Let R be the hyperfinite II\_1 or the hyperfinite III\_1 factor (pick which ever one you prefer), and let Bim(R) denote the tensor category of R-R-bimodules.


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This question is inspired by the recent article 
[The classification of subfactors of index at most five](http://arxiv.org/pdf/1304.6141.pdf)
by Jones, Morrison, and Snyder.
More precisely, I am interested in Thm 1.1 and Thm 2.10 of the above paper (both of them are older results).


Given the above,
I expect the following to be true:



> 
> (1) Any unitary fusion category can be embedded in Bim(R).  
> 
>  (2) Any two embeddings are conjugated by an automorphism of R.
> 
> 
> 


however, I am not sure if things have ever been formulated in this way.


My questions are:  

• What is the closest result to (1) and (2) available in the literature?  

• is it easy to adapt/use some existing proofs to get (1) and (2), and how?



realizing fusion categories as subfactors of the hyperfinite
1, and indeed its generalization to the amenable case, is in ["Amenable tensor categories and their realizations as AFD bimodules"](http://www.ams.org/mathscinet-getitem?mr=1749868) by Hayashi and Yamagami, see Section 7. I don't think 2 has appeared in the literature, though I'd expect that it's true.


In the fusion case, if the universal grading group is trivial, I think you can just look at the algebra object $\oplus\_x x \otimes x^\*$ in $\mathcal{C}$ and use the usual reconstruction and uniqueness for subfactors. It's possible I'm missing a technicality, but I think this should work (in particular, Popa doesn't assume irreducibility). If the grading group is non-trivial there still may be some trick that reduces it to the known cases, but it's not clear to me. 


