(\infty\)-categories and higher algebra. Along the way, we develop families of factorization systems for \((\infty,n)\)-categories, enriched \(\infty\)-categories, and \(\infty\)-operads, which may be of independent interest.

As a service aimed at readers less familiar with homotopy-coherent mathematics, we include a brief introduction to the necessary \(\infty\)-categorical technology in the form of an appendix.

1 Introduction[0001]

1.1 Overview[0002]

Braided monoidal categories of representations, particularly quantum group representations, serve as a powerful framework for understanding invariants of knots, links, and tangles. Via skein theory [Wal06, BK01, MW12] or (relatedly) factorization homology [AF15, AFR18], such braided monoidal categories are also at the heart of 3- and 4-dimensional topological quantum field theories, such as those by Witten–Reshetikhin–Turaev [Wit89, RT91] and Crane–Yetter–Kauffman [CKY97].

Following the paradigm of [CF94], our work is motivated by the desire to construct higher-dimensional TQFTs by categorifying these theories. To this end, we will provide a construction of a “categorified braided monoidal category”, which we believe encompasses all the necessary data to form the basis for future constructions of 4- and 5-dimensional analogs, respectively, of Witten–Reshetikhin–Turaev and Crane–Yetter–Kauffman theories of Lie type A, see also [MWW22, Str23] and references therein.

We begin our discussion in the decategorified context. The Reshetikhin–Turaev link invariants [RT90] give an interpretation of the Jones polynomial [Jon85] in terms of morphisms in the braided monoidal category of representations of quantum \(\mathfrak{sl}_2\). In fact, all of the Reshetikhin–Turaev invariants of type A are controlled via Schur–Weyl duality by a single braided monoidal category \(H\), just as the \(\mathfrak{sl}_N\) link polynomials are specializations of the HOMFLYPT link invariant. Recall for \(n \in \mathbb N_0 \coloneqq \{ 0 , 1, 2, \ldots \}\) the Hecke algebra \(H_n\), which is a quotient of the group algebra over \(\mathbb{Z}[q^{\pm 1}]\) of the Artin braid group \(\operatorname{Br}_n\) for the symmetric group \(S_n\). The braided monoidal category \(H\) consists of the following data:

1. The objects are given by natural numbers \(n \in \mathbb N_0\).

2. The endomorphism algebra of each object \(n \in H\) is \(H_n\). All other hom-sets are trivial.

3. The monoidal structure is given on objects by addition, i.e. \(m \otimes n \coloneqq m + n\), and on morphism as the map \(H_m \times H_n \rightarrow H_{m+n}\) corresponding to the parabolic subgroup \(S_m\times S_n\hookrightarrow S_{m+n}\).

4. The braiding—a natural isomorphism \(m \otimes n \xrightarrow{\sim} n \otimes m\) for each \(m,n \in H\)—is given by the image in \(\mathrm{End}_H(m+n) \coloneqq H_{m+n}\) of the positive \((m,n)\)-shuffle braid in \(\operatorname{Br}_{m+n}\).

In the pioneering work [Kho00], Khovanov defined a homology theory for knots and links that categorifies the Jones polynomial, leading to a plethora of categorifications of many other polynomials invariants; see e.g. [Kho06] for a survey. Despite the diverse flavors and contexts of these constructions (e.g. involving perverse or coherent sheaves, matrix factorizations, symplectic geometry, Lie theory, and diagrammatic calculus), these categorifications are all—whether explicitly, implicitly, or a posteriori—shadows of a universal categorification involving Soergel bimodules [Soe92]; see [Str23]. Specifically, there is a monoidal additive category \(\mathrm{Sbim}_n\) of Soergel bimodules, which categorifies the Hecke algebra \(H_n\), i.e. whose split Grothendieck ring is \(H_n\). Concretely, \(\mathrm{Sbim}_n\) is a certain full additive subcategory of the category of graded bimodules of the polynomial algebra \(R_n \coloneqq k[x_1, \ldots, x_n]\) over a \(\mathbb{Q}\)-algebra \(k\) with each \(x_i\) in degree \(2\). The \(\mathbb{Z}\)-action by grading shift categorifies the \(\mathbb{Z}[q^{\pm 1}]\)-action on \(H_n\). The image of braids in \(H_n\) can only be categorified to objects in the (monoidal) bounded chain homotopy category \(\mathrm{K}^b(\mathrm{Sbim}_n)\) of \(\mathrm{Sbim}_n\), the so called Rouquier complexes [Rou06], see subsection 2.2.

The natural next step is to assemble these categorifications into a “categorified braided monoidal category”, in which categorified braid invariants can be seen as morphisms. Our main Theorems A and B yield this as a consequence. Namely, they provide a braided monoidal 2-category \(\mathcal H\)—more precisely, an \(\mathbb E_2\)-monoidal \((2,2)\)-category—which decategorifies to \(H\) upon taking Grothendieck groups.

1. The objects are given by natural numbers \(n \in \mathbb N_0\).

2. The endomorphism monoidal category of each object \(n \in \mathcal H\) is \(\mathrm{K}^b(\mathrm{Sbim}_n)\). All other hom-categories are trivial.

3. The monoidal structure is given on objects by addition, i.e. \(m \otimes n \coloneqq m + n\), and on morphism by a parabolic induction functor \(\mathrm{K}^b(\mathrm{Sbim}_m)\times \mathrm{K}^b(\mathrm{Sbim}_n) \rightarrow\mathrm{K}^b(\mathrm{Sbim}_{m+n})\).

4. The braiding isomorphism \(m \otimes n \xrightarrow{\sim} n \otimes m\) is given by the Rouquier complex in \(\mathrm{K}^b(\mathrm{Sbim}_{m+n})\) corresponding to the \((m,n)\)-shuffle braid in \(\operatorname{Br}_{m+n}\).

In fact, we go much further: we work in a homotopy-theoretic context based on higher algebra in the sense of Lurie [Lur09, Lur17] and construct a braided monoidal \((\infty,2)\)-category¹ \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\); a truncation of which recovers the braided monoidal 2-category \(\mathcal H\) described above. Explicitly, the hom triangulated categories \(\mathrm{K}^b(\mathrm{Sbim}_n)\) are replaced by stable \(\infty\)-categories \({\mathbf K}^b(\mathrm{Sbim}_n)\) of chain complexes, chain maps and chain homotopies with the full hierarchy of higher homotopies. In contrast to \(\mathrm{K}^b(\mathrm{Sbim}_n)\), using the stable \(\infty\)-categories \({\mathbf K}^b(\mathrm{Sbim}_n)\) has an essentially advantage, which we crucially use in the construction of \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) and its braided monoidal structure. Namely, \({\mathbf K}^b\) has a universal property: It constructs the free idempotent-complete stable \(\infty\)-category on an idempotent-complete additive category, see subsection 3.4.

The braided monoidal \((\infty,2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) yields, in the spirit of Rouquier [Rou17], a suitable derived setting for defining invariants not just of braids but also of braid cobordisms with their isotopies and higher isotopies. Hints of such (coherent) braided monoidal 2-categories from link homology have appeared in the literature, see e.g. [MR20]².

A further motivation for such a derived setting, and for working \(\infty\)-categorically comes from TQFTs: passing from the triangulated categories \(\mathrm{K}^b(\mathrm{Sbim}_n)\) to their underlying stable \(\infty\)-categories \({\mathbf K}^b(\mathrm{Sbim}_n)\) yields better finiteness properties. These finiteness properties can be essential for extending TQFTs to manifolds of higher dimensions, and for ensuring that the resulting invariants are small enough to have well-defined decategorifications. See [MWW23, Thm. 1.5 and § 4.7] for an example of this phenomenon.

1.2 Monoidality theorem[0003]

Our first result is an upgrade of \(\mathcal H\) to a monoidal \((\infty,2)\)-category whose hom-\(\infty\)-categories are \(k\)-linear, stable, idempotent-complete, and equipped with a \(\mathbb{Z}\)-action given by the grading shift action on \({\mathbf K}^b(\mathrm{Sbim}_n)\).

To formulate the result, we let \(\mathrm{st}_k\) denote the \(\infty\)-category of small stable idempotent-complete \(k\)-linear \(\infty\)-categories³. Let \(\mathrm{st}^{B\mathbb{Z}}_{k}\) be the \(\infty\)-category \(\mathrm{Fun}(B\mathbb{Z}, \mathrm{st}_k)\) of such \(\infty\)-categories equipped with an additional compatible \(\mathbb{Z}\)-action. Then, Day convolution⁴ induces a symmetric monoidal structure on \(\mathrm{st}^{B\mathbb{Z}}_{k}\), and in turn on the \(\infty\)-category \(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\) of small \(\infty\)-categories enriched in \(\mathrm{st}^{B\mathbb{Z}}_{k}\) in the sense of Gepner-Haugseng [GH15].

The \((\infty,2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) from theorem A is constructed in section 6 via certain enriched variants of Morita \((\infty,2)\)-categories developed in section 4 following [Lur17], see also [Hau17, JS17].

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 \(U(1) \simeq B\mathbb{Z}\).⁵

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.

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.

1.4 Context and proof outline for the braiding theorem[0009]

Here we further contextualize Theorem B while discussing key aspects of its setup and proof.

1.4.1 Braided monoidal 2-categories and their generalizations[000A]

The problem of finding the right definition of braided monoidal (even ordinary) \(2\)-categories has a long history, going back to [KV94, KV94, BN96], see [Sch09] for a survey. One motivation [BD95] was the construction of surface invariants in \(4\)-manifolds, in particular \(2\)-knots, just like braided monoidal \(1\)-categories are related to link invariants. A challenge hereby was the specification of the extra data required for a braiding on a \(2\)-category, for example with respect to the naturality of the braiding—which is not a property, but extra structure.

An even larger challenge is the construction of interesting concrete examples of braided monoidal \(2\)-categories, at least as rich as the theory of braided monoidal categories built from quantum groups. First hints of such a landscape came into view with the invention of Khovanov homology and its various associated invariants of tangles and tangle cobordisms [Kho06]. However, the idea of extracting a braided monoidal \(2\)-category from these invariants, see e.g. [BL11], turned out to be hard. Indeed, this idea drove the study of functoriality properties of categorified link and tangle invariants with respect to tangle cobordisms. At the state of the art, the most well-behaved link homology theories assign homotopy classes of chain maps to isotopy classes of tangle cobordisms. As discussed in [MWW22, § 6], this is sufficient to satisfy the (classical) axioms for a braided monoidal \(2\)-category in a toy model that is combinatorial and discrete up to the level of \(1\)-morphisms and truncated at the level of \(2\)-morphisms.

It is however desirable to go beyond such a toy model, so that the 2-hom-objects are no longer just sets or vector spaces, but are of a more homological nature: higher homotopies give invariants of higher isotopies between braids and links. Indeed, systematically incorporating such higher homotopies in link homology theories is also highly relevant beyond the goal of constructing an enhancement of a braided monoidal \(2\)-category, e.g. towards skein algebra categorification [QW21, Discussion after Conjecture 1.8], cabling operations [GHW22, § 1.3], as well as wrapping and flatting functors [GW23, § 6.4], [Eli18, MMV24].

To formulate answers to such questions, the classical axioms for braided monoidal \(2\)-categories are no longer sufficient: For example the Rouquier complexes corresponding to braid group generators satisfy the braid relation up to homotopy equivalence, [Rou06]. These equivalences must be provided as additional data, which then should be subject to further coherence conditions, requiring higher homotopies ad infinitum. Even worse, the compatibilities to be checked at each level quickly explode—in both complexity and in number,—and hence become unfeasible beyond the first two well-known stages of Reidemeister moves and Carter–Saito movie moves.

To be able to address these problems, we find it essential to work within the homotopy-theoretic context of \((\infty,2)\)-categories, as explained further in the following.

The modern formalism of higher algebra, as in [Lur17], offers a robust answer to the problem of modelling braided monoidal structures in such a homotopical context: Namely in terms of the notion of an \(\mathbb E_2\)-algebra [BV73]. This notion applies in any symmetric monoidal \(\infty\)-category \(\mathcal V\), and in the \((2,1)\)-category of ordinary categories recovers the notion of a braided monoidal category by a folklore result that we recover in remark 7.7.7. Namely, an \(\mathbb E_2\)-algebra structure on an object \(V \in \mathcal V\) consists of a suitably compatible system of maps \(\mathrm{Conf}_n(\mathbb{R}^2) \rightarrow\mathrm{Hom}_\mathcal V(V^{\otimes n},V)\) from configuration spaces of points in \(\mathbb{R}^2\). For example, a chosen basepoint of \(\mathrm{Conf}_2(\mathbb{R}^2)\) selects a multiplication map \(V \otimes V \xrightarrow{\mu} V\), and the generator of \(\pi_1(\mathrm{Conf}_2(\mathbb{R}^2)) \simeq \mathbb{Z}\) selects a braiding isomorphism \(\mu \xrightarrow{\sim} \mu \circ \tau\) (where \(\tau\) denotes the symmetry isomorphism in \(\mathcal V\)). More generally, the requisite compatibilites as \(n\) varies collectively encode the homotopy coherent associativity of \(\mu\) as well as its compatibility with the braiding.

