I have found three different definitions of Modular Tensor Categories. I want to know if anybody can give a sketch of proof for their equivalences (some parts are easy of course)
A Modular Tensor Category $\mathscr{C}$ is a semisimple ribbon category with finite number of isomorphism classes of simple objects, such that (all three conditions are apparently equivalent) either
- (Turaev) The matrix $\tilde{s}$ with components $\tilde{s}_{ij}=\mathrm{tr}(\sigma_{{X_i}X_j}\circ \sigma_{{X_j}X_i})$ is invertible (trace is quantum trace, $\sigma$ is braiding isomorphism and $X_i$ are simple)
- (Bruguieres, Muger) Multiples of unit $\mathbf{1}$ are the only transparent objects of the category. (An object $X$ is called transparent if $\sigma_{XY}\circ\sigma_{YX}=\mathrm{id}_{Y\otimes X}$ for all objects $Y$.)
- Our premodular category $\mathscr{C}$ is factorizable. The fucntor $\mathscr{C}\boxtimes \mathscr{C}^\text{rev}\to \mathcal{Z}(\mathscr{C})$ is an equivalence of categories.
The first two definitions I understand very well. The last one though I have no idea what it even means and what are these symbols and what functor?
But in any case I'm mostly interested in a sketch of proof for equivalence of these definitions (and maybe an explanation of the meaning of the last one). For example it is quite trivial to show that Turaev results the second one. It is just some linear algebra basically and some graphical calculus. The converse though I couldn't figure out.
In his paper Non-degeneracy conditions for braided finite tensor categories, Shimizu shows the equivalence of these definitions in the non-semisimple case.
Of course, the $s$-matrix doesn't work there, but it is not hard to show (and I believe he does that) that in the semisimple setting, non-degeneracy of the $s$-matrix is equivalent to non-degeneracy of the Hopf pairing of Lyubashenko's coend.