1. Summary 1.1. axioms (from two persectives, math and physical); 1.2. applications
2. notations and context for axioms 2.1. categories 2.2. cobordism 2.2.1. manifolds 2.3 examples
Turaev Viro paper (Topology, 1992) here
arXiv:1106.6033 The full equivalence proof between string nets and the Turaev Viro model by Kirillov Jr.
arXiv:cond-mat/0506438, Appendix E, often with physical motivations for some of the axioms.
arXiv:hep-th/0401076 Freidel and Louapre , Ponzano-Regge model of 3D quantum gravity
arXiv:gr-qc/0410141 The same model as above. Amplitudes are proved to be given by the Reshetikhin-Turaev evaluation of a coloured chain mail links based on the D(SU(2)) quantum group.
Roberts (Topology, 1995) The proof that (square of absolute values of) amplitudes of SU(2) Chern-Simons theory are given by Turave-Viro invariants (based on SU_q(2).
Karowski and Schrader (Comm. Math. Phys. 1992) http://link.springer.com/article/10.1007%2FBF02096773?LI=true#page-1 Here observables are also in the game.
Ambjorn, Durhuus, Jonsoon: Quantum geometry: a statistical field theory approach
Karowski et al ;
and see ...