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Higher Witt Groups for 2-Categories I: Centralizers

Hao Xu

Abstract

In this article, we investigate monoidal, braided, sylleptic centralizers of monoidal, braided, sylleptic 2-functors. We specifically focus on multifusion 2-categories and show that monoidal, braided, sylleptic centralizers are multifusion again, via studying the corresponding enveloping algebras. We provide a characterization of the non-degeneracy condition for monoidal, braided, and sylleptic fusion 2-categories, via vanishing of their centers. Applying Double Centralizer Theorems, we establish the relationship between monoidal, braided, symmetric local modules and free modules. In particular, we obtain factorization properties of non-degenerate monoidal, braided, and sylleptic fusion 2-categories. Main results in this article will be used to study higher Witt equivalences of non-degenerate monoidal, braided, sylleptic 2-categories in the sequential articles.

Higher Witt Groups for 2-Categories I: Centralizers

Abstract

In this article, we investigate monoidal, braided, sylleptic centralizers of monoidal, braided, sylleptic 2-functors. We specifically focus on multifusion 2-categories and show that monoidal, braided, sylleptic centralizers are multifusion again, via studying the corresponding enveloping algebras. We provide a characterization of the non-degeneracy condition for monoidal, braided, and sylleptic fusion 2-categories, via vanishing of their centers. Applying Double Centralizer Theorems, we establish the relationship between monoidal, braided, symmetric local modules and free modules. In particular, we obtain factorization properties of non-degenerate monoidal, braided, and sylleptic fusion 2-categories. Main results in this article will be used to study higher Witt equivalences of non-degenerate monoidal, braided, sylleptic 2-categories in the sequential articles.
Paper Structure (23 sections, 72 theorems, 100 equations, 9 figures)

This paper contains 23 sections, 72 theorems, 100 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphical Calculus
  4. 2-Functors
  5. Algebras
  6. Modules, Local Modules
  7. Induction 2-Functors
  8. Module 2-Categories and Enrichment
  9. Monoidal Centralizers and Braided Centers
  10. Definitions and Basic Properties
  11. Factorization of Endofunctors into Free Modules and Monoidal Local Modules
  12. Braided Centralizers and Sylleptic Centers
  13. Definitions
  14. Enveloping Algebras
  15. Finite Semisimple Braided Centralizers
  16. ...and 8 more sections

Key Result

Lemma 2.2.7

One has the following 3-categories:

Figures (9)

  • Figure 1: Step 1
  • Figure 2: Step 2
  • Figure 3: Step 3
  • Figure 4: Step 4
  • Figure 5: Step 5
  • ...and 4 more figures

Theorems & Definitions (205)

  • Definition 2.2.1
  • Definition 2.2.2
  • Definition 2.2.3
  • Definition 2.2.4
  • Definition 2.2.5
  • Definition 2.2.6
  • Lemma 2.2.7
  • Definition 2.3.1
  • Definition 2.3.2
  • Definition 2.3.3
  • ...and 195 more