nLab braided infinity-group

Definition

Definition

An ∞-group $G$ is braided if it is equipped with the following equivalent structure

1. Regarded as a monoidal (∞,1)-category, $G$ is a braided monoidal (∞,1)-category.

2. The delooping ∞-groupoid $\mathbf{B}G$ has the structure of an ∞-group.

3. The double delooping ∞-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 (∞,0)-category.

Examples

See the examples at braided 2-group, braided 3-group.

Related concepts

• ∞-group, k-tuply groupal n-groupoid

• May recognition theorem

• braided ∞-group,

  • braided 2-group

  • braided 3-group

• sylleptic ∞-group

  • sylleptic 3-group

• abelian ∞-group

  • symmetric 2-group

References

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

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Def. 4.28 of: Principal ∞-bundles – General theory, Journal of Homotopy and Related Structures, Volume 10, Issue 4 (2015), pages 749-801 (doi:10.1007/s40062-014-0083-6, arXiv:1207.0248)