# nLab symmetric monoidal 2-category

Contents

### Context

#### Higher category theory

higher category theory

## 1-categorical presentations

#### Higher algebra

higher algebra

universal algebra

# Contents

## Definition

A symmetric monoidal (weak) 2-category is a monoidal 2-category with a categorified version of a symmetry.

That is, it is a 2-category $C$ equipped with a tensor product $\otimes : C \times C \to C$ 2-functor which satisfies all possible conditions for being commutative up to equivalence. In the language of k-tuply monoidal n-categories, a braided monoidal 2-category is a quadruply monoidal 2-category. As described there, this may be identified with a pointed 6-category with a single $k$-morphism for $k=0,1,2,3$. We can also say that it is a monoidal 2-category whose E1-algebra structure is refined to an E4-algebra structure.

## Examples

• A symmetric monoidal 2-category all whose objects are invertible under the tensor product is a symmetric 3-group.

## References

Last revised on July 21, 2021 at 12:33:58. See the history of this page for a list of all contributions to it.