# nLab braided monoidal 2-category

Contents

### Context

#### Monoidal categories

monoidal categories

## In higher category theory

#### 2-Category theory

2-category theory

# Contents

## Idea

A braided monoidal (weak) 2-category is a monoidal 2-category with a categorified version of a braiding.

That is, it is a 2-category $C$ equipped with a tensor product $\otimes : C \times C \to C$ 2-functor which satisfies the first in a hierarchy of conditions for being commutative up to equivalence. In the language of k-tuply monoidal n-categories, a braided monoidal 2-category is a doubly monoidal 2-category. As described there, this may be identified with a pointed 4-category with a single object and a single 1-morphism. We can also say that it is a monoidal 2-category whose E1-algebra structure is refined to an E2-algebra structure.

## Properties

### Picard 2-groupoid

The Picard 2-groupoid of a braided monoidal 2-category is a braided 3-group.

## References

Last revised on July 21, 2021 at 12:48:16. See the history of this page for a list of all contributions to it.