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Chapter 3172: Braiding on type A Soergel bimodules: semistrictness and naturality
Section B.3: Factorization systems for algebras over \(\infty\) -operads
Reference (00L5) (cite)

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

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

Original source: arXiv:2401.02956v2

In this chapter

Chapter overview

1 Introduction
1.1 Overview
1.2 Monoidality theorem
1.3 Braiding theorem
1.4 Context and proof outline for the braiding theorem
1.4.1 Braided monoidal 2-categories and their generalizations
1.4.2 Higher-categorical notions of faithfulness and homotopy categories
1.4.3 The fiber functor
1.4.4 The prebraiding
1.4.5 Prebraidings via \(\infty\)-operads
1.4.6 From prebraidings to braidings
1.5 Organization of the paper
2 A prebraiding on the homotopy category of Soergel bimodules
2.1 Review of Soergel bimodules and diagrammatics
2.2 Review of Rouquier complexes
2.3 Isomorphism classes of Soergel bimodules
2.4 Prebraidings
2.5 Prebraiding for Soergel bimodules
2.6 Centralizers and prebraidings
3 Stable linear algebra
3.1 Completions of \(\infty\)-categories
3.1.1 The Yoneda embedding
3.1.2 Presentable \(\infty\)-categories
3.1.3 The monoidal structure on \(\mathrm{Pr}^\mathrm{L}\) and presentably symmetric monoidal categories
3.1.4 Adjoining colimits
3.2 Compact generation and ind-completion
3.2.1 Compact and projectively generated \(\infty\)-categories
3.2.2 Ind-completion
3.3 Additive and stable \(\infty\)-categories
3.3.1 Definitions
3.3.2 Symmetric monoidal structure
3.4 From additive to stable \(\infty\)-categories
3.4.1 The \(\infty\)-category of chain complexes
3.4.2 The free stable category on an additive category
3.4.3 \({\mathbf K}^b\) as a left adjoint
3.5 \(\infty\)-categories of graded modules
3.5.1 \(\mathbb{K}\)-modules
3.5.2 Compact-projective \(\mathbb{K}\)-modules
3.5.3 \(\mathcal Z\)-graded \(\mathbb{K}\)-modules
3.6 Derived \(\infty\)-categories of graded modules
3.6.1 Derived \(\infty\)-categories
3.6.2 Derived \(\infty\)-categories of graded modules
4 Graded-linear \(\infty\)-categories and Morita theory
4.1 Presentably enriched \(\infty\)-categories
4.1.1 Closed monoidal \(\infty\)-categories and closed module \(\infty\)-categories
4.1.2 Presentably enriched \(\infty\)-categories
4.2 Graded linear \(\infty\)-categories
4.2.1 \(\mathbb{K}\)-linear \(\infty\)-categories
4.2.2 \(\infty\)-categories enriched in graded modules
4.3 From gradings to actions
4.4 \(\infty\)-Morita theory
4.5 Morita categories of discrete flat algebras
5 \((\infty,k)\)-categories and their factorization systems
5.1 Basic notions in \((\infty,k)\)-category theory
5.2 Truncatedness and connectedness
5.3 Factorization systems for \((\infty,k)\)-categories
5.4 Homotopy \((n, k)\)-categories of \((\infty, k)\)-categories
5.5 Faithful functors and homotopy categories
5.6 Proof of theorem 5.5.2
6 The monoidal \((\infty,2)\)-category of chain complexes of Soergel bimodules
6.1 The monoidal \((2,2)\)-category of Bott-Samelson bimodules
6.2 The monoidal \((2,2)\)-category of Soergel bimodules
6.3 The Soergel \((2,2)\)-category agrees with its classical variant
6.4 The monoidal \((\infty,2)\)-category of chain complexes of Soergel bimodules
6.5 The fiber functor on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\)
7 Prebraidings and braidings via \(\infty\)-operads
7.1 Recollections on unital \(\infty\)-operads
7.2 The operads \(\mathbb A_2\) and \(\mathbb{T}_2\)
7.3 Relative \(\mathbb T_2\)-structures
7.4 Centralizers and centers
7.5 A factorization system on the \(\infty\)-category of operads
7.6 Lifting operadic structure
7.7 From \(\mathbb A_2 \otimes \mathbb E_1\)- to \(\mathbb E_2\)-algebras
7.8 Lifting maps of algebras
8 The main theorem
8.1 Spaces of braidings and prebraidings
8.2 Statement of the main theorem
8.3 From prebraidings to braidings
8.4 Proof of the main theorem
A Recollections, notation, and conventions regarding higher category theory
A.1 From \(n\)-categories to \((\infty,n)\)-categories
A.2 Some basic notions in \(\infty\)-category theory
A.2.1 Basic notions
A.2.2 Monomorphisms and subcategories
A.2.3 Adjunctions
A.2.4 Simplicial objects
A.2.5 Localizations
A.2.6 Connected categories versus weakly contractible \(\infty\)-categories
A.3 Higher coherence
A.4 Straightening and unstraightening
A.5 Adjunctions revisited
A.6 Set-theoretic considerations
A.7 Presentable \(\infty\)-categories
A.8 Some basics of \(\infty\)-operads
A.8.1 Basic notions
A.8.2 Key examples
A.8.3 \(\mathcal O\)-monoidal \(\infty\)-categories
A.8.4 \(\mathcal O\)-algebra objects
A.8.5 \(\mathcal O\)-algebras of symmetric monoidal \(\infty\)-categories
A.8.6 Symmetric monoidal structure on overcategories
A.8.7 Boardman-Vogt tensor product and Dunn additivity
A.8.8 Laxly \(\mathcal O\)-monoidal functors
A.8.9 Localizations of \(\mathcal O\)-monoidal \(\infty\)-categories
A.8.10 Presentably \(\mathcal O\)-monoidal \(\infty\)-categories
A.8.11 Adjunctions of \(\mathcal O\)-algebras
A.9 Module \(\infty\)-categories
A.9.1 Left, right, and bimodules for associative algebras
A.9.2 The bar construction and relative tensor product of bimodules
A.9.3 Module categories of commutative algebras
A.10 Enriched \(\infty\)-categories
B Factorization systems for enriched \(\infty\)-categories
B.1 Recollections on factorization systems
B.1.1 Basics of factorization systems
B.1.2 Factorization systems on presentable \(\infty\)-categories
B.2 Induced factorization systems
B.3 Factorization systems for algebras over \(\infty\)-operads
B.4 Factorization systems for enriched \(\infty\)-categories
References

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Original source · 2401.02956v2