ScalingStacks

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.