The Reshetikhin-Turaev construction take as input a Modular Tensor Category (MTC) and spits out a 3D TQFT. I've been told that the other main construction of 3D TQFTs, the Turaev-Viro State sum construction, factors through the RT construction in the sense that for each such TQFT Z there exists a MTC M such that the RT construction of applied to M reproduces Z. Is this true for all known 3D TQFTs? Does anyone know any counter examples?
Edit: (1) I want to be flexible with what we call a TQFT, so anomalies are okay.
(2) There have been some good answers to the effect that more or less if I have an extended TQFT then it factors through the RT construction. But this is not really what I'm after. Are there any (non-extended) examples that people know about? Ones which might not come from the RT construction. Are all known 3D TQFTs extended TQFTs?
