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 $M \not\simeq N$**) :

- $(M \subset M) \sim (N \subset N) $
- $(P \bar\otimes M \subset Q \bar\otimes M) \sim (P \bar\otimes N \subset Q \bar\otimes N) $
- $[(P \subset Q) \sim (R \subset S) ]$ $\Leftrightarrow$ $[(P \bar\otimes M \subset Q \bar\otimes M) \sim (R \bar\otimes N \subset S \bar\otimes N) ]$
- $(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 ?