nLab
symmetric monoidal 2-category

Definition

A symmetric monoidal (weak) 2-category is a monoidal 2-category with a categorified version of a symmetry.

That is, it is a 2-category $C$ equipped with a tensor product $\otimes : C \times C \to C$ 2-functor which satisfies all possible conditions for being commutative up to equivalence. In the language of k-tuply monoidal n-categories, a braided monoidal 2-category is a quadruply monoidal 2-category. As described there, this may be identified with a pointed 6-category with a single $k$-morphism for $k=0,1,2,3$. We can also say that it is a monoidal 2-category whose E1-algebra structure is refined to an E4-algebra structure.

Examples

• A symmetric monoidal 2-category all whose objects are invertible under the tensor product is a symmetric 3-group.

Related concepts

• monoidal 2-category, 3-group

• braided monoidal 2-category, braided 3-group

• sylleptic monoidal 2-category, sylleptic 3-group

• symmetric monoidal 2-category, symmetric 3-group

• k-tuply monoidal (n,r)-category

References

• Brian Day, Ross Street, Section 5 of: Monoidal Bicategories and Hopf Algebroids, Advances in Mathematics Volume 129, Issue 1, 15 July 1997, Pages 99-157 (doi:10.1006/aima.1997.1649)

• Mike Shulman, Constructing symmetric monoidal bicategories (arXiv:1004.0993v1)