nLab
braided monoidal 2-category

Contents

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

2-Category theory

Contents

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 CC equipped with a tensor product :C×CC\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

Picard 2-groupoid

The Picard 2-groupoid of a braided monoidal 2-category is a braided 3-group.

References

  • Lawrence Breen, Une lettre à P. Deligne au sujet des 22-catégories tressées (1988) (pdf)

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

  • John Baez and Martin Neuchl, Higher-dimensional algebra I: braided monoidal 2-categories, Adv. Math. 121 (1996), 196-244. (pdf)

  • Sjoerd Crans, Generalized centers of braided and sylleptic monoidal 2-categories, Adv. Math. 136 (1998), 183-223. (pdf)

Last revised on October 10, 2017 at 17:05:23. See the history of this page for a list of all contributions to it.