nLab sylleptic 3-group

Definition

Definition

A sylleptic 3-group is a 3-group equipped with the following equivalent structure:

1. Regarded as a monoidal 2-category, $G$ is a sylleptic monoidal 2-category.

2. The A-∞ algebra/E1-algebra structure on $G$ refines to an E3-algebra structure.

3. $G$ is a 3-tuply monoidal 2-groupoid.

4. $G$ is a groupal 3-tuply monoidal (2,0)-category.

Examples

• If $(\mathcal{C}, \otimes)$ is a sylleptic monoidal 2-category, then its Picard 3-group is a sylleptic 3-group.

Related concepts

• sylleptic monoidal 2-category

• ∞-group, k-tuply groupal n-groupoid

• braided ∞-group,

  • braided 2-group

  • braided 3-group

• sylleptic ∞-group

  • sylleptic 3-group

• abelian ∞-group

  • symmetric 2-group

  • abelian 3-group

References

• Nick Gurski, Angélica M. Osorno, Section 2.2 of: Infinite loop spaces, and coherence for symmetric monoidal bicategories, Adv. Math. 246 (2013) 1-32 (arXiv:1210.1174)