To apply this formalism of \(\mathbb E_2\)-algebras to describe a braiding on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\), we use \(\mathcal V= \mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\). Spelled out, \(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\) is the \(\infty\)-category of \(\mathrm{st}^{B\mathbb{Z}}_{k}\)-enriched \(\infty\)-categories, i.e. \((\infty,2)\)-categories whose hom-\((\infty,1)\)-categories are stable, idempotent complete, and equipped with a \(k\)-linear structure and an action by \(\mathbb{Z}\), all appropriately compatible with the composition operations. Indeed, a \(\mathrm{st}^{B\mathbb{Z}}_{k}\)-enriched \(\infty\)-category \(\mathcal C\in \mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\) has composition morphisms \(\mathrm{Hom}_\mathcal C(c_0,c_1) \otimes \mathrm{Hom}_\mathcal C(c_1,c_2) \rightarrow\mathrm{Hom}_\mathcal C(c_0,c_2)\) in \(\mathrm{st}^{B\mathbb{Z}}_{k}\), where \(\otimes\) denotes the Day convolution symmetric monoidal structure on \(\mathrm{st}^{B\mathbb{Z}}_{k}\), rather than merely functors \(\mathrm{Hom}_\mathcal C(c_0,c_1) \times \mathrm{Hom}_\mathcal C(c_1,c_2) \rightarrow\mathrm{Hom}_\mathcal C(c_0,c_2)\): The tensor product in \(\mathrm{st}^{B\mathbb{Z}}_{k}\) implicitly enforces our desired compatibility.

theorem A and Theorem B construct \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) as an \(\mathbb E_2\)-algebra⁶ object in \(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\). In particular, its monoidal structure and braiding are automatically homotopy-coherently compatible with the \(k\)-linearity and \(\mathbb{Z}\)-actions on its hom-\((\infty,1)\)-categories.

To construct this \(\mathbb E_2\)-algebra structure, we take inspiration from the homotopy-theoretic machinery of obstruction theory, in which context one often finds it (somewhat paradoxically) easier to prove a stronger theorem. Specifically, we prove Theorem B by establishing not just the existence of an \(\mathbb E_2\)-algebra structure on the \((\infty,2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\), but also its homotopy-theoretic uniqueness along with its compatibility with \(k\)-linearity, \(\mathbb{Z}\)-action, and with the fiber functor \(H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\).

1.4.2 Higher-categorical notions of faithfulness and homotopy categories[000B]

To prove our main theorems, we develop machinery to reduce the construction of algebraic structures on \((\infty,2)\)-categories to corresponding structures on their underlying ordinary homotopy categories, as well as higher-dimensional variants. We outline some of these results, which might be of independent interest.

As a toy case, observe that given an \(\infty\)-category \(\mathcal C\), a full subcategory \(\mathcal C'\) is determined entirely by the subset \(h_0\mathcal C'\) of the set \(h_0\mathcal C\) of equivalences classes of objects in \(\mathcal C\) that are contained in \(\mathcal C'\). Furthermore, if \(\mathcal C\) is endowed with some multiplicative structure (e.g. monoidal, braided monoidal, or symmetric monoidal), then \(\mathcal C'\) inherits such a structure if and only if \(h_0\mathcal C'\) inherits the resulting structure from \(h_0\mathcal C\).

Given an \((\infty,2)\)-category \(\mathcal C\), we write \(h_1\mathcal C\) for the ordinary category obtained by first discarding its noninvertible 2-morphisms and then passing to the homotopy category, i.e. taking \(\pi_0\) of its hom-spaces. Moreover, we say that a functor between \((\infty,2)\)-categories is faithful if it is fully faithful on hom-\((\infty,1)\)-categories.⁷ Then, analogously to the above situation, we show that a faithful functor \(\mathcal C' \rightarrow\mathcal C\) is determined entirely by their corresponding faithful functor \(h_1\mathcal C' \rightarrow h_1\mathcal C\). Moreover, we show that if \(\mathcal C\) is braided monoidal equipped with the faithful functor \(\mathcal C' \rightarrow\mathcal C\), then endowing \(\mathcal C'\) together with a (compatible) braided monoidal structure is equivalent to endowing \(h_1\mathcal C'\) and \(h_1\mathcal C' \rightarrow h_1\mathcal C\) with such a structure, see corollary 5.5.5. The task of defining an \(\mathbb E_2\)-algebra structure on \(\mathcal C'\) therefore reduces in such a situation to the task of defining a braiding (in the classical sense!) on its homotopy 1-category \(h_1\mathcal C'\).

In fact, we prove the above results in much broader generality: In section 5, we study the notion of \(n\)-faithfulness for functors between \((\infty,k)\)-categories, and show in subsection 5.3 that they define the right classes in factorization systems on \(\mathrm{Cat}_{(\infty,k)}\), whose corresponding left classes are given by the \(n\)-surjective functors, which are surjective on objects, and on parallel morphisms up to level \((n+1)\), see definition 5.3.1. Using the iterative definition of higher categories, \(\mathrm{Cat}_{(\infty,k)} \coloneqq \mathrm{Cat}[\mathrm{Cat}_{(\infty,k-1)}]\), we deduce this from general results that we prove regarding factorization systems on enriched \(\infty\)-categories in subsection B.4, also see [Hau23] for related recent result. Generalizing the above, we establish in corollary 5.5.3 that for any \(n\) and any \(k\), \((n-1)\)-faithful functors to an \((\infty,k)\)-category \(\mathcal C\) are equivalently determined by \((n-1)\)-faithful functors to its homotopy \(n\)-category.

1.4.3 The fiber functor[000C]

The fiber functor \(H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\) on which Theorem B is built, can be viewed as a categorified and graded analog of the forgetful functor from a category of quantum group representations to \(\mathrm{Vec}\). Since its target \(\mathrm{st}^{B\mathbb{Z}}_{k}\) is braided monoidal (in fact symmetric monoidal), it is tempting to apply the machinery of subsection 1.4.2 to obtain a braiding on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\).

However, this does not work because the analogy with the classical situation breaks down in an important way. In the classical setting, the fiber functor is faithful and monoidal, but in general not braided. By contrast, our categorified fiber functor will be braided monoidal but not faithful (in the sense of Subsection 1.4.2). This lack of faithfulness prevents us from directly using the reduction results from Subsection 1.4.2. In Subsection 1.4.4, we will discuss a restricted version of the fiber functor that is faithful, which allows us to leverage the results indicated in Subsection 1.4.2.

In the end, the non-faithfulness turns out to be an essential feature of our fiber functor. This feature is what allows \(H_{\mathrm{loc}}\) and its source category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) to be equipped with a non-symmetrically braided monoidal structure, even though the target category \(\mathrm{st}^{B\mathbb{Z}}_{k}\) is symmetric monoidal. A faithful braided fiber functor would force the source category to be symmetric!

In the classical situation, the existence of a fiber functor arises from the fact that quantum group representations are ultimately categories of modules for a quasitriangular Hopf algebra, which can be recovered from the fiber functor via Tannakian reconstruction. We would be very interested to see an application of Tannakian reconstruction to our categorified fiber functor.

We expect our fiber functor to be important for future applications and computations. For instance, the fact that it is braided may give a means of recursively computing all the algebraic data implicit in the braided monoidal structure on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) on a cell-by-cell basis.

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 \(H_{\mathrm{loc}}\colon {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\).

1.4.5 Prebraidings via \(\infty\)-operads[000E]

To lift our \(1\)-categorical prebraiding to braided monoidal structures on \((\infty,2)\)-categories, we first generalize prebraidings themselves to this \((\infty,2)\)-categorical setting.

We implement the notion of a prebraiding in the context of \(\infty\)-operads as indicated in the diagram \[\mathbb T_2 \otimes \mathbb E_1 \longrightarrow \mathbb A_2 \otimes \mathbb E_1 \longrightarrow \mathbb E_1 \otimes \mathbb E_1 \stackrel{\sim}{\longrightarrow} \mathbb E_2\] thereof, as we now explain. For more details see subsection A.8 and section 7.

• As indicated in Subsection 1.4.1, we formalize the notion of a braided monoidal structure via the notion of an \(\mathbb E_2\)-algebra (with \(k\)-ary operations parametrized by \(\mathrm{Conf}_k(\mathbb{R}^2)\)).

• Much simpler is the notion of an \(\mathbb E_1\)-algebra, which has \(k\)-ary operations parametrized by \(\mathrm{Conf}_k(\mathbb{R}^1)\). These spaces are discrete, and this \(\infty\)-operad is equivalent to the ordinary associative operad.

• The equivalence \(\mathbb E_1 \otimes \mathbb E_1 \xrightarrow{\sim} \mathbb E_2\) is an instance of Dunn additivity. Hence, an \(\mathbb E_2\)-algebra structure is equivalent to two compatible \(\mathbb E_1\)-algebra structures: \(\mathrm{Alg}_{\mathbb E_2}(\mathcal V) \simeq \mathrm{Alg}_{\mathbb E_1}(\mathrm{Alg}_{\mathbb E_1}(\mathcal V))\).

• Whereas \(\mathbb E_1\) parametrizes (homotopy-coherently) associative and unital binary operations, by forgetting associativity one arrives at the operad \(\mathbb A_2\), which parametrizes binary operations that are merely unital.

• The \(\infty\)-operad \(\mathbb T_2\) is a two-colored ordinary operad, and parametrizes pairs of pointed objects \(\mathcal A,\mathcal B\in \mathrm{Alg}_{\mathbb E_0}(\mathcal V)\)⁹ together with pointed morphisms \(\mathcal A\xrightarrow{F} \mathcal B\), and \(\mathcal A\otimes \mathcal A\xrightarrow{\mu} \mathcal B\), along with pointed homotopies \(\mu(\eta_\mathcal A\otimes (-)) \simeq F \simeq \mu((-) \otimes \eta_\mathcal A)\) in \(\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal V)}(\mathcal A,\mathcal B)\). In particular, we view a \(\mathbb T_2\)-algebra as having not an underlying object but having an underlying morphism, namely the morphism \(F\). The map of \(\infty\)-operads \(\mathbb T_2 \rightarrow\mathbb A_2\) corepresents the operation carrying an \(\mathbb A_2\)-algebra to the evident \(\mathbb T_2\)-algebra with \(\mathcal A= \mathcal B\) and \(F= \mathrm{id}_{\mathcal A}\).

We define a prebraiding on an \(\mathbb E_1\)-algebra morphism \(F\) in a symmetric monoidal \(\infty\)-category \(\mathcal V\) to be a \(\mathbb T_2\)-algebra structure on the correponding morphisms in \(\mathrm{Alg}_{\mathbb E_1}(\mathcal V)\), or equivalently a lift of the corresponding \([1] \otimes \mathbb E_1\)-algebra in \(\mathcal V\) to a \(\mathbb T_2 \otimes \mathbb E_1\)-algebra, see subsection 8.1. In the context of ordinary categories, we show in example 8.1.2 that this notion coincides with the one described in Subsection 1.4.4.

1.4.6 From prebraidings to braidings[000F]

In Subsection 1.4.4 we outlined the construction of a prebraiding on the monoidal functor \(h_1\mathrm{BSbim}\hookrightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) between ordinary monoidal \(1\)-categories, which is compatible with the fiber functor \(h_1 H_{\mathrm{loc}}: h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow h_1 \mathrm{st}^{B\mathbb{Z}}_{k}\). To complete the proof of Theorem B, it remains to show that this admits a unique lift to a braiding on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) such that the fiber functor is braided.

We prove this in steps, combining the machinery described above, as follows, see section 8.

Applying a version of the techniques described in Subsection 1.4.2, it follows that our prebraiding on \(h_1\mathrm{BSbim}\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) over \(h_1\mathrm{st}^{B\mathbb{Z}}_{k}\) lifts uniquely to a prebraiding on the \((\infty,2)\)-functor \(\mathrm{BSbim}\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) over \(\mathrm{st}^{B\mathbb{Z}}_{k}\).

In order to proceed, we note a crucial property of prebraidings, see corollary 8.3.7: given an adjunction \(F \colon \mathcal C\rightleftarrows \mathcal D\colon G\) in which the left adjoint is symmetric monoidal, the data of a prebraiding on an \(\mathbb E_1\)-algebra morphism \(c \rightarrow G(d)\) is equivalent to the data of a prebraiding on its adjunct \(F(c) \rightarrow d\). We use this to extend our prebraiding over \(\mathrm{st}^{B\mathbb{Z}}_{k}\) from one on \(\mathrm{BSbim}\hookrightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) to one on \(\mathrm{Sbim}\hookrightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\), and then we use it again to extend the latter to one on the defining equivalence \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \xrightarrow{\sim} {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\).

So, we have obtained a prebraiding, i.e. a \(\mathbb T_2\)-structure on the identity morphism of \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \in \mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathrm{st}^{B\mathbb{Z}}_{k}})\). Because the identity morphism is invertible, this is equivalent to an \(\mathbb A_2 \otimes \mathbb E_1\)-structure on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \in \mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathrm{st}^{B\mathbb{Z}}_{k}}\).

A final task is to lift this \(\mathbb A_2 \otimes \mathbb E_1\)-structure to an \(\mathbb E_2 \simeq \mathbb E_1 \otimes \mathbb E_1\) structure. Recall that prebraidings on identity functors between ordinary monoidal \(1\)-categories, i.e. \(\mathbb A_2\otimes \mathbb E_1\)-structures in \(\mathrm{Cat}_1\), are precisely the same as braidings, i.e. \(\mathbb E_2\)-structures. More generally, we show in subsection 7.7 that \(\mathbb A_2 \otimes \mathbb E_1\)- and \(\mathbb E_2\)-algebras agree in general \((2,1)\)-operads.

