nLab braided monoidal 2-category

Idea

A braided monoidal (weak) 2-category is a monoidal 2-category with a categorified version of a braiding.

That is, it is a 2-category $C$ equipped with a tensor product $\otimes : C \times C \to C$ 2-functor which satisfies the first in a hierarchy of conditions for being commutative up to equivalence. In the language of k-tuply monoidal n-categories, a braided monoidal 2-category is a doubly monoidal 2-category. As described there, this may be identified with a pointed 4-category with a single object and a single 1-morphism. We can also say that it is a monoidal 2-category whose E1-algebra structure is refined to an E2-algebra structure.

Properties

Coherence theorem

• coherence theorem for braided monoidal bicategories

Picard 2-groupoid

The Picard 2-groupoid of a braided monoidal 2-category is a braided 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

• Lawrence Breen, Une lettre à P. Deligne au sujet des $2$-catégories tressées (1988) (pdf)

• André Joyal, Ross Street, Section 5 of: Braided tensor categories, Adv. Math. 102 (1993), 20–78 (doi:10.1006/aima.1993.1055)

• Mikhail Kapranov, Vladimir Voevodsky, 2-Categories and Zamolodchikov tetrahedra equations, in: Proc. Symp. Pure Math. 56 Part 2 (1994), AMS, Providence, pp. 177–260 (doi:10.1090/pspum/056.2)

• Mikhail Kapranov, Vladimir Voevodsky, Braided monoidal 2-categories and Manin-Schechtman higher braid groups, Journal of Pure and Applied Algebra Volume 92, Issue 3, 25 March 1994, Pages 241-267 (doi:10.1016/0022-4049(94)90097-3)

• John C. Baez, Martin Neuchl, Higher-Dimensional Algebra I: Braided Monoidal 2-Categories, Adv. Math. 121 (1996), 196-244 (arXiv:q-alg/9511013)

• Sjoerd E. Crans, Section 2.2. of: Generalized Centers of Braided and Sylleptic Monoidal 2-Categories, Advances in Mathematics, Volume 136, Issue 2, 25 June 1998, Pages 183-223 (doi:10.1006/aima.1998.1720)

• Thibault D. Décoppet, Hao Xu, Local Modules in Braided Monoidal 2-Categories [arXiv:2307.02843]