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Higher-Dimensional Algebra I: Braided Monoidal 2-Categories

John C. Baez, Martin Neuchl

TL;DR

This paper develops a conceptual, semistrict framework for braided monoidal 2-categories by defining semistrict monoidal and braided monoidal 2-categories and introducing a center construction $\mathcal{Z}(\mathcal{C})$ that yields a semistrict braided monoidal 2-category. It shows how, starting from a monoidal 2-category $\mathcal{C}$, the center $\mathcal{Z}(\mathcal{C})$ captures a higher analogue of the quantum double and provides a natural setting for 4D TQFTs and 2-tangle invariants, closely related to Tannaka–Krein reconstruction in the higher-categorical context. The paper also establishes an embedding of any braided semistrict monoidal 2-category into its center via a braided monoidal 2-functor, offering a strictification viewpoint and connecting to potential future theories of Hopf 2-algebras and higher quantum doubles. Collectively, these results advance the program of higher-dimensional algebra by linking semistrict braided structures, center constructions, and topological quantum field theory in four dimensions.

Abstract

We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their applications to 4d topological quantum field theories and 2-tangles (surfaces embedded in 4-dimensional space). Then we give concise definitions of semistrict monoidal 2-categories and braided monoidal 2-categories, and show how these may be unpacked to give long explicit definitions similar to, but not quite the same as, those given by Kapranov and Voevodsky. Finally, we describe how to construct a semistrict braided monoidal 2-category Z(C) as the `center' of a semistrict monoidal category C. This is analogous to the construction of a braided monoidal category as the center, or `quantum double', of a monoidal category. As a corollary, our construction yields a strictification theorem for braided monoidal 2-categories.

Higher-Dimensional Algebra I: Braided Monoidal 2-Categories

TL;DR

This paper develops a conceptual, semistrict framework for braided monoidal 2-categories by defining semistrict monoidal and braided monoidal 2-categories and introducing a center construction that yields a semistrict braided monoidal 2-category. It shows how, starting from a monoidal 2-category , the center captures a higher analogue of the quantum double and provides a natural setting for 4D TQFTs and 2-tangle invariants, closely related to Tannaka–Krein reconstruction in the higher-categorical context. The paper also establishes an embedding of any braided semistrict monoidal 2-category into its center via a braided monoidal 2-functor, offering a strictification viewpoint and connecting to potential future theories of Hopf 2-algebras and higher quantum doubles. Collectively, these results advance the program of higher-dimensional algebra by linking semistrict braided structures, center constructions, and topological quantum field theory in four dimensions.

Abstract

We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their applications to 4d topological quantum field theories and 2-tangles (surfaces embedded in 4-dimensional space). Then we give concise definitions of semistrict monoidal 2-categories and braided monoidal 2-categories, and show how these may be unpacked to give long explicit definitions similar to, but not quite the same as, those given by Kapranov and Voevodsky. Finally, we describe how to construct a semistrict braided monoidal 2-category Z(C) as the `center' of a semistrict monoidal category C. This is analogous to the construction of a braided monoidal category as the center, or `quantum double', of a monoidal category. As a corollary, our construction yields a strictification theorem for braided monoidal 2-categories.

Paper Structure

This paper contains 13 sections, 7 theorems, 45 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Center Construction
  3. Applications to 4-Dimensional TQFT
  4. Applications to 2-Tangles
  5. Definitions
  6. Semistrict Monoidal 2-Categories
  7. Semistrict Braided Monoidal 2-Categories
  8. The Center Construction
  9. $\mathcal{Z}(\mathcal{C})$ as a 2-Category
  10. The Monoidal Structure
  11. The Braiding
  12. Embedding $\mathcal{C}$ in $\mathcal{Z}(\mathcal{C})$
  13. Conclusions

Key Result

Lemma 3

Suppose $\tilde{\mathcal{C}}$ is a semistrict 3-category with one object, and let $(\mathcal{C}, \otimes, i)$ be defined as above. Then $\mathcal{C}$ is a 2-category, $\otimes \colon \mathcal{C} \otimes_{\rm G} \mathcal{C} \to \mathcal{C}$ and $i \colon \mathcal{I} \to \mathcal{C}$ are 2-functors, a Conversely, for any $(\mathcal{C},\otimes,i)$ with these properties, there is a unique semistrict 3

Figures (4)

  • Figure 1: Weak $k$-tuply monoidal $n$-categories: expected results
  • Figure 2: Vertical and horizontal product in ${\rm hom}(\emptyset, S^1)$
  • Figure 3: The Eckmann--Hilton argument
  • Figure 4: The Yang--Baxter equation

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Definition 6
  • Lemma 7
  • ...and 12 more