# nLab braided 2-group

Contents

group theory

### Cohomology and Extensions

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

(∞,1)-category theory

# Contents

## Definition

###### Definition

A 2-group $G$ is braided if it is equipped with the following equivalent structure:

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

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

3. The double delooping 3-groupoid $\mathbf{B}^2 G$ exists.

4. The groupal A-∞ algebra/E1-algebra structure on $G$ refines to an E2-algebra structure.

5. $G$ is a doubly groupal groupoid.

6. $G$ is a groupal doubly monoidal (1,0)-category.

## References

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

As a special case of k-tuply groupal n-groupoids:

In the generality of braided ∞-group stacks the notion appears in

A discussion of ∞-group extensions by braided 2-groups is in

Last revised on May 16, 2022 at 07:04:26. See the history of this page for a list of all contributions to it.