nLab braided 3-group

Definition

A braided 3-group is a braided ∞-group which is a 3-group. For $G$ a 3-group, a braiding on it is the following equivalent structure

1. the structure of a 2-group on the delooping $\mathbf{B}G$;

2. a doudle delooping $\mathbf{B}^2 G$;

3. a lift of tha A-∞=E-1-algebra structure on $G$ to an E-2 algebra structure.

Examples

For $R$ a commutative ring, and $Alg_R \simeq 2 Vect_R$ the braided monoidal 2-category of $R$-algebras, bimodules and bimodule homomorphism, the maximal 3-group

$$\mathbf{Br}(R) \hookrightarrow Core(Alg_R)$$

inside is a braided 3-group. Its homotopy groups are the Brauer group, the Picard group and the group of units of $R$. See at Brauer group – Relation to category of modules for more on this.

Related concepts

• ∞-group, k-tuply groupal n-groupoid

• braided ∞-group,

  • braided 2-group

  • braided 3-group

• sylleptic ∞-group

  • sylleptic 3-group

• abelian ∞-group

  • symmetric 2-group

References

As a special case of k-tuply groupal n-groupoids the notion is at least implicit in:

• Antonio R. Garzón, Jesus G. Miranda, Section 6 of: Serre homotopy theory in subcategories of simplicial groups, Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123 (doi:10.1016/S0022-4049(98)00143-1)

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)