nLab symmetric 2-group

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.

Related concepts

• symmetric monoidal groupoid

• ring groupoid, symmetric ring groupoid

• ∞-group, k-tuply groupal n-groupoid

• braided ∞-group,

  • braided 2-group

  • braided 3-group

• sylleptic ∞-group

  • sylleptic 3-group

• abelian ∞-group

  • abelian group

  • symmetric 2-group

  • abelian 3-group

References

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:

• M. Bullejos, Pilar Carrasco, Antonio M. Cegarra, Cohomology with coefficients in symmetric cat-groups. An extension of Eilenberg–MacLane’s classification theorem, Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 163 - 189 (doi:10.1017/S0305004100071498)