# 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

• Lawrence Breen, Une lettre à P. Deligne au sujet des $2$-catégories tressées (1988) (pdf)

• Mikhail Kapranov and Vladimir Voevodsky, 2-Categories and Zamolodchikov tetrahedra equations, in Proc. Symp. Pure Math. 56 Part 2 (1994), AMS, Providence, pp. 177–260.

• John Baez and Martin Neuchl, Higher-dimensional algebra I: braided monoidal 2-categories, Adv. Math. 121 (1996), 196-244. (pdf)

• Sjoerd Crans, Generalized centers of braided and sylleptic monoidal 2-categories, Adv. Math. 136 (1998), 183-223. (pdf)

Last revised on October 10, 2017 at 17:05:23. See the history of this page for a list of all contributions to it.