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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.

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  • $\begingroup$ 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. $\endgroup$
    – Theo Johnson-Freyd
    Commented Dec 2, 2016 at 14:54

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For the case of vector spaces graded by an abelian group (with braiding determined by an abelian 3-cocycle following Joyal-Street), this was done by Dong and Lepowsky in their 1993 book "Generalized Vertex Algebras and Relative Vertex Operators". The object has the name "abelian intertwining algebra", and standard examples come from applying the lattice vertex algebra construction to rational lattices. A greater supply of examples was given in van Ekeren, Möller, Scheithauer by an orbifold construction in 2015.

As far as I can tell, the general definition has not appeared explicitly in the literature. Some variants have appeared under the name "quantum vertex algebra", though. In particular, I remember working out that Borcherds's definition of (A,H,S)-vertex algebra in his "Quantum Vertex Algebras" paper can be modified to fit the braided setting without too much trouble.

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  • $\begingroup$ Thanks! Right now the examples I think I want are in grouplike cateogries, so the Dong-Lepowsky book suffices. The vEMS paper is one I didn't have and am looking forward to reading. $\endgroup$
    – Theo Johnson-Freyd
    Commented Dec 2, 2016 at 23:35

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