This observation applies to our situation due to a crucial truncatedness result regarding the endomorphism \(\infty\)-operad of the object \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \in \mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathrm{st}^{B\mathbb{Z}}_{k}}\). Namely, while we expect it to be quite complicated in general, we prove in corollary 8.4.4 that the maximal sub-\(\infty\)-operad in the image of its \(\mathbb E_1\)-structure is in fact just a \((2,1)\)-operad. This suffices for our purposes since the map of \(\infty\)-operads \(\mathbb E_1 \rightarrow\mathbb E_2\) is surjective on path components of mapping spaces.

Thus, our prebraiding on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) over \(\mathrm{st}^{B\mathbb{Z}}_{k}\) extends uniquely to an \(\mathbb E_2\)-algebra structure establishing our main goal.

After now having introduced the key ideas and concepts, we finish this introduction by giving an outline of the organization of the paper.

1.5 Organization of the paper[000G]

In section 2 we start by reviewing the basics of Soergel bimodules, Bott–Samelson bimodules, and their diagrammatics. Next we proceed to the bounded homotopy category of Soergel bimodules and discuss Rouquier complexes for braids. The homotopy equivalence classes of bounded chain complexes of Soergel bimodules can then be organized into a monoidal \(1\)-category \(h_1\mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\). In subsection 2.5, we define the notion of a prebraiding and construct such a prebraiding on the functor \(h_1\mathrm{BSbim}\rightarrow h_1\mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) that includes isomorphism classes of Bott–Samelson bimodules into \(h_1\mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\). This requires explicit diagrammatic computations. Finally, we relate the notion of prebraiding to centers and centralizers in theorem 2.6.4, which will be important for \(\infty\)-categorical aspects later on.

From section 3 we move into the world of \(\infty\)-categories. Section section 3 mostly recalls and collects results from [Lur17]. We begin in Subsection 3.1 by reviewing various colimit completion procedures, such as the \(\operatorname{Ind}\)-completion under filtered and the \(\mathcal P^{\Sigma}\)-completion under sifted colimits, and symmetric monoidal structures on \(\infty\)-categories with certain colimits (and functors preserving those colimits). Next, we recall the notion of compactly/projectively generated \(\infty\)-categories in subsection 3.2. Furthermore, in proposition 3.2.8, we show that \(\operatorname{Ind}\) gives an equivalence between small idempotent-complete \(\infty\)-categories with finite colimits, and presentable compactly generated \(\infty\)-categories. Similarly, \(\mathcal P^{\Sigma}\) gives an equivalence between small idempotent-complete \(\infty\)-categories with coproducts, and presentable projectively generated \(\infty\)-categories. We review the theory of additive and stable \(\infty\)-categories in subsection 3.3 and give an \(\infty\)-categorical interpretation of a process that is ubiquitous in the categorification literature: passing from an additive category to its bounded chain homotopy category, in subsection 3.4. In particular we show as corollary 3.4.10 that for a small idempotent-complete additive \(1\)-category \(\mathcal A\), the stable \(\infty\)-category \({\mathbf K}^b(\mathcal A)\) of chain complexes in \(\mathcal A\) is the free stable \(\infty\)-category on \(\mathcal A\). We introduce the \(\infty\)-category of graded \(k\)-modules and review their Day convolution symmetric monoidal structures in subsection 3.5. Lastly, in subsection 3.6 we review the theory of derived \(\infty\)-categories developed in [Lur17, § 1.3] and prove as proposition 3.6.9 that the module \(\infty\)-category of a discrete graded algebra is the derived \(\infty\)-category of graded modules.

In section 4, we define the \(\infty\)-categories \(\mathrm{add}_{k}^{B\mathbb{Z}}\) and \(\mathrm{st}^{B\mathbb{Z}}_{k}\) of additive and stable \(k\)-linear \(\infty\)-categories with a \(\mathbb{Z}\)-action, as well as their self-enrichment. Furthermore, we construct full symmetric monoidal enriched subcategories \(\mathrm{Mor}^{\mathrm{flat}, \mathrm{gr-proj}}(\mathrm{mod}_{k}^{Z})\) and \(\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{Z})\) of \(\mathrm{add}_{k}^{B\mathbb{Z}}\) and \(\mathrm{st}^{B\mathbb{Z}}_{k}\) respectively, whose objects are labelled by flat discrete graded \(k\)-algebras, and whose enriched homs are given by (chain complexes) of discrete graded bimodules satisfying suitable finiteness condition. These Morita categories facilitate our homotopy coherent construction of the Soergel \((2,2)\)-category in section 6. We begin in subsection 4.1 with a review of morphism objects in various module categories as well as the self-enrichment of presentably symmetric monoidal \(\infty\)-categories. In subsection 4.2, we define \(\mathrm{add}_{k}^{B\mathbb{Z}}\) and \(\mathrm{st}^{B\mathbb{Z}}_{k}\) as well as the symmetric monoidal left adjoint \({\mathbf K}^b\) in the graded-linear context. Concerning gradings, we prove in subsection 4.3 an \(\infty\)-categorical version of the familiar equivalence between categories enriched in graded modules and categories with an action of a monoid.¹⁰ To finish the section, we construct the desired enriched Morita categories \(\mathrm{Mor}^{\mathrm{flat}, \mathrm{gr-proj}}(\mathrm{mod}_{k}^{Z})\) and \(\mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{Z})\) in subsection 4.5.

In section 5, we enter the world of \((\infty, k)\)-categories. The purpose of this section is to introduce and study various factorization systems on the \(\infty\)-category of \((\infty, k)\)-categories. We prove as theorem 5.3.7 the existence of the (\(n\)-surjective, \(n\)-faithful) factorization system on \(\mathrm{Cat}_{(\infty, {k})}\), generalizing the familiar (surjective-on-objects, fully-faithful) factorization system on \(1\)-categories. Furthermore, we define the homotopy \(n\)-category functor and prove as theorem 5.5.2 that \((n-1)\)-faithful functors into a \((\infty, k)\)-category \(\mathcal C\) are controlled by its homotopy \(n\)-category \(h_n \mathcal C\). This is crucial in both the construction of the Soergel \((2,2)\)-category in section 6 as well as reducing braidings to prebraidings in section 8. We begin in subsection 5.1 with a quick recollection on the basics of \((\infty, k)\) category theory. In subsection 5.2, we recall the familiar (\(n\)-connected, \(n\)-truncated) factorization system on the \(\infty\)-category of spaces. We then proceed in subsection 5.3 to inductively define the classes of (\(n\)-surjective, \(n\)-faithful)-morphism and prove as theorem 5.3.7 that they form factorization system. Lastly, we define the \(n\)-homotopy category functor in subsection 5.4, state theorem 5.5.2 (regarding faithful functors and homotopy categories) and its consequences in subsection 5.5, and prove theorem 5.5.2 in subsection 5.6.

In section 6, we define the relevant higher categories of Bott–Samelson bimodules and (chain complexes of) Soergel bimodules. We construct the monoidal \((2,2)\)-categories \(\mathrm{BSbim}\) in subsection 6.1 and \(\mathrm{Sbim}\) in subsection 6.2, whose hom categories are the categories of type A Bott–Samelson and Soergel bimodules, respectively. By construction, \(\mathrm{Sbim}\) carries all the desired extra structure such as local \(k\)-linearity and \(\mathbb{Z}\)-action. In subsection 6.3, we verify that the hom-categories of \(\mathrm{Sbim}\) indeed agree with the categories \(\mathrm{Sbim}_n\). To finish the section, and to prove theorem A, we construct in subsection 6.4 the monoidal \((\infty, 2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) whose hom categories are stable \(\infty\)-categories of chain complexes of type A Soergel bimodules, together with all its structure, and define the fiber functor \(H_{\mathrm{loc}}\) from \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) to \(\mathrm{st}^{B\mathbb{Z}}_{k}\) in subsection 6.5.

In section 7, we introduce the operadic machinery necessary for constructing the braiding on the monoidal \((\infty, 2)\)-category \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\). In particular, we formulate prebraidings using the language of \(\infty\)-operads and prove as corollary 7.7.8 that braidings and prebraidings coincide when the ambient symmetric monoidal \(\infty\)-category (and more generally \(\infty\)-operad) is suitably truncated, generalizing the well-known statement that an \(\mathbb E_2\)-algebra structure on a 1-category is the same as a braided monoidal structure. We extensively use Lurie’s theory of \(\infty\)-operads [Lur17, § 2] and refer the reader to subsection A.8 for an overview.

After a quick recollection on unital \(\infty\)-operads in subsection 7.1, we construct the \(\mathbb{T}_2\) and \(\mathbb A_2\) operads, which are the main players of this section, in subsection 7.2. After defining a relative version of \(\mathbb{T}_2\) structure in subsection 7.3, we review the \(\infty\)-categorical versions of centralizers and center [Lur17, § 5.2] in subsection 7.4. Using the theory of centralizers, we prove as corollary 7.4.15 that \(\mathbb{T}_2\)-structures are the \(\infty\)-categorical generalization of the notion of prebraiding. We construct the (\(n\)-surjective, \(n\)-faithful) factorization system on \(\infty\)-operads in subsection 7.5 and prove a surjectivity result in subsection 7.6. In subsection 7.7 we prove corollary 7.7.8 (concerning prebraidings and braidings), which is the main result of this section. Finally, in subsection 7.8 we end with an easy but useful result regarding lifting maps of algebras.

In section 8, we finally prove theorem B. After defining the spaces of braidings and prebraidings in subsection 8.1, in subsection 8.2 we state our main theorem 8.2.1, an abstract theorem about lifting prebraidings on homotopy \(1\)-categories to braidings, assuming the existence of a fiber functor. We expect this theorem can be used to build braidings on other interesting \((\infty,2)\)-categories. Applying theorem 8.2.1 to the prebraiding on \(h_1\mathrm{BSbim}\rightarrow h_1\mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) constructed in section 2, we prove a precise version of theorem B as corollary 8.2.2. The remainder of the section is concerned with the proof of theorem 8.2.1. The space of braidings is shown to be equivalent, through a series of reduction steps, to spaces of prebraidings in progressively less-structured situations, terminating in a set (rather than a space) of prebraidings between the homotopy \(1\)-categories. Assuming various truncatedness conditions, we prove those reduction steps are equivalences in subsection 8.3. Finally, in subsection 8.4 we prove those truncatedness hypothesis and complete the proof of theorem 8.2.1.

In section A, we provide a leasurely introduction to various aspects of \(\infty\)-categories and higher algebra.

In section B, we review the theory of factorization systems on \(\infty\)-categories and prove a number of ways of obtaining new factorization systems from existing ones, with a focus on presentable \(\infty\)-categories. In particular, we prove two important results for constructing factorization systems, Theorem B.3.1 (concerning algebras over \(\infty\)-operads) in Subsection B.3 and Theorem B.4.1 (concerning enriched \(\infty\)-categories) in Subsection B.4.

Acknowledgements. We would like to thank Markus Zetto for helpful comments on an early draft of this paper. YLL would like to thank David Ben-Zvi, Tom Gannon, Rune Haugseng, Theo Johnson-Freyd, Sam Raskin, Tomer Schlank, and Reuben Stern for helpful discussions, and Mike Hopkins for his continuous support and encouragement. AMG gratefully acknowledges inspiration from David Ayala, Clark Barwick, David Ben-Zvi, Justin Campbell, Tom Gannon, Tyler Lawson, Sam Raskin, Nick Rozenblyum, Matt Stoffregen, and Mike Willis. DR would like to thank Christopher Douglas, Theo Johnson-Freyd, and Kevin Walker for many illuminating discussions on higher categories, link homologies and TQFTs. PW would like to acknowledge the pioneering work of Scott Morrison and Kevin Walker towards TQFTs based on link homology theories and thank them for many illuminating conversations on the topic. CS would like to thank Gustavo Jasso and Stefan Schwede for many very useful discussions around higher categories.

Funding. YLL is supported by the Simons Collaboration on Global Categorical Symmetries. AMG acknowledges the support of NSF grant DMS-2105031. DR acknowledges support by the Emmy Noether program of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 493608176. DR and PW acknowledge support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2121 “Quantum Universe” - 390833306 and the Collaborative Research Center - SFB 1624 “Higher structures, moduli spaces and integrability”. CS is supported by the Gottfried Wilhelm Leibniz Prize of the German Research Foundation.

Finally, the authors acknowledge the important role of the Spring 2020 MSRI programs “Higher Categories and Categorification” and “Quantum Symmetries” supported by the National Science Foundation grant DMS-1440140 and the 2019 Erwin–Schrödinger institute workshop “Categorification in quantum topology and beyond” in catalyzing their collaboration.

