If we restrict to a class of subfactors $(N \subset M)$ wherein all the factors are isomorphic, we easily see how to define an equivalence $\sim_{1}$ of subfactors.

But in general, it seems natural to have an equivalence $\sim$ such that, even if $P \not\simeq Q$ : $(P \subset P) \sim (Q \subset Q) $ or $(N\bar\otimes P \subset M\bar\otimes P) \sim (N \bar\otimes Q \subset M\bar\otimes Q) $ or anything else...

How to define the equivalence of subfactors in general ?