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March 2, 2009

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 GG acting on a category XX, and let’s study the the stabilizer of some object xx in XX. 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)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)Stab(x) is a 2-group with the following objects and morphisms. An object of Stab(x)Stab(x) is an object gg of GG together with an isomorphism

a:gxx a: g x \to x

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

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

g,a:gxx g, a: g x \to x

to

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

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

a:gxx a: g x \to x

should equal the composite of

fx:gxgx f x: g x \to g' x

and

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

It really looks much prettier as a triangle!

With some work one makes Stab(x)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 GG is a weak 2-group and its action on XX is also weak - the general case. I also feel sure we get a 2-group homomorphism

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

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

But then if we look at the stabilizer of larger subcategories of XX, what does the counterpart of aa look like?

Posted at March 2, 2009 11:26 AM UTC

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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 2S_2 at each point.

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

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

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

Posted by: David Corfield on March 2, 2009 11:58 AM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

One way to approach such questions is to get the solution in terms of general abstract nonsense and then turn the crank in any given concrete realization and work out what it looks like in detail.

Here it seems the general situation is:

we are in some higher category CC (of 2-groups, say) which we can safely assume to be groupoidal (no non-invertible morphisms in sight) and in CC we have a morphism

f:KG. f : K \to G \,.

Now we are looking, it seems to me, for the thing of automorphisms of ff in the under-category KCK \downarrow C.

Suppose we have managed to say under-category KCK \downarrow C in our context, then the object Stab G(f)Stab_G(f) in question would be the suitable weak limit

Stab G(f) pt f pt f KC. \array{ Stab_G(f) &\to& pt \\ \downarrow && \downarrow^{f} \\ pt &\stackrel{f}{\to}& K \downarrow C } \,.

Posted by: Urs Schreiber on March 2, 2009 4:19 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

How, if at all, does this picture relate to Mike’s?

Posted by: David Corfield on March 3, 2009 10:33 AM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

I thought about this for a bit, and maybe I’m just being stupid, but I don’t even understand how what Urs wrote answers the question. Where is the object XX that GG is acting on? What is KK?

Posted by: Mike Shulman on March 3, 2009 4:15 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

I chose bad symbols due to not carefully reading the entry.

What I described was supposed to be the full automorphism 2-group of a category fixing a subcategory.

So my GG was your XX and my KK your {x}\{x\} and Mike’s YY. My ff is Mike’s ii.

So let me say it again, with different symbols:

let XX be a category and i:YXi : Y \to X a morphism, which can be regarded as an object of YCatY \downarrow Cat.

Notice that the endomorphism 2-monoid of XX itself is the weak pullback

AUT(X) pt X pt X Cat. \array{ AUT(X) &\to& pt \\ \downarrow && \downarrow^X \\ pt &\stackrel{X}{\to}& Cat } \,.

Similarly for the automorphism 2-group with CatCat replaced by the core of CatCat.

Analogously, the weak pullback

Stab(i) pt i pt i YCat \array{ Stab(i) &\to& pt \\ \downarrow && \downarrow^i \\ pt &\stackrel{i}{\to}& Y\downarrow Cat }

computes endo/auto–morphisms of XX that weakly fix YY.

Here, too, an object is a 2-cell

Y i i X X \array{ && Y \\ & {}^{i}\swarrow &\Rightarrow& \searrow^i \\ X &&\to && X }

etc.

Posted by: Urs Schreiber on March 3, 2009 4:49 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

You have to be careful here. The 3-categorical pullback

Aut(X) 1 X 1 X Cat\array{Aut(X) & \to & 1\\ \downarrow & & \downarrow ^X\\ 1 & \underset{X}{\to} & Cat}

is the automorphism 2-group of the category XX, meaning the auto-equivalences and isomorphisms between them. It doesn’t change if you replace CatCat by its core. The 3-categorical comma object

Core(End(X)) 1 X 1 X Cat\array{Core(End(X)) & \to & 1\\ \downarrow & \Downarrow & \downarrow^X\\ 1 & \underset{X}{\to} & Cat}

is the core of the endomorphism 2-monoid of XX: it consists of endofunctors XXX\to X and isomorphisms between them. Generally, comma objects in nn-categories for n>2n\gt 2 seem not to be quite right; see for instance the bottom of this page. This is currently tripping me up in my attempts to define 3-toposes.