2 A prebraiding on the homotopy category of Soergel bimodules[000H]

2.1 Review of Soergel bimodules and diagrammatics[000I]

We let \(k\) denote the rationals \(\mathbb{Q}\) or, more generally, a commutative \(\mathbb{Q}\)-algebra. We consider the \(k\)-linear monoidal categories \(\mathrm{Sbim}_n\) of Soergel bimodules for the symmetric group \(S_n\) acting on its natural representation. In this section, if not specified otherwise, categories mean ordinary categories (in contrast to \(\infty\)-categories used later) and functors mean ordinary functors. For a fixed nonnegative integer \(n\), let \(R_n=k[x_1,x_2,\ldots, x_n]\) denote the polynomial ring over \(k\) in \(n\) variables viewed as polynomial functions on \(\mathfrak{h}^*=(k^n)^*\) in the standard way. Permuting the basis vectors of \(k^n\) induces a left action of the symmetric group \(W=S_n\) on \(R_n\) such that the simple transposition \(s_i=(i,i+1)\) acts by swapping the variables \(x_i\) and \(x_{i+1}\). Denote \(\check\alpha_i=x_{i}-x_{i+1}\) for \(1\leq i\leq n-1\). Then restriction to the span of the \(\check\alpha_i\)’s gives the usual geometric representation of \(W\) viewed as the Coxeter group generated by the simple transpositions. For any subgroup \(G\) of \(W\) let \(R_n^G\) be the subalgebra of \(G\)-invariants in \(R_n\). In case \(G=\langle s_i\rangle\) for some \(1\leq i\leq n-1\) we abbreviate \(R_n^G=R_n^i\). We will view \(R_n\) as a graded (by which we mean \(\mathbb{Z}\)-graded) algebra by putting the generators \(x_i\) in degree \(2\). Note that \(R_n^i\) is a graded subalgebra and we have a canonical, grading-preserving decomposition \[ R_n= R_n^i\oplus \check\alpha_iR_n^i\simeq R_n^i\oplus R_n^i\langle 2\rangle\] as graded \(R_n^i\)-bimodules. Here and in the following we denote for \(j\in \mathbb{Z}\) and a graded (bi)module \(M=\oplus_{i\in\mathbb{Z}} M_i\) by \(M\langle j\rangle\) the graded (bi)module which equals \(M\) as (bi)module but with the grading shifted up by \(j\), i.e. \(M\langle j\rangle_i=M_{i-j}\). The grading shifting functors \(\langle j\rangle\), \(j\in\mathbb{Z}\) equip the category of graded \((R_n,R_m)\)-bimodules for fixed \(n,m\) with an action of the group \(\mathbb{Z}\).

By a graded \(k\)-linear category¹¹ we mean a category enriched in \(\mathbb{Z}\)-graded \(k\)-modules. As an example we can take as objects graded \(R_n\)-bimodules with all \(R_n\)-bimodule maps, denoted \(\mathrm{Hom}^{\mathrm{gr}}\). In this case, the grading shift functors are compatible with the grading on morphisms as follows: \[ \mathrm{Hom}^{\mathrm{gr}}(M\langle k\rangle,N\langle l\rangle) = \mathrm{Hom}^{\mathrm{gr}}(M,N)\langle l-k\rangle\]

Using this shorthand, ([000M]) may also be expressed as: \[ B_{\mathbf{i}}\langle j\rangle \simeq B_{i_k} \otimes_{R} B_{i_{k-1}}\otimes_R \cdots\otimes_R B_{i_1}\langle j\rangle\]

The version \(\mathrm{BSbim}^{\mathrm{gr}}_n\) in Definition [000J] is the most flexible one, but with one caveat: when making statements about isomorphism, idempotents, and categorical constructions such as (co)products, we tacitly require that the structure morphisms are of degree zero, see e.g. ([000J]), i.e. we work in the underlying category. For this reason, we will henceforth almost exclusively work with \(\mathrm{BSbim}_n\). The only exception is subsection 2.5, where we use the version \(\overline{\mathrm{BSbim}}^{\mathrm{gr}}_n\) to connect to the diagrammatic Hecke category.

For later use, we also record how Bott–Samelson and Soergel bimodules for various \(n\) can be related.

It is straightforward to see directly from Definition 2.1.1 that Bott–Samelson bimodules are sent to (bimodules isomorphic to) Bott–Samelson bimodules under parabolic induction. To distinguish the two kinds of tensor product, we will use the convention: \[\begin{aligned} \circ_1:= \otimes_{R_n} \colon \mathrm{BSbim}_n\times \mathrm{BSbim}_n&\rightarrow\mathrm{BSbim}_n\\ \boxtimes := \otimes_{k} \colon \mathrm{BSbim}_m\times \mathrm{BSbim}_n&\rightarrow\mathrm{BSbim}_{m+n} \end{aligned}\] and write \(f\circ_2g \colon M\rightarrow P\) for the composition of morphisms \(f\colon M\rightarrow N\), \(g\colon N\rightarrow P\) in \(\mathrm{BSbim}_n\). We use the same notation for \(\mathrm{Sbim}_n\).

The symbols \(\circ_2\), \(\circ_1\) and \(\boxtimes\) are meant to foreshadow that these operations should form the \(2\)- and \(1\)-morphism composition and the tensor product in a monoidal bicategory, see remark 2.3.3.

2.2 Review of Rouquier complexes[000V]

The complexes \(F(\underline{\beta})\) are called Rouquier complexes. They were first thoroughly studied by Rouquier who proved in [Rou06] that, up to canonical homotopy equivalence, these complexes are independent of the chosen braid word representing \(\beta\). More precisely the following holds:

This rephrasing of Rouquier’s results from [Rou06] is a slight strengthening of [EH17, Prop. 2.19]. As a consequence we may abuse notation and write \(F(\beta)\) instead of \(F(\underline{\beta})\).

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.

In corollary 6.4.3, we match ([001D]) with its \(\infty\)-categorical version.

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]).

2.4 Prebraidings[001J]

As we will see, the cabled crossing complexes from Definition 2.2.7 supply part of the data of a braiding on (an \(\infty\)-categorical version of) chain complexes of Soergel bimodules. What these complexes themselves do not yet encode is the naturality of the braiding—informally speaking, how chain complexes of Soergel bimodules slide through cabled crossings up to coherent homotopy. To capture the naturality of the braiding, we start with a threefold simplified situation: we only aim to slide bimodules (instead of complexes thereof!) through cabled crossings and in fact only Bott–Samelson bimodules, and even simpler, it will be enough to do this on the level of isomorphism classes. To this end, we introduce the crucial notion of a prebraiding.

We have the following trivial observation:

We also want to introduce a relative notion of prebraiding, over a braided monoidal \(1\)-category:

2.5 Prebraiding for Soergel bimodules[001V]

We are now prepared to construct a prebraiding on the functor \(h_1 K_{\mathrm{loc}}\colon h_1 \mathrm{BSbim}\rightarrow h_1 \mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) from ([001D]).

Considering \(h_1 K_{\mathrm{loc}}\colon h_1 \mathrm{BSbim}\rightarrow h_1 \mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) as a functor over \(h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\) via the functor \(h_1H_{\mathrm{loc}}\colon h_1 \mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\), we obtain the following refined version:

The following result is the crucial naturality part for the proof of Theorem 2.5.2. The result is in fact stronger than needed, since it is a statement on the chain level.

We also use the notation \(\mathrm{slide}_{Y}\coloneqq\mathrm{slide}_{Y_1,Y_2}\), when \(m\) and \(n\) are clear from the context.

To establish theorem 2.5.4 we construct, after some preparation, the chain maps, which we then call slide maps, explicitly. For this we work with the Hecke category \(\mathcal{DS}_n\), i.e. the diagrammatical presentation of the monoidal \(1\)-category \(\mathrm{BSbim}_n\) from [EW16], [EK10]. The construction of the chain maps \(\mathrm{slide}_{Y_1,Y_2}\) proceeds in two steps. The first step is specific to the setting of Bott–Samelson and Soergel bimodules and uses \(\mathcal{DS}\). It establishes the existence of atomic slide chain maps, namely \(\mathrm{slide}_{\mathbf{1}_1,B_1}\) for \((m,n)=(1,2)\) and \(\mathrm{slide}_{B_1,\mathbf{1}_1}\) for \((m,n)=(2,1)\); see Lemma 2.5.5. The second step uses that every Bott–Samelson bimodule is a composition (monoidal and horizontal) of Bott–Samelson bimodules on two strands, and so knowing the atomic slide chain maps is sufficient to construct general slide maps along the following scheme: Essentially the same argument would work for any monoidal bicategory generated by a single object and one endomorphism of its tensor square.

To formulate the statements, we need at least a rough description of the Hecke category \(\mathcal{DS}_n\) and the fact that it is equivalent to \(\overline{\mathrm{BSbim}}^{\mathrm{gr}}_n\) as graded monoidal \(k\)-linear category. For details we refer to [EW16], [EK10]. Each simple reflection \(s_i\in S_n\) is encoded by a colour. Objects in \(\mathcal{DS}_n\) are finite ordered sequences of such colours and they encode the Bott–Samelson bimodules in the form ([000N]) (with \(j=0\) since we consider \(\overline{\mathrm{BSbim}}^{\mathrm{gr}}_n\)). For instance if \(n=2\) and we encode \(s_1\) as red and \(s_2\) as blue, then the Bott–Samelson bimodule \(B_{\mathbf{i}}\) from ([000N]) is encoded as a sequence of colors red and blue according to \({\mathbf{i}}\), for instance \({\mathbf{i}}=(1,1,2,1,1)\) corresponds to the object given by the color sequence \((red, red, blue, red, red)\). Morphisms in \(\mathcal{DS}_n\) are \(k\)-linear combinations of isotopy classes of certain decorated graphs embedded in the plane. A morphism from \(B_{\mathbf{i}}\) to \(B_{\mathbf{i}'}\) will have the colour sequence for \({\mathbf{i}}\) as bottom boundary and that for \({\mathbf{i}}'\) at the top boundary; for instance the first two diagrams in ([0023]) represent morphism from \((2,2)\) to \((2)\) and vice versa, the third goes from the unit to \((2)\), etc.

The monoidal structure is given on objects by concatenating sequences and, on morphisms, by (the bilinear extension of) placing diagrams horizontally next to each other. The composition of morphisms is, likewise, given by (the bilinear extension of) stacking diagrams on top of each other. The empty sequence is the unit object.

Apart from multiplication with polynomials, the generating morphisms (in the monoidal sense) are exactly the following, where blue represents any color/number which is neighbored to red and not neighbored to orange.

For the rest of this section we identify \(\mathrm{BSbim}_n\) with \(\mathcal{DS}_n\) as monoidal \(1\)-categories (via \(\overline{\mathrm{BSbim}}^{\mathrm{gr}}_n\)) and perform computations using the diagrammatic calculus.

The Rouquier complexes ([000Z]) are translated into the diagrammatics as where we encode \(s_i\) by blue. (Instead of remembering the grading shifts from ([000Z]), it is more convenient in the diagrammatic setting to consider the maps as homogeneous of degree one). To keep track of the monoidal unit (corresponding to \(R\)) appearing in the complexes, we mostly indicate them by a colored dot as shown in ([0024]).

Observe that the relevant chain complexes and chain maps are for the two cases are related by swapping the colours red and blue.

Now that we have obtained the atomic slide chain maps in lemma 2.5.5, we can construct all remaining slide chain maps in an essentially formal way.

2.6 Centralizers and prebraidings[002C]

In the following, given a monoidal \(1\)-category \(\mathcal C\), we will identify \(1\boxtimes x\) and \(x\boxtimes 1\) with \(x\) for any object \(x\in\mathcal C\), see [Eti+15, Rk. 2.2.9], and will suppress writing associators, as they can be recovered from context.

The category \(Z(F)\) is monoidal with \((b,\gamma)\boxtimes (b,\gamma')\coloneqq (b\boxtimes b',\gamma\boxtimes\mathrm{id}\circ\mathrm{id}\boxtimes\gamma')\) on objects, and with the tensor product from \(\mathcal B\) on morphisms. The unit object \(1_{Z(F)}\in Z(F)\) is \((1_{\mathcal B}\in\mathcal B, \gamma)\) with \(\gamma_a\colon 1\boxtimes F(a)=F(a)=F(a)\boxtimes 1\). We leave the coherence isomorphisms and their compatibility to the reader.

Together with the unit \(1_{Z(F)} \in Z(F)\) and \(F \colon \mathcal A\rightarrow\mathcal B\), the evaluation functor fits into the following commuting diagram of monoidal functors:

In example 7.4.4, we will discuss the universal property satisfied by \(Z(F)\) with its monoidal evaluation functor.

The functor \(\mathrm{ev}\) is completely determined by the monoidal functor \[\mathrm{ev}_{1_{\mathcal A}} \coloneqq \mathrm{ev}(-, 1_{\mathcal A}) \colon Z(F) \rightarrow\mathcal B,\] which sends an object \((b, \gamma)\in Z(F)\) to the underlying object \(b\) and a morphism in \(Z(F)\) to the underlying morphism in \(\mathcal B\). In these terms, \(\mathrm{ev}(-, ?) = \mathrm{ev}_{1_{\mathcal A}}(-) \boxtimes F(?)\).

We obtain a classification of prebraidings on a monoidal functor \(F\), Definitions 2.4.1 and 2.4.8, in terms of the centralizer \(Z(F)\) of \(F\):

3 Stable linear algebra[002M]

This section introduces the \(\infty\)-categorical foundations of our work. Throughout, we adopt standard conventions and notations in higher category theory, which we recall in detail in section A. We follow [Lur09] and use Grothendieck universes — here called small, large, and huge — to deal with set-theoretic issues (see subsection A.6 for more details). In particular, we write \(\mathrm{Cat}_{\infty}\) for the large \(\infty\)-category of small \(\infty\)-categories and \(\mathcal S\) for the (large) \(\infty\)-category of small spaces, both of which are objects of the huge \(\infty\)-category \(\widehat{\mathrm{Cat}}_{\infty}\) of large \(\infty\)-categories.

