Braided monoidal categories - more precisely, C*-categories of such kind - are the basic mathematical tool to encode the structure of superselection sectors in low-dimensional (<4) QFT. The low dimensionality forces the braiding coming from permutation statistics to be non-trivial. There is a large literature on the subject, but two fundamental papers are the following:

- K. Fredenhagen, K.-H. Rehren, B. Schroer, *Superselection Sectors with Braid Group Statistics and Exchange Algebras. I. General Theory.* [Commun.Math.Phys. **125** (1989) 201-226][1];
- K. Fredenhagen, K.-H. Rehren, B. Schroer, *Superselection Sectors with Braid Group Statistics and Exchange Algebras. II. Geometric Aspects and Conformal Invariance.* Rev.Math.Phys. Special Issue in honor of Rudolf Haag (1992) 113-157.

There is a recent, short review by Y. Kawahigashi on the subject, centered around chiral conformal QFT's on the circle (*Conformal Field Theory, Tensor Categories and Operator Algebras*, J.Phys. **A** Math. Theor. **48** (2015) 303001, [arXiv:1503.05675 [math-ph]][2], specially Section 3), which can be classified to a large extent using such methods.


  [1]: http://projecteuclid.org/euclid.cmp/1104179464
  [2]: http://arxiv.org/abs/1503.05675