nLab
symmetric monoidal 2-category

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Higher algebra

Contents

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 CC equipped with a tensor product :C×CC\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 kk-morphism for k=0,1,2,3k=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.

References

Revised on October 10, 2017 17:05:56 by Mike Shulman (76.167.222.204)