3.1 Completions of \(\infty\)-categories[002N]

3.1.1 The Yoneda embedding[002P]

Any small \(\infty\)-category \(\mathcal C\) has a Yoneda embedding into its \(\infty\)-category of (\(\mathcal S\)-valued) presheaves \(\mathcal P(\mathcal C)=\mathrm{Fun}(\mathcal C^\mathrm{op}, \mathcal S)\), [Lur09, Prop. 5.1.3.1], see also [Cis19]. It is characterized by the universal property that \(\mathcal P(\mathcal C)\) has all small colimits (i.e. is cocomplete) [Lur09, Cor. 5.1.2.4], and that for any cocomplete \(\infty\)-category \(\mathcal D\) the restriction along the Yoneda embedding induces an equivalence \[ \mathrm{Fun^L}(\mathcal P(\mathcal C),\mathcal D) \rightarrow\mathrm{Fun}(\mathcal C,\mathcal D)\] where \(\mathrm{Fun^L}\) denotes the full subcategory of the \(\infty\)-category of functors on those functors which preserve all small colimits (i.e. the cocontinuous functors) [Lur09, Thm. 5.1.5.6], see also [Cis19, Thm. 6.3.13]. An \(\infty\)-category \(\mathcal C\) is called idempotent complete if its image under the Yoneda embedding \(\mathcal C\rightarrow\mathcal P(\mathcal C)\) is closed under retracts (see [Lur09, Proof of Prop. 5.1.4.2]). We refer to [Lur09, § 4.4.5] for a discussion of retracts and idempotents in \(\infty\)-categories.

3.1.2 Presentable \(\infty\)-categories[002S]

In general, the presheaf category \(\mathcal P(\mathcal C)\) of a small \(\infty\)-category \(\mathcal C\) is a large category. The sense in which \(\mathcal P(\mathcal C)\) is nevertheless still controlled by a small amount of data is formalized by the notion of a presentable \(\infty\)-category. We recall the definition and some basic facts from [Lur09, Sec. 5.4 and 5.5].

A main application of the notion of presentable \(\infty\)-category is the adjoint functor theorem:

For more details, we refer to [Lur09, § 5.5.3] and [Cis19, § 7].

3.1.3 The monoidal structure on \(\mathrm{Pr}^\mathrm{L}\) and presentably symmetric monoidal categories[002X]

The tensor unit of this symmetric monoidal structure is given by the presentable \(\infty\)-category \(\mathcal S\) of spaces. The tensor product \(\mathcal C_1\otimes \mathcal C_2\) of two presentable \(\infty\)-categories \(\mathcal C_1, \mathcal C_2\) comes equipped with a functor \(\mathcal C_1 \times \mathcal C_2 \rightarrow\mathcal C_1\otimes \mathcal C_2\) which preserves small colimits separately in both variables and is characterized by the universal property that for any presentable \(\infty\)-category \(\mathcal D\), the induced functor \[\mathrm{Fun^L}(\mathcal C_1\otimes \mathcal C_2, \mathcal D) \rightarrow\mathrm{Fun^{L \times L}}(\mathcal C_1 \times \mathcal C_2, \mathcal D)\] is an equivalence. Here, \(\mathrm{\mathrm{Fun}^{L \times L}}(\mathcal C_1 \times \mathcal C_2,\mathcal D)\) denotes the full subcategory of \(\mathrm{Fun}(\mathcal C_1\times \mathcal C_2, \mathcal D)\) on those functors which preserve small colimits separately in both variables. Abusing notation, given objects \(c_1\in \mathcal C_1\) and \(c_2\in \mathcal C_2\) we denote the image of \((c_1, c_2) \in \mathcal C_1 \times \mathcal C_2\) under the functor \(\mathcal C_1\times \mathcal C_2 \rightarrow\mathcal C_1\otimes \mathcal C_2\) by \(c_1\boxtimes c_2\in \mathcal C_1 \otimes \mathcal C_2\) and call it their external tensor product.

It follows from [Lur17, Prop. 4.8.1.17, Prop. 4.8.1.16] after taking adjoints, that the tensor product of presentable \(\infty\)-categories can be expressed as the following functor category (which is in particular presentable): \[ \mathcal C\otimes \mathcal D\simeq \mathrm{Fun^L}(\mathcal D, \mathcal C^\mathrm{op})^\mathrm{op}= \mathrm{Fun^R}(\mathcal C^\mathrm{op}, \mathcal D)\]

The symmetric monoidal structure on \(\mathrm{Pr}^\mathrm{L}\) allow us to study (commutative) algebras therein, see subsection A.8.4.

More explicitly, a presentably symmetric monoidal \(\infty\)-category is a symmetric monoidal \(\infty\)-category whose underlying \(\infty\)-category \(\mathcal C\) is presentable and so that the tensor product functor \(-\otimes-\colon \mathcal C\times \mathcal C\rightarrow\mathcal C\) preserves small colimits separately in both variables.

If \(A\) is a commutative algebra object in a symmetric monoidal \(\infty\)-category \(\mathcal C\), consider the \(\infty\)-category \(\mathrm{Mod}_A(\mathcal C)\) of left \(A\)-modules in \(\mathcal C\) (see subsection A.9).

Since \(\mathcal S\) is the tensor unit of the symmetric monoidal structure on \(\mathrm{Pr}^\mathrm{L}\), it is initial in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) and any \(\mathcal C\in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) comes equipped with a unique symmetric monoidal left adjoint functor \(\iota_{\mathcal C} \colon \mathcal S\rightarrow\mathcal C\).

3.1.4 Adjoining colimits[0039]

The \(\infty\)-categories \(\mathrm{Cat}_{\infty}^\mathrm{idem}, \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) and \(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) from notation 3.1.1 are instances of \(\mathrm{Cat}_{\infty}^\mathcal K\) for \(\mathcal K\) consisting of the ‘walking idempotent’ of [Lur09, § 4.4.5], or the walking idempotent together with the set of finite (discrete) sets or the set of finite simplicial sets, respectively.

Just like the presheaf \(\infty\)-category \(\mathcal P(\mathcal C)\) is the free completion of a small \(\infty\)-category \(\mathcal C\) under small colimits, we may complete under other classes of colimits. The following combines [Lur17, Lem. 4.8.4.2, Rem. 4.8.1.8] and [Lur09, Cor. 5.3.6.10]:

The second statement of proposition 3.1.11 implies that \(\mathcal C\hookrightarrow \mathcal P_{\mathcal K}^{\mathcal K'}(\mathcal C)\) may be thought of as a generalized Yoneda embedding: It is the free cocompletion of \(\mathcal C\) under \(\mathcal K'\)-shaped colimits subject to the relation that \(\mathcal K\)-shaped colimits in \(\mathcal C\) are preserved.

3.2 Compact generation and ind-completion[003E]

3.2.1 Compact and projectively generated \(\infty\)-categories[003F]

Many presentable \(\infty\)-categories are generated by \(\omega\)-compact objects, where \(\omega\) is the cardinality of the natural numbers. We will henceforth refer to \(\omega\)-filtered diagrams simply as filtered diagrams and to \(\omega\)-compact objects as compact objects. Hence, an object \(c\) of an \(\infty\)-category \(\mathcal C\) with filtered colimits is compact if \(\mathrm{Hom}_{\mathcal C}(c,-)\colon \mathcal C\rightarrow\mathcal S\) preserves filtered colimits.

Similarly, we recall the following definitions:

We refer to definition 3.6.1 and example 3.6.3 for a comparison with classical notions of compactness and projectivity in ordinary (abelian) categories.

In proposition 3.2.8 we will show that these functors are in fact equivalences.

3.2.2 Ind-completion[003U]

The ind-completion \(\operatorname{Ind}(\mathcal C)\) of a small \(\infty\)-category \(\mathcal C\) is defined to be the smallest full subcategory of \(\mathcal P(\mathcal C)\) which contains the image of the Yoneda embedding and is closed under filtered colimits, [Lur09, Rem. 5.3.5.2, Prop. 5.3.5.3]. Then, \(\operatorname{Ind}(\mathcal C)\) has filtered colimits and the inclusion \(\mathcal C\rightarrow\operatorname{Ind}(\mathcal C)\) is characterized by the universal property that for any \(\infty\)-category \(\mathcal D\) with filtered colimits, it induces an equivalence \[ \mathrm{Fun}^{\omega}(\operatorname{Ind}(\mathcal C),\mathcal D) \rightarrow\mathrm{Fun}(\mathcal C,\mathcal D),\] where \(\mathrm{Fun}^{\omega}\) denotes the full subcategory of functors which preserve filtered colimits.

If \(\mathcal C\) moreover has finite colimits, then \(\operatorname{Ind}(\mathcal C)\) is equivalent to the full subcategory of \(\mathcal P(\mathcal C)\) on those functors \(\mathcal C^{\mathrm{op}} \rightarrow\mathcal S\) which send finite colimits in \(\mathcal C\) to finite limits of spaces by [Lur09, Cor. 5.3.5.4]. In this case, \(\operatorname{Ind}(\mathcal C)\) is presentable, the inclusion \(\mathcal C\rightarrow\operatorname{Ind}(\mathcal C)\) preserves finite colimits, and for any presentable \(\infty\)-category \(\mathcal D\), the induced functor \[\mathrm{Fun^L}(\operatorname{Ind}(\mathcal C),\mathcal D) \rightarrow\mathrm{Fun}^{\mathrm{rex}}(\mathcal C, \mathcal D)\] is an equivalence by [Lur09, Cor. 5.3.5.10], where \(\mathrm{Fun}^{\mathrm{rex}}\) denotes the full subcategory of functors which preserve finite colimits. For more details see [Lur09, Section 5.3].

Similarly, the \(\mathcal P^{\Sigma}\)-completion \(\mathcal P^{\Sigma}(\mathcal C)\) of a small \(\infty\)-category \(\mathcal C\) is the smallest full subcategory of \(\mathcal P(\mathcal C)\) that contains the image of the Yoneda embedding and is closed under sifted colimits. The \(\infty\)-category \(\mathcal P^{\Sigma}(\mathcal C)\) has sifted colimits and the inclusion \(\mathcal C \rightarrow\mathcal P^{\Sigma}(\mathcal C)\) is characterized by the universal property that for any \(\infty\)-category \(\mathcal D\) with sifted colimits, it induces an equivalence \[\mathrm{Fun}^{\Sigma}(\mathcal P^{\Sigma}(\mathcal C),\mathcal D) \rightarrow\mathrm{Fun}(\mathcal C,\mathcal D),\] where \(\mathrm{Fun}^{\Sigma}\) denotes the full subcategory of functors which preserve sifted colimits, [Lur09, Prop. 5.5.8.15].

If \(\mathcal C\) moreover has finite coproducts, then \(\mathcal P^{\Sigma}(\mathcal C)\) is equivalent to the full subcategory of \(\mathcal P(\mathcal C)\) on those functors \(\mathcal C^{\mathrm{op}} \rightarrow\mathcal S\) which send finite coproducts in \(\mathcal C\) to finite products of spaces [Lur09, Def. 5.5.8.8 and Rem. 5.5.8.16.(1)]. In this case, \(\mathcal P^{\Sigma}(\mathcal C)\) is presentable, the inclusion \(\mathcal C\rightarrow\mathcal P^{\Sigma}(\mathcal C)\) preserves finite coproducts and for any presentable \(\infty\)-category \(\mathcal D\), the induced functor \[\mathrm{Fun^L}(\mathcal P^{\Sigma}(\mathcal C), \mathcal D) \rightarrow\mathrm{Fun}^{\sqcup}(\mathcal C, \mathcal D)\] is an equivalence, where \(\mathrm{Fun}^{\sqcup}\) denotes the full subcategory of functors which preserve finite coproducts [Lur09, Rem. 5.5.8.16(iii)], see  [Lur09, § 5.5.8] for details.

Given a presentable \(\infty\)-category projectively/compactly generated by a small set \(S\) of objects, then the objects in this set also generate the full subcategories \(\mathcal C^{\mathrm{cp}}\), or \(\mathcal C^\mathrm{c}\), respectively:

Recall that any presentably symmetric monoidal \(\infty\)-category \(\mathcal C\) comes equipped with a unique symmetric monoidal left adjoint functor \(\iota_{\mathcal C}\colon \mathcal S\rightarrow\mathcal C\). We now show that the unique symmetric monoidal left adjoint \(\iota_{\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}} \colon \mathcal S\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) sends a space \(X\) to the presheaf category \(\mathcal P(X)\).

It follows from lemma 3.2.9 that for \(\mathcal C, \mathcal D\in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) (resp. in \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\)), the compact (resp. compact-projective) objects of the tensor product \(\mathcal C\otimes \mathcal D\) are retracts of iterated finite colimits (retracts of finite coproducts) of external tensor products of compact (resp. compact-projective) objects.

Explicitly, a commutative algebra \(\mathcal C\in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) (and analogously for \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\)) therefore amounts to a presentably symmetric monoidal \(\infty\)-category \(\mathcal C\), whose underlying presentable \(\infty\)-category is compactly generated, so that its unit is compact, and if \(c, d\) are compact objects in \(\mathcal C\), then \(c\otimes d\) is also compact.

3.3 Additive and stable \(\infty\)-categories[004L]

Here, we briefly review the theory of additive and stable \(\infty\)-categories. For more details, we refer to [Lur17],  [BFN10] and [GGN15].

