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 ?

Let $A$ be a factor and $\mathcal{C}_{A}$ be the category of all the subfactors $(M \subset N)$ such that $M$ and $N$ are isomorphic to $A$. The most famous of them is perhaps $\mathcal{C}_{R}$ with $R$ the hyperfinite $II_{1}$ factor.
One can define an equivalence relation called $\sim_{1}$, available on each such category, as follows:

Definition : Let $A$ be a factor, $(P \subset Q)$ and $(R \subset S)$ be two subfactors of $\mathcal{C}_{A}$, then:
$(P \subset Q) \sim_{1} (R \subset S)$ if it exists an isomorphism $\phi : Q \to S$ such that $\phi(P) = R$.
Temporary remark : I hope this is the most common definition (also called isomorphism of subfactors), because I don't find it written explicitly in the literature (certainly because it's obviously this one).

The purpose of this issue is to ask how naturally generalize $\sim_{1}$ into an equivalence relation $\sim$ available on the category $\mathcal{C}$ of all the subfactors, in order to verify the following specifications (even if $M \not\simeq N$) :

1. $(M \subset M) \sim (N \subset N) $
2. $(P \bar\otimes M \subset Q \bar\otimes M) \sim (P \bar\otimes N \subset Q \bar\otimes N) $
3. $[(P \subset Q) \sim (R \subset S) ]$ $\Leftrightarrow$ $[(P \bar\otimes M \subset Q \bar\otimes M) \sim (R \bar\otimes N \subset S \bar\otimes N) ]$
4. $(P^{G} \subset P) \sim (Q^{G} \subset Q) $ such that $G$ embeds into $Out(P)$ and $Out(Q)$.

Is it coherent ? (of course $3 \Rightarrow 2 \Rightarrow 1$)

Motivation (Jones' philosophy) : The purpose of this equivalence relation is that any equivalence class $[M \subset N]_{\sim}$ captures only the information (or symmetry) contained in the relative position of $M$ inside $N$ (forgetting the factors themselves), in order to obtain a kind of strictly group-like object.

Ambiguities : It appears that such a relation $\sim$ retricted to $\mathcal{C}_{R}$ would be coaser than $\sim_{1}$ :

• After Bisch-Nicoara-Popa and Owen Sizemore's comments (see below), one can have $(P \subset Q) \not\sim_{1} (P \subset Q)^{t}$ while $(P \subset Q) \sim (P \subset Q)^{t}$.
  Examples: uncountably many non-isomorphic subfactors at index $6$ would be equivalent !
  Is there a relevant difference between the relative position of $P$ inside $Q$, and $P^{t}$ inside $Q^{t}$ ?
• After Ocneanu and Jones' works, an amenable group $G$ acts outerly on $R$ by only one manner, but a non-amenable one, by at least two manners (see here).

Conclusion : the relation $\sim$ would be coaser, but maybe more natural regarding to the motivation.

The existence of what I have called ambiguities reinforces the purpose of the following question :

What's the natural equivalence of subfactors in general ?

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

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

How to define the isomorphism of subfactors in general ?

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 ?