nLab
braided monoidal 2-category

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Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

  • trace

  • traced monoidal category?

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Examples

Theorems

In higher category theory

2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

Contents

Idea

A braided monoidal 2-category is a 2-category C equipped with a tensor product ⊗:C×C→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.

Properties

Coherence theorem

Picard 2-groupoid

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

References

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Revised on April 8, 2013 22:28:06 by Urs Schreiber (82.113.99.18)
Edit | Back in time (6 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: Day convolution, monoidal category, tensor product, symmetric monoidal category, braided monoidal category, multicategory, periodic table, cartesian monoidal category, k-tuply monoidal n-category, cartesian closed category, locally cartesian closed category, Crans-Gray tensor product, monoidal model category, pushout-product axiom, closed monoidal category, strict monoidal category, dagger-compact category, trace, dualizable object, string diagram, compact closed category, k-tuply monoidal (n,r)-category, star-autonomous category, closed monoidal structure on presheaves, exponential object, distributive category, monoidal (infinity,1)-category, symmetric monoidal (infinity,1)-category, fusion category, rigid monoidal category, semisimple category, modular tensor category, symmetric monoidal functor, semicartesian monoidal category, cartesian bicategory, braided monoidal 2-category, tensor category, monoidal functor, oplax monoidal functor, internal category in a monoidal category, coherence theorem for monoidal categories, tensor power, monoidal categories - contents, closed functor, monoidal bicategory, pivotal category, bicartesian closed category, finite quantum mechanics in terms of dagger-compact categories, compact double category, symmetric monoidal dagger-category, symmetric algebra, ribbon category, category with duals, twist, bimonoidal category, spherical category, monoid axiom in a monoidal model category, symmetric monoidal (infinity,n)-category, model structure on monoids in a monoidal model category, premonoidal category, cartesian functor, monoidal structure map, cartesian closed (infinity,1)-category, category of monoids, closed monoidal (infinity,1)-category, braided monoidal functor, Hopf adjunction, evaluation map, Frobenius monoidal functor, bilax monoidal functor, monoidal Quillen adjunction, commutative monoid in a symmetric monoidal (infinity,1)-category, fully dualizable object, (infinity,n)-category with duals, Spanier-Whitehead duality, monoidal derivator, Picard group, contents of contents, Mac Lane's proof of the coherence theorem for monoidal categories, Picard 3-group, cartesian closed functor, Deligne tensor product of abelian categories, symmetric multicategory, coherence theorem for symmetric monoidal categories, permutative category, bipermutative category, tensor product of modules, internal hom of chain complexes, tensr product of chain complexes, tensor product of chain complexes, tensor product of abelian groups, coherence theorem for braided monoidal categories, coherence theorem for monoidal bicategories, coherence theorem for braided monoidal bicategories, coherence theorem for symmetric monoidal bicategories, braided 3-group, sylleptic monoidal 2-category, symmetric monoidal 2-category, line 2-bundle, Picard 2-group, Picard infinity-group, cartesian monoidal (infinity,1)-category, monoidal (2,1)-category, Drinfeld center