# nLab symmetric 2-group

Contents

group theory

### Cohomology and Extensions

#### $(\infty,1)$-Category theory

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# Contents

## Definition

###### Definition

A symmetric 2-group is a 2-group equipped with the following equivalent structure:

1. Regarded as a monoidal category, $G$ is a symmetric monoidal category.

2. Regarded as a 1-truncated ∞-group it has the structure of an abelian ∞-group.

3. The delooping 2-groupoid $\mathbf{B}G$ is a braided 3-group.

4. The double delooping 3-groupoid $\mathbf{B}^2 G$ is a 4-group.

5. The triple delooping 4-groupoid $\mathbf{B}^4 G$ exists.

6. The A-∞ algebra/E1-algebra structure on $G$ refines to an E3-algebra structure.

7. $G$ is a 3-tuply monoidal groupoid.

8. $G$ is a groupal 3-tuply monoidal (1,0)-category.

Under the name “symmetric cat-groups” and thought of as a sub-class of braided monoidal categories, the notion of braided 2-groups is considered in: