# What are braided vertex algebras?

The notion of vertex algebra, like any reasonable algebraic notion, makes sense inside any (sufficiently linear) symmetric monoidal category. The standard pictures of the operator product, however, suggest that symmetry is more than is needed: one should be able to formulate the axioms of "vertex algebra" internal to any (sufficiently linear) braided monoidal category. Has this been done in the literature? Are there standard examples?

For vertex operator algebras, I expect you need your braided monoidal category to be balanced.

• I should say: there is a purely abstract-nonsense definition of VA in a braided category, which is that a VA, according to Huang and Kong, is a (holomorphic) symmetric monoidal functor from a certain category of genus-zero complex cobordisms. A braided monoidal category determines a functor from the same category, and so a "VA in a BMC" is a lax natural transformation of symmetric monoidal functors. What I'm looking for is an "unpacked" or "hands on" version of the definition, with examples, and hopefully I will also learn where such a construction first appeared. – Theo Johnson-Freyd Dec 2 '16 at 14:54