### The Stabilizer of a Subcategory

#### Posted by David Corfield

A very long time ago, John gave us a definition of a stabilizer of an object in a category on which some 2-group is acting, having had a moment of insight near the Nine Zigzag Bridge in Shanghai.

I was desperately trying to understand sub-2-groups. So, I thought: in Klein geometry, the conceptual meaning of “subgroup” is really “stabilizer of some point in a set on which a group acts”.

So, let’s take a 2-group $G$ acting on a category $X$, and let’s study the the stabilizer of some object $x$ in $X$. Whatever this stabilizer is like, maybe this should become the definition of a sub-2-group!

(Or, maybe not - there are also stabilizers of things more complicated and interesting than a mere object. But never mind! - it’s still an interesting exercise.)

Of course we need to define the stabilizer, say $Stab(x)$. There’s an obvious way to do this if you’re careful not to be evil. I’ll just sketch it.

The stabilizer $Stab(x)$ is a 2-group with the following objects and morphisms. An object of $Stab(x)$ is an object $g$ of $G$ together with an isomorphism

$a: g x \to x$

Nota bene: we’re not evilly demanding that $g x = x$; we’re specifying an isomorphism between them!

A morphism of $Stab(x)$, say from

$g, a: g x \to x$

to

$g', a': g' x \to x$

is a morphism $f: g \to g'$ in $G$ making the obvious triangle commute. Namely,

$a: g x \to x$

should equal the composite of

$f x: g x \to g' x$

and

$a': g' x \to x.$

It really looks much prettier as a triangle!

With some work one makes $Stab(x)$ into a 2-group - I didn’t check everything here, but I’m following the tao of mathematics so I’m sure everything works, even when $G$ is a weak 2-group and its action on $X$ is also weak - the general case. I also feel sure we get a 2-group homomorphism

$i: Stab(x) \to G.$

Presumably we can think of what John’s doing here as finding the stabilizer of the subcategory of $X$ composed of $x$ and its identity morphism. What I’m not so clear about is which $a$, isomorphisms between $g x$ and $x$, we’re allowed. Presumably just any such isomorphism in $X$, and not just those coming from restrictions of natural equivalences arising from the action of $G$.

But then if we look at the stabilizer of larger subcategories of $X$, what does the counterpart of $a$ look like?

## Re: The Stabilizer of a Subcategory

Let me see if I have things right. Consider the category (with structure) whose objects are points on the Euclidean plane, and with a vertex group $S_2$ at each point.

There’s a 2-group acting on this whose objects are $E(2)$ and whose morphisms again make up a vertex group $S_2$ at each point.

The Baezian stabilizer of a point in the plane has as objects $O(2) \times S_2$, and for each element, $\phi$ of $O(2)$, and each $g$ and $h$ in $S_2$ there is a single arrow between $\langle \phi, g \rangle$ and $\langle \phi, h \rangle$.

So the stabilizer is equivalent to $O(2)$ with trivial morphisms, and the quotient of the original 2-group by the stabilizer is the plane with internal $S_2$ symmetry, just as we started with.