Higher-Dimensional Algebra I: Braided Monoidal 2-Categories

TL;DR

This paper introduces definitions and constructions for braided monoidal 2-categories, explores their applications to 4d topological quantum field theories, and proves a strictification theorem for these higher categories.

Contribution

It provides concise definitions, explicit unpackings, and a construction of the center for semistrict braided monoidal 2-categories, advancing the understanding of higher categorical structures.

Findings

Constructed a semistrict braided monoidal 2-category as the center of a monoidal category

Provided explicit definitions similar to existing frameworks by Kapranov and Voevodsky

Proved a strictification theorem for braided monoidal 2-categories

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.