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A braided monoidal $(\infty,2)$-category of Soergel bimodules

Yu Leon Liu, Aaron Mazel-Gee, David Reutter, Catharina Stroppel, Paul Wedrich

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.

A braided monoidal $(\infty,2)$-category of Soergel bimodules

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 -monoidal -category whose hom--categories are -linear, stable, idempotent-complete, and equipped with -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 -categories and higher algebra. Along the way, we develop families of factorization systems for -categories, enriched -categories, and -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 -categorical technology in the form of an appendix.
Paper Structure (117 sections, 123 theorems, 282 equations, 2 figures)

This paper contains 117 sections, 123 theorems, 282 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Monoidality theorem
  4. Braiding theorem
  5. Context and proof outline for the braiding theorem
  6. Braided monoidal 2-categories and their generalizations
  7. Higher-categorical notions of faithfulness and homotopy categories
  8. The fiber functor
  9. The prebraiding
  10. Prebraidings via $\infty$-operads
  11. From prebraidings to braidings
  12. Organization of the paper
  13. A prebraiding on the homotopy category of Soergel bimodules
  14. Review of Soergel bimodules and diagrammatics
  15. Review of Rouquier complexes
  16. ...and 102 more sections

Key Result

Theorem A

There is a monoidal $(\infty,2)$-category ${\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})$ with objects labelled by natural numbers $n \in \mathbb{N}_0$ and whose endomorphim $\infty$-categories are the $k$-linear, stable, idempotent-complete $\infty$-categories ${\mathbf K}^b(\mathrm{Sbim}_n)$ of chai

Figures (2)

  • Figure 1: Cabled crossings $X_{2,3}$, $X'_{3,2}$, and Coxeter braids $X_{1,4}$, $X'_{1,4}$.
  • Figure 2: Graphical illustration of $\mathrm{slide}_{Y_1,Y_2}$---here in case $(m,n)=(2,3)$.

Theorems & Definitions (358)

  • Theorem A: \ref{['prop:Kbloc-Sbim-explicit']}
  • Theorem B: \ref{['cor:main-corollary']}
  • Definition 2.1.1
  • Definition 2.1.2
  • Remark 2.1.3
  • Definition 2.1.4
  • Definition 2.1.5
  • Definition 2.2.2
  • Definition 2.2.3
  • Theorem 2.2.4: Rouquier canonicity
  • ...and 348 more