nLab symmetric 2-group



Group Theory

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




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

  1. Regarded as a monoidal category, GG 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 BG\mathbf{B}G is a braided 3-group.

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

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

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

  7. GG is a 3-tuply monoidal groupoid.

  8. GG 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:

Last revised on May 16, 2022 at 07:27:55. See the history of this page for a list of all contributions to it.