This post was inspired by [this answer](https://mathoverflow.net/questions/155950/isomorphism-theorem-for-subfactors/155968#155968) of Dave Penneys. 


In the category of (irreducible hyperfinite II$\_1$) [subfactors](http://en.wikipedia.org/wiki/Subfactor), 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 ?
> 
> 
> 



The category of subfactors extending the category of groups?
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...