3.3.1 Definitions[004M]

An \(\infty\)-category is called zero-pointed if it has an initial and a terminal object and if the unique morphism from the initial to the terminal object is an isomorphism. In this case, we call the initial/terminal object a zero object. A zero-pointed \(\infty\)-category is called semi-additive if it furthermore has finite products and finite coproducts and if the canonical morphism \(x\sqcup y \rightarrow x \times y\) is an isomorphism. In this case, we write the product/coproduct as \(x\oplus y\) and refer to it as a direct sum. A semi-additive \(\infty\)-category is called additive if furthermore the shear map \(( \pi_1, \nabla)\colon x\oplus x \rightarrow x \oplus x\) is an isomorphism, where \(\pi_1\colon x \oplus x \rightarrow x\) denotes the projection to the first factor (using that \(x \oplus x\) is a product) and \(\nabla\colon x \oplus x \rightarrow x\) is the fold map (using that \(x \oplus x\) is a coproduct). A functor between additive \(\infty\)-categories is called additive if it preserves finite coproducts. We denote the \(\infty\)-category of additive functors between two additive \(\infty\)-categories \(\mathcal A, \mathcal B\) by \(\mathrm{Fun}^{\sqcup}(\mathcal A,\mathcal B)\). This \(\infty\)-category is itself an additive \(\infty\)-category [GGN15, Cor. 2.9]. The notion of an additive \(\infty\)-category is a direct generalization of the ordinary \(1\)-categorical notion, and indeed an ordinary \(1\)-category is additive in the usual sense if and only if it(s nerve) is additive in the \(\infty\)-categorical sense. Conversely, a semi-additive \(\infty\)-category \(\mathcal C\) is additive if and only if its homotopy category \(h_1\mathcal C\) is additive as an ordinary \(1\)-category [GGN15, Prop. 2.8].

A zero-pointed \(\infty\)-category is called stable if it admits finite colimits and if any square is a pullback square if and only if it is a pushout square. A functor between stable \(\infty\)-categories is called exact if it preserves finite colimits. Given two stable \(\infty\)-categories \(\mathcal C, \mathcal D\), we denote the \(\infty\)-category of exact functors between them by \(\mathrm{Fun}^{\mathrm{ex}}(\mathcal C,\mathcal D)\). This \(\infty\)-category \(\mathrm{Fun}^{\mathrm{ex}}(\mathcal C,\mathcal D)\) is itself stable since it is a full subcategory of \(\mathrm{Fun}(\mathcal C, \mathcal D)\) (which is stable by [Lur17, Prop. 1.1.3.1]) that contains the zero object and is stable under forming fibers and cofibers, as a straightforward computation shows. In what follows, we will only consider idempotent complete stable categories.

Since stable \(\infty\)-categories are additive and exact functors preserve finite coproducts, there is a forgetful functor \(\mathrm{st}\rightarrow\mathrm{add}\).

As in subsection 3.1 (in particular proposition 3.2.8), it will be useful to characterize small additive or stable \(\infty\)-categories in terms of projectively resp. compactly generated presentable \(\infty\)-categories.

3.3.2 Symmetric monoidal structure[004V]

The universal example of an stable presentable \(\infty\)-category is the \(\infty\)-category \(\mathrm{Sp}\) of spectra. Likewise, the universal example of an additive presentable \(\infty\)-category is the \(\infty\)-category \(\mathrm{Sp}_{\geq 0}\) of connective spectra, equivalent to the \(\infty\)-category \(\mathrm{Grp}_{\mathbb{E}_{\infty}}(\mathcal S)\) of grouplike \(\mathbb{E}_{\infty}\)-spaces, see [GGN15]. Both \(\mathrm{Sp}\) and \(\mathrm{Sp}_{\geq 0}\) are idempotent algebras in \(\mathrm{Pr}^\mathrm{L}\), i.e. commutative algebras \(A\in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) so that the multiplication \(A\otimes A \rightarrow A\) is an isomorphism. It is shown in [Lur17, Prop. 4.8.2.18] and [GGN15, Cor. 4.8] that the full subcategories \(\mathrm{Mod}_{\mathrm{Sp}}(\mathrm{Pr}^\mathrm{L})\) and \(\mathrm{Mod}_{\mathrm{Sp}_{\geq 0}}(\mathrm{Pr}^\mathrm{L})\) of \(\mathrm{Pr}^\mathrm{L}\) are equivalent to \({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}\) and \({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}}\), respectively. As categories of modules of a commutative algebra, this induces symmetric monoidal structures on \({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}\) and \({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}}\) respectively by proposition 3.1.8.([0032]).

By  [Lur17, Prop. 1.4.3.7], the \(\infty\)-category \(\mathrm{Sp}\) is compactly generated (by the single object \(\mathbb{S}\), the sphere spectrum). It follows from lemma 3.2.9 that the compact objects in \(\mathrm{Sp}\) are finite spectra, i.e. finite colimits of the sphere spectrum (note that a retract of a finite spectrum is again finite). However, \(\mathrm{Sp}\) is not projectively generated (its only projective object is the zero spectrum, cf. [Lur17, Rem. 7.2.2.5]). On the other hand, \(\mathrm{Sp}_{\geq 0}\) is projectively generated by the sphere spectrum [Lur17, Cor. 7.1.4.13]. It therefore follows from lemma 3.2.9 that the compact-projective objects in \(\mathrm{Sp}_{\geq 0}\) are finite sums of the sphere spectrum (note that a retract of a finite sum of sphere spectra is again a finite sum of sphere spectra).

Using the theory of commutative algebras in presentable categories, we immediately obtain the following stable and additive analogues of the first half of proposition 3.2.10 concerning symmetric monoidal structures on subcategories of \(\mathrm{Pr}^\mathrm{L}\). The passage from \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\simeq \mathrm{add}\) to \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}} \simeq \mathrm{st}\) will be treated in the next section.

Tracing through the proof, the symmetric monoidal structures on \(\mathrm{add}\) respectively \(\mathrm{st}\) may be characterized as follows (c.f. [BFN10, Prop. 4.4]): For \(\mathcal C, \mathcal D\in \mathrm{add}\) the tensor product \(\mathcal C\otimes \mathcal D\) is equipped with a functor \(\mathcal C\times \mathcal D\rightarrow\mathcal C\otimes \mathcal D\), additive in both variables, and satisfies the universal property that for any \(\mathcal E\in \mathrm{add}\) the induced functor \[\mathrm{Fun}^{\mathrm{add}}(\mathcal C\otimes \mathcal D, \mathcal E) \rightarrow\mathrm{Fun}^{\mathrm{add}\times\mathrm{add}}(\mathcal C\times \mathcal D, \mathcal E)\] is an equivalence, where \(\mathrm{Fun}^{\mathrm{add}\times\mathrm{add}}(\mathcal C\times \mathcal D, \mathcal E)\) denotes the full subcategory of \(\mathrm{Fun}(\mathcal C\times \mathcal D, \mathcal E)\) on the functors which are additive in both variables (i.e. which preserve finite coproducts separately in either variable).

For \(\mathcal C, \mathcal D\in \mathrm{st}\), the tensor product \(\mathcal C\otimes \mathcal D\) is characterized analogously in terms of functors \(\mathcal C\times \mathcal D\rightarrow\mathcal E\) which are exact in both variables (i.e. which preserve finite colimits separately in both variables).

3.4 From additive to stable \(\infty\)-categories[0051]

Given an ordinary additive \(1\)-category \(\mathcal A\), one may form a stable \(\infty\)-category \({\mathbf K}^b(\mathcal A)\) of bounded (in both directions) chain complexes, chain homomorphisms, and (higher) chain homotopies between these. In this section, we review this construction and prove that it satisfies a universal property: the stable \(\infty\)-category \({\mathbf K}^b(\mathcal A)\) is the free stable \(\infty\)-category on the additive category \(\mathcal A\).

3.4.1 The \(\infty\)-category of chain complexes[0052]

Given an ordinary additive \(1\)-category \(\mathcal A\), the \(\infty\)-category \({\mathbf K}^b(\mathcal A)\) can be defined, see [Lur17, § 1.3.1], using the technology of dg nerves as follows.

Viewing \(\mathcal A\) as an additive \(\infty\)-category, there is a canonical additive functor \(\mathcal A\rightarrow{\mathbf K}^b(\mathcal A)\) induced from the functor that interprets objects of \(\mathcal A\) as chain complexes concentrated in degree zero.

In corollary 3.4.10, we will prove that \({\mathbf K}^b(\mathcal A)\) is the universal stable \(\infty\)-category associated to \(\mathcal A\). To do so, we express \({\mathbf K}^b\) in terms of the functors constructed in the previous sections.

3.4.2 The free stable category on an additive category[0057]

Recall that \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{c}}\) is a symmetric monoidal subcategory, and hence that \(\mathrm{Sp}_{\geq 0}\in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\) may also be considered an algebra in \(\mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\). Notice also that the full subcategory inclusion \(\mathrm{Sp}_{\geq 0}\rightarrow\mathrm{Sp}\) is symmetric monoidal, has a right adjoint (namely the \(0\)-th connective cover functor \(\tau_{\geq 0}\)) and sends compact objects in \(\mathrm{Sp}_{\geq 0}\) to compact objects in \(\mathrm{Sp}\), as the sphere spectrum compactly generates \(\mathrm{Sp}_{\geq 0}\) and \(\mathrm{Sp}\).

As \(\mathrm{Sp}_{\geq 0}\) is an idempotent algebra in \(\mathrm{Pr}^\mathrm{L}\), the second functor \(-\otimes_{\mathrm{Sp}_{\geq 0}} \mathrm{Sp}\) here is equivalent to the composite \[ \mathrm{Mod}_{\mathrm{Sp}_{\geq 0}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}) \xrightarrow{\mathrm{forget}} \mathrm{Pr}^{\mathrm{L},\mathrm{c}}\xrightarrow{-\otimes \mathrm{Sp}} \mathrm{Mod}_{\mathrm{Sp}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}).\]

Recall now the symmetric monoidal left adjoint functor from construction 3.4.4 and the equivalences \({\mathcal P^{\Sigma}}\) and \((-)^{\mathrm{c}}\) from proposition 3.3.4. Then the following holds.

3.4.3 \({\mathbf K}^b\) as a left adjoint[005E]

We now show that for an ordinary additive idempotent-complete \(1\)-category \(\mathcal A\), the universal stable \(\infty\)-category \(\mathcal A^{\mathrm{fin}}\) from proposition 3.4.5 is equivalent to \({\mathbf K}^b(\mathcal A)\).

Consider the symmetric monoidal left adjoint \((-)^{\mathrm{fin}} \colon \mathrm{add}\rightarrow\mathrm{st}\) of the forgetful functor from proposition 3.4.5. Categories in the image of \((-)^{\mathrm{fin}}\) carry so-called weight structures, which were originally introduced independently by Bondarko in [Bon10] and (under the name of co-t-structures) by Pauksztello in [Pau08], and afterwards adopted to the \(\infty\)-categorical setting by Elmanto and Sosnilo [ES22], whose exposition we closely follow.

By [ES22, Def. 2.2.1] a weight structure on an idempotent complete stable \(\infty\)-category \(\mathcal D\) is a pair \((\mathcal D_{\leq 0}, \mathcal D_{\geq 0})\) of two full idempotent complete subcategories fulfilling the following conditions:

1. \(\Sigma \mathcal D_{\geq 0} \subset \mathcal D_{\geq 0}, \Sigma^{-1} \mathcal D_{\leq 0} \subset \mathcal D_{\leq 0}\). We write \(\mathcal D_{\geq n} = \Sigma^n \mathcal D_{\geq 0}\), \(\mathcal D_{\leq n} = \Sigma^n \mathcal D_{\leq 0}\).

2. For \(x \in \mathcal D_{\leq 0}\) and \(y \in \mathcal D_{\geq 1}\), we have \(\pi_0(\mathrm{Hom}_{\mathcal D}(x,y)) \simeq 0.\)

3. For any object \(x\), there is a fiber sequence \(x_{\leq 0} \rightarrow x \rightarrow x_{\geq 1}\) with \(x_{\leq 0} \in \mathcal D_{\leq 0}, x_{\geq 1} \in \mathcal D_{\geq 1}\).

Note that condition (3) merely requires the existence of such a fiber sequence, neither is it unique nor functorially associated to \(x\). A weight structure is called bounded if \(\mathcal D= \bigcup_{n}(\mathcal D_{\geq -n} \cap \mathcal D_{\leq n})\). For any weight structure, the weight heart \(\mathcal D^{\heartsuit} \coloneqq \mathcal D_{\geq 0} \cap \mathcal D_{\leq 0}\) is additive and idempotent complete.

Just like t-structures, weight structures only depend on and may be constructed in terms of the underlying (triangulated) homotopy category of \(\mathcal D\).

A functor \(F\colon\mathcal C\rightarrow\mathcal D\) between stable, idempotent-complete \(\infty\)-categories with weight structure is weight exact if it is exact and the restriction of \(F\) to the full subcategory \(\mathcal C_{\geq 0} \subseteq \mathcal C\) factors through the full subcategory \(\mathcal D_{\geq 0} \subseteq \mathcal D\) and the restriction of \(F\) to \(\mathcal C_{\leq 0} \subseteq \mathcal C\) factors through \(\mathcal D_{\leq 0} \subseteq \mathcal D\). Weight exact functors \(F \colon \mathcal C\rightarrow\mathcal D\) restrict to additive functors \(F^{\heartsuit}\colon \mathcal C^{\heartsuit} \rightarrow\mathcal D^{\heartsuit}\) between the weight hearts.