If you want to recover the full endomorphism 2-monoid End(X)End(X) of endofunctors XXX\to X and all transformations between them, then you have to consider the universal 2-category equipped with a lax natural transformation

End(X) 1 lax X 1 X Cat\array{End(X) & \to & 1\\ \downarrow & \Downarrow_{lax} & \downarrow ^X\\ 1 & \underset{X}{\to} & Cat}

which doesn’t live in a 3-category but rather in the biclosed non-symmetric monoidal category (2Cat, Gray,lax)(2Cat,\otimes_{Gray,lax}) where Gray,lax\otimes_{Gray,lax} is the lax version of the Gray tensor product.

I’m still thinking about whether&why your definition of the stabilizer is the same as mine. They’ve definitely got the same intuition behind them.

Posted by: Mike Shulman on March 3, 2009 10:15 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

For amplification:

taking endomorphisms means forming loops:

for XX an object of an \infty-category CC we have generally

End C(X):=Ω XC, End_C(X) := \Omega_{X} C \,,

where on the right we have the based (at XX) loop space object, which is the weak pullback

Ω X(C) pt X pt X C \array{ \Omega_X(C) &\to& pt \\ \downarrow && \downarrow^X \\ pt &\stackrel{X}{\to}& C }

(you know this – but for the record I mention that this is described for instance in the examples at nnLab: homotopy limit )

And endomorphisms fixing a map i:YXi : Y \to X are endomorphisms of ii in YCY \downarrow C.

Posted by: Urs Schreiber on March 3, 2009 5:05 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

Let GG be an nn-group acting on an object XX of an nn-category CC (such as a 2-group acting on a category = object of the 2-category CatCat), and let i:YXi:Y\to X be a morphism (thought of as the “inclusion of a subobject”). Then I would define the stabilizer of ii under the action of GG to be the (nn-categorical) pullback Stab(i) 1 i G Aut C(X) i C(Y,X).\array{Stab(i) && \to && 1\\ \downarrow &&&&\downarrow i\\ G & \to & Aut_C(X) & \underset{-\circ i}{\to} & C(Y,X)}. in (n1)Cat(n-1)Cat. Thus, an object of Stab(i)Stab(i) consists of a gGg\in G and a 2-cell isomorphism giig i \cong i, a morphism of Stab(i)Stab(i) consists of h:ggh:g\cong g' in GG and a triangle commuting up to a 3-cell isomorphism, and so on.

Posted by: Mike Shulman on March 2, 2009 8:08 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

That looks good!

So now can we choose a topological category for XX with a nice continuous transitive action of GG, and a shape YY a subcategory of XX, so that the category of cosets G/Stab(i)G/Stab(i) (perhaps //) is equivalent to the functor category Fun(Y,X)Fun(Y, X)?

One thing that’s given me some concern is that GG acting on the morphisms of XX can’t be transitive, having to send identity maps to other such.

Posted by: David Corfield on March 3, 2009 8:56 AM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

What is the statement you are trying to categorify here?

Posted by: Mike Shulman on March 3, 2009 4:33 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

I’m trying to get us back on the chase of Klein 2-geometry. Here I’m trying to categorify homogeneous spaces. Things like the Euclidean group acting on the plane with the stabilizer of a point being O(2)O(2) and the coset space E(2)/O(2)E(2)/O(2) being isomorphic to the plane.

Posted by: David Corfield on March 3, 2009 4:50 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

I’m not sure when I’ll have the time to read all the stuff you guys have been doing with Klein 2-geometry. But this particular question sounds like a job for regularity and exactness.

Let me start by rephrasing the 1-categorical version. Let GG be an internal group object in a regular category KK, and let GG act on the object XKX\in K via a map act:G×XXact:G\times X\to X. Let’s say this action is transitive if

  1. XX has a global element x:1Xx:1\to X, and
  2. the map act x:G(1 G,x)G×XactXact_x:G\overset{(1_G,x)}{\to} G\times X \overset{act}{\to} X, which sends gg to gxg\cdot x, is a regular epi.

