# nLab symmetric monoidal 2-category

### Context

#### Higher category theory

higher category theory

## 1-categorical presentations

#### Higher algebra

higher algebra

universal algebra

# Contents

## Definition

A symmetric monoidal 2-category is a monoidal 2-category whose E1-algebra structure is refined to an E3-algebra structure.

A symmetric monoidal (infinity,n)-category for $n = 2$.

A (fully) strict symmetric monoidal 2-category is simply a strict $2$-category $\mathcal{C}$ together with a strict 2-functor $\otimes : \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$ and an object $1$ of $\mathcal{C}$ which satisfy the same axioms as for a strict symmetric monoidal category (which in turn are exactly the same as those for a commutative monoid).

## Examples

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

## References

Revised on May 22, 2017 15:19:53 by Richard Williamson (84.202.247.109)