A key result of [ES22] is that \((-)^{\mathrm{fin}}\colon\mathrm{add}\rightarrow\mathrm{st}\) factors as an equivalence \(\mathrm{add}\rightarrow\mathrm{st}^{bw}\) followed by the functor \(\mathrm{st}^{bw} \rightarrow\mathrm{st}\) which forgets the weight structure, see [ES22, Const. 2.2.7]. An inverse of this equivalence \(\mathrm{add}\rightarrow \mathrm{st}^{bw}\) is given by the functor \((-)^{\heartsuit}\colon \mathrm{st}^{bw} \rightarrow\mathrm{add}\) taking the weight heart, see [ES22, Theorem 2.2.9]. The following corollary is a consequence of this theorem:

Extending corollary 3.4.10, we think of \((-)^{\mathrm{fin}}:\mathrm{add}\rightarrow\mathrm{st}\) as the correct generalization of \({\mathbf K}^b(\mathcal A)\) from ordinary additive, idempotent-complete \(1\)-categories to additive, idempotent-complete \(\infty\)-categories \(\mathcal A\).

3.5 \(\infty\)-categories of graded modules[005L]

The categories appearing in this paper will not just be additive or stable, but will typically be enriched in chain complexes of \(k\)-modules for a commutative ring \(k\), equipped with an additional \(\mathbb{Z}\)-grading. In this section, we recall the necessary technical machinery to address this coherently. This machinery will apply more generally to \(\mathbb E_{\infty}\)-ring spectra, i.e. commutative algebra objects in \(\mathrm{Sp}\). In §3.6, we relate these structures with possibly more familiar variants of derived categories. Similar definitions are discussed in [Lur18].

3.5.1 \(\mathbb{K}\)-modules[005M]

We now discuss the \(\infty\)-categorical analog of \(\mathrm{mod}_k\). It follows from lemma 3.2.12.([004J]) that for an \(\mathbb E_{\infty}\)-ring spectrum \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), the \(\infty\)-category \(\mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp})\) is a compactly generated stable, presentably symmetric monoidal category, i.e. \(\mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp}) \in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}})\).

The symmetric monoidal equivalence \(\operatorname{Ind}\colon \mathrm{st}\rightarrow\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\) transports \(\mathrm{Perf}_{\mathbb{K}}\) to \(\mathrm{Mod}_{\mathbb{K}}\) and vice versa.

Since example 3.5.3 is the situation relevant to our paper, the reader can safely view \(\mathbb{K}\) as a classical ring \(k\) and \(\mathrm{Mod}_{\mathbb{K}}\) as \(\mathcal D(\mathrm{mod}_k)\). The situation of example 3.5.3 will be discussed in more detail in §3.6.

3.5.2 Compact-projective \(\mathbb{K}\)-modules[005R]

As above, we will be concerned with the additive variants of the notions in §3.5.1. Let \(\mathbb{K}\) be a connective \(\mathbb E_{\infty}\)-ring spectrum, i.e. a commutative algebra \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\). Since \(\mathrm{Sp}_{\geq 0}\in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}})\), it follows from lemma 3.2.12 that the category \(\mathrm{Mod}_{\mathbb{K}}(\mathrm{Sp}_{\geq 0})\in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}})\).

Note that \(\mathrm{Mod}^{\geq 0}_{\mathbb{K}}\) is a full subcategory of \(\mathrm{Mod}_{\mathbb{K}}\).

If \(k\) is an ordinary ring, then an object of \(\mathrm{Perf}_{Hk}\) can be represented by a bounded chain complex of finitely generated projective \(k\)-modules. This generalizes to any \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\):

3.5.3 \(\mathcal Z\)-graded \(\mathbb{K}\)-modules[0061]

Given an ordinary monoid \(Z\) and a commutative ring \(k\), the category \(\mathrm{Fun}(Z, \mathrm{mod}_k)\) of \(Z\)-graded \(k\)-modules admits a convolution monoidal structure, for which the tensor product of \(Z\)-graded modules \((M_z)_{z \in Z}\) and \((N_z)_{z\in Z}\) is given by the \(Z\)-graded module which in degree \(z\in Z\) is \(\oplus_{z_1z_2 = z} M_{z_1} \otimes N_{z_2}\). This construction is a special case of the Day convolution monoidal structure on a functor category [Lur17, § 2.2.6]. Here, we focus on the symmetric monoidal case.

We briefly recall this construction of a symmetric monoidal structure on \(\mathrm{Fun}(J, \mathcal C)\) in the case where \(J\) is a small symmetric monoidal \(\infty\)-category and \(\mathcal C\) is a presentably symmetric monoidal \(\infty\)-category.

Assume \(J\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\) and \(\mathcal C\in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\). Since \(\mathcal P\colon \mathrm{Cat}_{\infty}\rightarrow\mathrm{Pr}^\mathrm{L}\) is symmetric monoidal by proposition 3.1.6, it follows that for \(J \in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\), the \(\infty\)-category \(\mathcal P(J)\) inherits a presentably symmetric monoidal structure, i.e. \(\mathcal P(J) \in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\). Then ([0064]) provides the following Day convolution monoidal structure on \(\mathrm{Fun}(J^{\mathrm{op}}, \mathcal C)\).

We will particularly focus on gradings by a homotopy coherent abelian monoid, i.e. a \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\).

3.6 Derived \(\infty\)-categories of graded modules[006F]

Many of the constructions of section 2 center around discrete (i.e. ordinary) graded \(k\)-algebras, and derived graded bimodules between them. In this section, we therefore focus on the case where \(\mathbb{K}\) is a discrete commutative ring \(k\) and \(\mathcal Z\) is a discrete commutative monoid \(Z\) and unpack our constructions in terms of homological algebra, generalizing Examples 3.5.3, 3.5.5 and 3.5.15.

3.6.1 Derived \(\infty\)-categories[006G]

We quickly review the basics of the theory of derived \(\infty\)-categories; we refer the reader to  [Lur17, § 1.3] for more details.

Given an abelian \(1\)-category \(\mathcal A\), its (unbounded) derived \(\infty\)-category \(\mathcal D(\mathcal A)\) is the \(\infty\)-categorical localization of the \(\infty\)-category of unbounded chain complexes in \(\mathcal A\) (constructed as the dg nerve [Lur17, § 1.3.1] of the corresponding differential graded category) at the quasi-isomorphisms. In particular, the homotopy 1-category \(h_1\mathcal D(\mathcal A)\) agrees with the ordinary derived 1-category of \(\mathcal A\) in the usual sense.

Let \(\mathcal D(\mathcal A)_{\geq 0}\) denote the full subcategory of \(\mathcal D(\mathcal A)\) on the chain complexes with vanishing homology in negative degrees. When \(\mathcal A\) is particularly well-behaved, the \(\infty\)-categories \(\mathcal D(\mathcal A)_{\geq 0}\) and \(\mathcal D(\mathcal A)\) can be expressed in terms of completions (of the type introduced throughout section 3), as we discuss now.

Recall the following classical analogues of definition 3.2.1:

The following statements are well-known and can be gathered from various parts of [Lur17, § 1.3]:

3.6.2 Derived \(\infty\)-categories of graded modules[006R]

We return to the main goal of this subsection to give a homological perspective on the constructions of the last sections. Let \(\mathbb{K}\) be a discrete commutative ring \(k\) and \(\mathcal Z\) a discrete commutative monoid \(Z\). Recall the notation \(\mathrm{mod}_{k}^Z\) for the ordinary category of \(Z\)-graded \(k\)-modules. Throughout this subsection, we also fix an ordinary (not necessarily commutative) \(Z\)-graded \(k\)-algebra \(A \in \mathrm{Alg}(\mathrm{mod}_{k}^Z)\).

This category \(\mathrm{grmod}_A\) is a \(1\)-projectively generated, in the sense of definition 3.6.1, presentable abelian \(1\)-category. A standard computation shows that its compact \(1\)-projective objects (i.e. its compact projective objects in the usual abelian sense) are precisely given by the graded-compact projective modules, defined as follows.

In the notation of definition 3.6.1, \(\mathrm{grmod}_A^{\mathrm{gr-cp}} = \left(\mathrm{grmod}_A\right)^{\mathrm{c}1\mathrm{p}}\).

Using proposition 3.6.6, we can identify the \(\infty\)-category \(\mathrm{RMod}_{HA}(\mathrm{Mod}_{Hk}^{\geq 0, Z})\) as well as its various subcategories in terms of homological algebra:

Motivated by proposition 3.6.9, we call the objects in the full subcategory \(\mathcal D(\mathrm{grmod}_A)^{\mathrm{c}} \subseteq \mathcal D(\mathrm{grmod}_A)\) graded-perfect.

4 Graded-linear \(\infty\)-categories and Morita theory[0072]

The goal of this section is twofold: in the first half of this section we introduce the relevant notions of graded and linear \(\infty\)-categories, and prove an \(\infty\)-categorical version of the familiar equivalence between categories enriched in graded modules and categories with an action. We have seen a concrete \(1\)-categorical instance already in form of the categories \(\overline{\mathrm{BSbim}}^{\mathrm{gr}}_n\) and \(\mathrm{BSbim}_n\) in section 2. In the second half of this section, we introduce the Morita categories relevant for the construction of our monoidal \((2,2)\)-category \(\mathrm{Sbim}\).

4.1 Presentably enriched \(\infty\)-categories[0073]

4.1.1 Closed monoidal \(\infty\)-categories and closed module \(\infty\)-categories[0074]

We generalize example 4.1.4 to module categories using the following terminology.

By [Lur17, Rem. 4.8.4.14], \(\mathrm{Fun}^{\mathcal K}_{\mathbb V}(\mathcal C, \mathcal D)\) is closed under \(\mathcal K\)-colimits, thus an object of \(\mathrm{Cat}_{\infty}^{\mathcal K}\).

4.1.2 Presentably enriched \(\infty\)-categories[007H]

Let \(\mathbb V\in \mathrm{Alg}(\mathrm{Pr}^\mathrm{L})\) and \(\mathcal C\in \mathrm{LMod}_{\mathbb V}(\mathrm{Pr}^\mathrm{L})\), i.e. \(\mathcal C\) is a presentable \(\infty\)-category with an action \(- \otimes -\colon \mathbb V\times \mathcal C\rightarrow\mathcal C\) by a presentable monoidal \(\infty\)-category \(\mathbb V\) which is cocontinuous in both variables. As in example 4.1.3, it follows from the adjoint functor theorem that the action is closed, i.e. for any pair of objects \(x, y \in \mathcal C\), there exists a morphism object \(\underline{\mathrm{Hom}}_{\mathcal C}(x, y ) \in \mathbb V\). It is shown in [GH15, Cor. 7.4.13] that these morphism objects assemble \(\mathcal C\) into a \(\mathbb V\)-enriched \(\infty\)-category with space of objects \(\mathcal C^{\simeq}\), and which we will also denote by \(\mathcal C\). By [Hei23, Thm. 7.21, Thm. 1.2], this construction is functorial and multiplicative in the following sense:

The functor ([007J]) will be our meain tool to construct symmetric monoidal enriched \(\infty\)-categories and symmetric monoidal enriched functors between them. In particular, if \(\mathbb V\in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\), then \(\mathbb V\) itself may be considered as self-enriched, i.e. \(\mathbb V\in \mathrm{CAlg}(\widehat{\mathrm{Cat}}[\mathbb V])\).

4.2 Graded linear \(\infty\)-categories[007M]

To incorporate \(\mathbb{K}\)-linearity for \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\) into the setup, we could define small \(\mathbb{K}\)-linear \(\infty\)-categories as small \(\infty\)-categories enriched in the presentably symmetric monoidal \(\infty\)-category \(\mathrm{Mod}_{\mathbb{K}}\) from notation 3.5.2. Due to proposition 4.1.7 and remark 4.1.9, it is technically easier to work with presentably enriched \(\infty\)-categories instead, as these can be expressed purely in the language of module categories. Our ‘presentable \(\mathbb{K}\)-linear’ terminology is justified by remark 4.2.3.

4.2.1 \(\mathbb{K}\)-linear \(\infty\)-categories[007N]

We start with some definitions which are crucial throughout the paper.

The following justifies the terminology ‘stable/additive presentable \(\mathbb{K}\)-linear’ in definition 4.2.1.

The next remark covers our main case of interest and connects to the framework from section 2.

4.2.2 \(\infty\)-categories enriched in graded modules[007X]

remark 4.1.9 motivates the following terminology:

As \(\infty\)-categories of modules of commutative algebras in presentably symmetric monoidal categories, both, \(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}\) and \(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}\), are presentably symmetric monoidal \(\infty\)-categories.

4.3 From gradings to actions[0081]

Given a ring \(k\), and a \(k\)-linear category \(\mathcal C\) with an action by a discrete monoid \(Z\), then the category \(\mathcal C\) is canonically enriched in the ordinary category \(\mathrm{mod}_k^Z\) of \(Z\)-graded \(k\)-modules. Indeed, given objects \(c, d\in \mathcal C\) we define the \(k\)-module of degree-\(z\) morphisms to be \[\mathrm{Hom}_{\mathcal C}(c,d)_z \coloneqq \mathrm{Hom}_{\mathcal C}(c, d[z])\] where \((-)[z] \colon \mathcal C\rightarrow\mathcal C\) denotes the action of \(z\in Z\) on \(\mathcal C\). Conversely, if \(\mathcal C\) is a category enriched in \(Z\)-graded \(k\)-modules and if moreover for every \(z\in Z\), the inner-hom functor \(\underline{\mathrm{Hom}}_{\mathcal C}(c,-) \colon \mathcal C\rightarrow\mathrm{mod}_k^Z\) is corepresentable (e.g. if \(\mathcal C\) is presentably enriched), then the enrichment arises from a \(Z\)-action on the underlying category. These constructions provide an equivalence between the category of presentable \(k\)-linear categories with an \(Z\)-action and the category of categories presentably enriched in \(Z\)-graded \(k\)-modules. In this section, we generalize these constructions to our \(\infty\)-categorical setting.