It’s not unreasonable to ask that XX have a global element, since any homogeneous space G/HG/H will have one, namely the image of the identity e:1Ge:1\to G. Here if HH is a sub-group-object of GG, then G/HG/H is the coequalizer G×HGG\times H \;\rightrightarrows\; G, on which GG clearly acts transitively in the usual way. This coequalizer will exist if we also assume that KK is exact, since G×HGG\times H \;\rightrightarrows\; G is an equivalence relation.

Now, if GG acts transitively on XX, then the regular epi act x:GXact_x:G\to X, like any regular epi in a regular category, is the quotient of its kernel pair. Intuitively, this kernel pair KG×GK\to G\times G consists of pairs (g 1,g 2)(g_1,g_2) such that g 1x=g 2xg_1 x = g_2 x. It’s easy to see that in fact we have KG×Stab(x)K\cong G\times Stab(x), where Stab(x)Stab(x) is the pullback of act x:GXact_x:G\to X along x:1Xx:1\to X; the isomorphism takes (g 1,g 2)(g_1,g_2) to (g 1,g 2g 1 1)(g_1,g_2 g_1^{-1}). Thus the quotient of KK is precisely the quotient G/Stab(x)G/Stab(x), so we have G/Stab(x)XG/Stab(x)\cong X whenever GG acts transitively on XX. And with a little more work we can show that transitive actions of GG are equivalent to subgroups, at least if KK is exact so that we have quotients. (In fact, KK being exact means that equivalence relations on an object GG are equivalent to regular epis out of GG, so all that remains is to identify those regular epis coming from transitive actions with those equivalence relations coming from subgroups.)

Now let’s push it up a notch. Let GG be an internal group object in a regular 2-category KK (you can think of K=CatK=Cat or GpdGpd). Thus, we have morphisms mult:G×GGmult:G\times G\to G, id:1Gid:1\to G, and inv:GGinv:G\to G satisfying the usual axioms up to coherent isomorphism. Clearly any group object in GpdGpd is a 2-group in the usual sense. The converse follows because we can construct the functor inv:GGinv:G\to G by choosing, for each object gGg\in G, an adjoint inverse inv(g)inv(g), and for a morphism γ:gg\gamma:g\to g' defining inv(γ)inv(\gamma) to be the composite inv(g)inv(g)ginv(g)1γ 11inv(g)ginv(g)inv(g). inv(g) \overset{\cong}{\to} inv(g) \cdot g' \cdot inv(g') \overset{1\cdot \gamma^{-1} \cdot 1}{\to} inv(g)\cdot g \cdot inv(g') \overset{\cong}{\to} inv(g'). Thus, 2-groups are the same as group objects in GpdGpd. I’m pretty sure they are also the same as group objects in CatCat as well; you can reverse the above argument to show that given the functor invinv, the category GG must be a groupoid. Have 2-group-theorists written this down already?

Moving on, suppose our group object GG acts on the object XKX\in K, and let’s say that this action is transitive if

  1. XX has a global element x:1Xx:1\to X, and
  2. The morphism act x:GXact_x:G\to X is eso.

Note that we’re only requiring the action to be transitive on objects. “Transitivity on arrows” turns out to be unnecessary, “because” the “2-inclusion” of a 2-subgroup is only required to be faithful, not full. In particular, if HGH\to G is a faithful 2-group homomorphism, then the (weak) quotient G//HG//H has a transitive action of GG in this sense, but it might not be transitive on arrows. As an extreme example, if G=1G=1 is the trivial (2-)group and HH is any 1-group, considered as a 2-group with only identity morphisms, then HGH\to G is faithful and its quotient G//HG//H is BHB H; and GG certainly doesn’t act transitively on the arrows of BHB H.

