The category of subfactors extending the category of groups?

Question

This post was inspired by this answer of Dave Penneys.

In the category of (irreducible hyperfinite II$_1$) subfactors, the morphisms of $(N \subset M)$ to $(N' \subset M')$ are usually defined as the $W^*$-morphisms $\phi: M \to M'$ with $\phi (N) \subset N'$.

Unfortunately, through this definition, the category of finite group is $\underline{not}$ a (natural) subcategory of the category of subfactors.
In fact, let $G$ and $G'$ be finite groups and $f: G \to G'$ be a surjective group-morphism, then in general, $f$ does $\underline{not}$ extend into a (usual) subfactor-morphism of $(R \subset R \rtimes G)$ to $(R \subset R \rtimes G')$.

Here is the explanation in the answer of Dave:
A II$_1$-factor is algebraically simple, so each morphism of II$_1$-factors is either injective or zero.
Thus every non-zero morphism is an isomorphism onto its image.
I don't think the canonical surjection $G\to G'=G/\ker(f)$ actually gives you a map of factors $R\rtimes G\to R\rtimes G'$. In particular, if we denote the implementing unitaries as $u_g$ for $g\in G$, the map $u_g\mapsto u_{g\ker(f)}$ does not extend to a non-zero map of II$_1$-factors if $\ker(f)$ is non-trivial. The element $u_g-u_{g'}$ would map to zero if $g,g'\in \ker(f)$, and a non-trivial map of II$_1$-factors must be injective.

Question: Is there an $\underline{other}$ (natural) definition of subfactor-morphisms such that the category of finite groups is a (natural) subcategory of this "new" category of subfactors ?

Answer 1

Take $C$ to be the category of dualizable $N$-$N$-bimoduls, $N$ a factor. A subfactor $N\subset M$ (or $N_0\subset N$) with finite index and finite depth gives an algebra object $A$ in $C$, namely $A={}_NM_N$ (or ${}_NL^2M_N$ if you prefer) and conversely an algebra object (more precisely a Q-system) gives a subfactor $N\subset M$. Instead of building artificially a category of subfactors, you take the category of (simple) algebra objects (Q-systems) in $C$. Each finite group gives an object $A_G=\bigoplus_{g\in G} {}_NN^{\circ\alpha_g}_N$ with $\alpha_g$ automorphisms on $N$ such that $\alpha_g\alpha_h=\alpha_{gh}$ for $g,h\in G$ and ${}_NN^{\circ\alpha_g}_N$ is ${}_NN_N$ seen as a $N$-$N$ bimodule, where the right action is composed with $\alpha$. A morphism $H\to G$ gives a morphism $A_H\to A_G$ between algebra object. This category contains also finite groups, their duals, Kac-algebras and weak-C${}^\ast$ Hopf algebras.

If you want irreducible subfactors, you ask $A$ to be haploid, then you lose weak-C${}^\ast$ Hopf algebras.

This also tells you how a "category of subfactors" should work...

Answer 2

This is an artificial answer, I'm looking for something more natural.

In this paper, T. Teruya introduced the notion of normal intermediate subfactors, generalizing exactly the notion of normal subgroups (see the post Jordan-Hölder theorem for subfactors for more details).

So we can generalize the group-morphisms to the subfactors as follows :
A (group-like) morphism for $(A \subset B)$ to $(C \subset D)$ is the data of:

• a normal intermediate subfactor $(A \subset P \subset B)$
• an intermediate subfactor $(C \subset Q \subset D)$
• a $W^*$-isomorphism $\phi_l : (A \subset P) \to (Q \subset D)$ or $\phi_r : (P \subset B) \to (C \subset Q)$

Remark: This notion generalizes by construction the group-morphisms, unfortunately, it's a bit artificial, I would prefer a more natural definition of morphisms, without using 'ad hoc' the notion of normal intermediate subfactors, but such that the kernel of these natural morphisms are exactly the normal intermediate subfactors.