We need the following compatibilities of structures with the functor \({\mathbf K}^b\) from proposition 4.2.7.

We are interested in \(\infty\)-categories with an action by a commutative monoid. For \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\), let \(B \mathcal Z\in \mathrm{CAlg}(\mathrm{Cat}_{\infty})\) denote its delooped symmetric monoidal \(\infty\)-category¹⁸. Since \(\mathcal Z\) is commutative, there is a symmetric monoidal equivalence \(B\mathcal Z\simeq B\mathcal Z^{\mathrm{op}}\). Then an object of \(\mathrm{add}_{\mathbb{K}}^{B\mathcal Z} = \mathrm{Fun}(B\mathcal Z, \mathrm{add}_{\mathbb{K}})\) is precisely a small additive, idempotent complete \(\mathbb{K}\)-linear \(\infty\)-category with an action by \(\mathcal Z\) via \(\mathbb{K}\)-linear additive functors (and similarly for \(\mathrm{st}_{\mathbb{K}}^{B\mathcal Z}\)).

The equivalence between gradings and actions derives from the following proposition:

We now prove the main proposition of this subsection: For a homotopy coherent abelian monoid \(\mathcal Z\), the \(\infty\)-category of compactly generated presentably \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\)-enriched \(\infty\)-categories is equivalent to the category \(\mathrm{Fun}(B\mathcal Z, \mathrm{st}_{\mathbb{K}})\), i.e. to the category of \(\mathbb{K}\)-linear stable \(\infty\)-categories with an action by \(\mathcal Z\). This equivalence is symmetric monoidal for the Day convolution structure on \(\mathrm{Fun}(B\mathcal Z, \mathrm{st}_{\mathbb{K}})\).

4.4 \(\infty\)-Morita theory[008D]

For any monoidal \(1\)-category \(\mathbb V\) with reflective coequalizers distributing over the tensor product, one may construct a Morita \(2\)-category whose objects are algebras in \(\mathbb V\), whose \(1\)-morphisms are bimodules and whose \(2\)-morphisms are bimodule maps. If \(\mathbb V\) is moreover presentably symmetric monoidal, and hence self-enriched, then also the categories of bimodules \(_{A}\mathrm{BMod}_B(\mathbb V)\) will inherit a \(\mathbb V\)-enrichment and thus the Morita 2-category inherits a \(\mathbb V\)-enrichment at the level of \(2\)-morphisms. The goal of this section it to establish \(\infty\)-categorical variants of these statements to be used for our homotopy coherent construction of the Soergel \((2,2)\)-category in section 6.

Various \(\infty\)-categorical constructions of \((\infty,2)\)-Morita categories exist in the literature, see e.g.  [Lur17], [Hau17], [JS17] and references therein. Due to their compatibility with enrichment, we follow ideas from [Lur17]. Our starting point is the following:

We can therefore think of the full subcategory of \(\mathrm{Mod}_{\mathbb V}(\mathrm{Pr}^\mathrm{L})\) on those presentable \(\mathbb V\)-module categories which are of the form \(\mathrm{RMod}_A(\mathbb V)\) for algebra objects \(A\) in \(\mathbb V\), as an \(\infty\)-categorical Morita category with objects algebras, morphisms given by bimodules and composition given by relative tensor product (cf. [Lur17, Rem. 4.8.4.9]). In the following, we will be interested in versions of the Morita category where we further restrict our bimodules requiring certain compactness or projectivity properties:

For \(\mathbb V\in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) and \(A\in \mathrm{Alg}(\mathbb V)\), the \(\infty\)-category \(\mathrm{RMod}_{A}(\mathbb V)\) is compactly generated and the \(\mathbb V\)-action preserves compact generators, see lemma 3.2.12.([004G])-([004H]); the functor \(\mathrm{RMod}_{-}(\mathbb V) \colon \mathrm{Alg}(\mathbb V) \rightarrow\mathrm{Mod}_{\mathbb V}(\mathrm{Pr}^\mathrm{L})\) thus factors through \(\mathrm{Mod}_{\mathbb V}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\). The analogous statement holds for \(\mathbb V\in \mathrm{CAlg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\) by lemma 3.2.12.([004C])-([004D]).

We are now ready to define our Morita categories of interest.

We unpack the relevant properties of these Morita categories:

4.5 Morita categories of discrete flat algebras[008U]

For the rest of this section, we focus on the case where \(\mathbb{K}= Hk\) for an ordinary commutative ring \(k\) and where \(\mathcal Z\) is a discrete commutative monoid \(Z\). Our goal of this subsection is to restrict our Morita categories from corollary 4.4.5 to certain symmetric monoidal full subcategories which only contain discrete (i.e. ordinary) \(k\)-algebras and whose hom-categories are given by ordinary categories of discrete graded bimodules, or their derived variants.

The following two observations and example are crucial when connecting back to section 2.

Most of the constructions in section 6 will build on the following \(\infty\)-categories.

The following justifies the terminology ’Morita categories’, see also example 6.0.2.

Explicitly, the functor in observation 4.5.10 sends a flat \(Z\)-graded \(k\)-algebra \(A\) to the stable \(\infty\)-category \[\mathrm{RMod}_{HA}\left(\mathrm{Mod}_{Hk}^{BZ}\right)^{\mathrm{c}}~~ \stackrel{\mathrm{Prop.}~\href{/tag/006U}{3.6.9}}{\simeq} ~~\mathcal D(\mathrm{grmod}_A)^{\mathrm{gr-perf}}\] of graded-perfect right \(A\)-modules, with \(Z\)-action given by grading shift.

5 \((\infty,k)\)-categories and their factorization systems[0099]

In this section, we introduce our \((\infty,k)\)-categorical framework, and establish the existence of various factorization systems generalizing the familiar (surjective-on-objects, fully faithful)-factorization system on \(\mathrm{Cat}_{\infty}\). We refer the reader to section A, especially A.1, for motivation and a leisurely introduction to \((\infty,k)\)-categories.

5.1 Basic notions in \((\infty,k)\)-category theory[009A]

Throughout, we will use the theory of enriched \(\infty\)-categories developed in [GH15], see also subsection A.10.

5.2 Truncatedness and connectedness[009L]

Here we recall the standard definition of the (\(n\)-connected, \(n\)-truncated) factorization system on the \(\infty\)-category \(\mathcal S\) of spaces. This will be the base case for our factorization systems on \((\infty, k)\)-categories.

Throughout, we will use the following cancellation properties generalizing well-known facts about surjections and injections of sets.

5.3 Factorization systems for \((\infty,k)\)-categories[009Y]

In this subsection we define \(n\)-surjective and \(n\)-faithful morphisms in \(\mathrm{Cat}_{(\infty, {k})}\) and prove in theorem 5.3.7 that they form factorization systems, using the main result theorem B.4.1 of section B. Furthermore, we establish various properties of these factorization systems.

Recall the cancellation property lemma 5.2.4 of truncated and connected maps of spaces. The second part of lemma 5.2.4 generalizes to functors of \((\infty,k)\)-categories:

Just as with the (\(n\)-connected, \(n\)-truncated) factorization system on the \(\infty\)-category of spaces, the \(n\)-surjective and \(n\)-faithful functors form a factorization system on the \(\infty\)-category of \((\infty,k)\)-categories:

In fact, the maximal sub-\((\infty,k)\)-category functor \(\mathrm{Cat}_{(\infty, {j})} \xrightarrow{\iota_k} \mathrm{Cat}_{(\infty, {k})}\) also preserves \(n\)-surjectivity provided \(n\) lies outside of the interval \([k, j)\):

Throughout, we will also repeatedly use the following simple observation:

We end this subsection with the following useful proposition, generalizing [SY19, Prop. 4.2.8].

5.4 Homotopy \((n, k)\)-categories of \((\infty, k)\)-categories[00AP]

In this subsection, we define the homotopy \((n,k)\)-category of an \((\infty,k)\)-category.

In general, the hom-categories in \(\tau_n \mathcal C\) can be computed by applying \(\tau_{n-1}\) to hom-categories of \(\mathcal C\):

For any \(j> k \geq 0\) and \(n\geq -2\), since the inclusion \(i_j \colon \mathrm{Cat}_{(\infty, {k})} \hookrightarrow \mathrm{Cat}_{(\infty, {j})}\) preserves \(n\)-factorizations (see observation 5.3.10), the diagram commutes. By adjunction, this induces for any \((\infty,j)\)-category \(\mathcal C\) a canonical functor of \((n,k)\)-categories \[ \tau_n \iota_k \mathcal C\rightarrow\iota_k \tau_n \mathcal C.\]

Now we are ready to define the \(n\)-homotopy category functor:

5.5 Faithful functors and homotopy categories[00B9]

For an \((\infty,1)\)-category \(\mathcal C\), the \(\infty\)-category of full subcategories of \(\mathcal C\) is equivalent to the poset of subsets of the set \(h_0(\mathcal C)\) of isomorphism classes of objects in \(\mathcal C\), or equivalently to the \(\infty\)-category of ‘full subcategories’ of the set \(h_0(\mathcal C)\). In the next two subsections, we generalize this to arbitrary \(n, k\geq 0\) and characterizes \((n-1)\)-faithful functors of \((\infty,k)\)-categories into some \((\infty,k)\)-category \(\mathcal C\) in terms of \((n-1)\)-faithful functors of \((n,n)\)-categories into the homotopy \(n\)-category \(h_{n}\mathcal C\).

The goal of the next subsection §5.6 will be to prove the following theorem.

Before proving theorem 5.5.2 in §5.6, we record a few corollaries. First, taking fibers at a \(\mathcal D\in \mathrm{Cat}_{(\infty, {k})}\) immediately leads to the following corollary:

Hence, corollary 5.5.3 is indeed a generalization of the statement at the beginning of this subsection: The \(\infty\)-category of \((n-1)\)-faithful functors into \(\mathcal D\) is equivalent to the \(\infty\)-category of \((n-1)\)-faithful functors into \(h_n\mathcal D\).

For later use, we need an analogous statement for \(\mathcal O\)-monoidal \(\infty\)-categories for a given \(\infty\)-operad \(\mathcal O\). For any small \(\infty\)-operad \(\mathcal O\) (see subsection A.8 and §7 for definitions and notation), we can extend this to a statement about categories of \(\mathcal O\)-algebras:

We record a further straight-forward consequence:

Taking fibers at some \(G\in \mathrm{Hom}_{\mathrm{Cat}_{(\infty, {k})}}(\mathcal X, \mathcal D)\), corollary 5.5.6 immediately implies the following corollary which will play a key role in the proof of our main theorem:

5.6 Proof of theorem 5.5.2[00BM]

This subsection is devoted to the proof of theorem 5.5.2.

We first consider the special case that \(k=n\). This will be a consequence of the following observation relating faithfulness and homotopy categories, well known in the case \(k=n=1\) of \((\infty,1)\)-categories.

We use proposition 5.6.1 to prove the \(n\geq k \geq 0\) case of theorem 5.5.2.

To generalize lemma 5.6.2 to also allow for the case \(k>n\), we will use the following lemma.

We can combine lemma 5.6.2 and lemma 5.6.3 into a proof of theorem 5.5.2.

6 The monoidal \((\infty,2)\)-category of chain complexes of Soergel bimodules[00C2]

Throughout this section, we let \(k\) be a \(\mathbb{Q}\)-algebra.³³ Recall from definition 4.3.1 the presentably symmetric monoidal \(\infty\)-categories \(\mathrm{add}_{k}^{B\mathbb{Z}}\coloneqq \mathrm{Fun}(B\mathbb{Z}, \mathrm{add}_k)\) and \(\mathrm{st}^{B\mathbb{Z}}_{k}\coloneqq \mathrm{Fun}(B \mathbb{Z}, \mathrm{st}_k)\) of small \(k\)-linear³⁴ additive, resp. stable, idempotent complete \(\infty\)-categories equipped with a \(\mathbb{Z}\)-action. These categories will always be understood as equipped with the Day convolution symmetric monoidal structure.

Unpacked, an ordinary category with local shifts is a \(1\)-category \(\mathcal C\) equipped with, for every \(c, c' \in \mathcal C\) a \(\mathbb{Z}\)-action on the hom-set \(\mathrm{Hom}_{\mathcal C}(c,c')\), denoted by \([n]\colon \mathrm{Hom}_{\mathcal C}(c,c') \rightarrow\mathrm{Hom}_{\mathcal C}(c,c')\) for \(n \in \mathbb{Z}\), which is compatible with composition in the sense that the following diagram commutes: Similarly, a \(k\)-linear (stable) \((\infty,2)\)-category with local shifts is an \((\infty,2)\)-category whose hom-categories are additive (stable), \(k\)-linear, and have a homotopy coherent \(\mathbb{Z}\)-action which is compatible with composition.