Now if GG acts transitively on XX, then the eso act x:GXact_x:G\to X, like any eso in a regular 2-category, is the quotient of its kernel. This kernel is the comma object

K G act x G act x X\array{K & \overset{}{\to} & G\\ \downarrow & \Downarrow & \downarrow act_x\\ G& \underset{act_x}{\to} & X}

In particular, in CatCat, an object of KK is a triple (g 1,g 2,α)(g_1,g_2,\alpha) where α:g 1xg 2x\alpha: g_1 x \to g_2 x is not necessarily invertible! Still, using the fact that GG is a group object, we can identify KK with the product G×Stab 2(x)G\times Stab_2(x) where now an object of Stab 2(x)Stab_2(x) is an object gGg\in G equipped with a not-necessarily-invertible morphism xgxx\to g x, and a morphism in Stab 2(x)Stab_2(x) is a morphism g 1g 2g_1\to g_2 in GG making the obvious triangle commute. Then XX being the quotient of KK means that we have a lax codescent diagram

G×Stab 2(x)×Stab 2(x) G×Stab 2(x) G X.\array{ G\times Stab_2(x) \times Stab_2(x)& \underoverset{\to}{\to}{\to}& G\times Stab_2(x) & \underoverset{\to}{\to}{\leftarrow} & G & \to & X. }

Thus, in a certain sense, we have XG//Stab 2(x)X\simeq G//Stab_2(x), but it doesn’t look quite like the usual weak quotient of GG by the action of Stab 2(x)Stab_2(x). Rather, it’s a sort of lax quotient in which we only glue in a morphism, rather than an isomorphism, from gg to ghg h for hStab 2(x)h\in Stab_2(x). This is connected with the fact that Stab 2(x)Stab_2(x) is not necessarily itself a 2-group! It’s a monoidal groupoid, but its objects need not be invertible. So perhaps for the purposes of Klein 2-geometry, a “2-subgroup” of a 2-group need not be itself a 2-group?

However, if XX is itself a groupoid, then the above comma object becomes a pullback, Stab 2(x)Stab_2(x) becomes a 2-group, and the lax quotient becomes an ordinary (weak) quotient. So in the case of a 2-group acting on a groupoid, things seem to work out just as we would expect, but if we act on a category then some laxness creeps in. In either case, I guess that if KK is also exact, then the appropriate type of “2-subgroups” can be shown to be equivalent to transitive actions.

Finally, we can guess a generalization of this to \infty-groups acting on \infty-groupoids. The action of GG on XX should be transitive if XX has a point x:1Xx:1\to X and the map act x:GXact_x:G\to X is surjective on π 0\pi_0, and in this case we should have XG//Stab (x)X\simeq G//Stab_\infty(x), setting up an equivalence between transitive actions of GG and “\infty-subgroups.” But now, an “\infty-subgroup” just means any \infty-group homomorphism HGH\to G.

Posted by: Mike Shulman on March 4, 2009 9:06 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

Wow, excellent!

What happens in the case of Lie groups? In your description at the 1-categorical level, you required KK to be regular. The category of manifolds isn’t regular, is it? But Klein geometry works there.

A Klein Geometry requires a Lie group and closed Lie subgroup. Can we just ask for a closed Lie sub-2-group at the 2-level?

Do we have a Cartan’s 2-theorem: every closed sub-2-group of a Lie 2-group is a Lie sub-2-group?

Posted by: David Corfield on March 5, 2009 9:13 AM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

No, of course the category of manifolds isn’t regular or exact, and doesn’t even have finite limits. But looking at the argument I gave, to show that a transitive GG-object is the quotient of GG by its stabilizer, I think all you really need is that the action map act x:GXact_x:G\to X has a kernel pair of which it is the quotient. For that, it’s probably good enough for act xact_x to be a surjective submersion. Presumably that always happens when the action is transitive on underlying sets.

Going in the other direction, to make a homogenous space G/HG/H you just need the equivalence relation G×HG×GG\times H \to G\times G to be effective. I’d guess that HH being closed is exactly what you need to make that true, probably partly because of Cartan’s theorem.

I’ll leave the other questions to the Lie 2-group theorists. But one way to avoid them entirely would be to work in the 2-category of stacks on the category of manifolds, which is certainly regular and exact.

Posted by: Mike Shulman on March 5, 2009 4:28 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

What’s a nice nontrivial example of an \infty-group?

Posted by: David Corfield on March 5, 2009 10:38 AM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

…setting up an equivalence between transitive actions of G and “\infty-subgroups.” But now, an “\infty-subgroup” just means any \infty-group homomorphism HGH \to G.

So take the fundamental \infty-groups of the 2-sphere and the 3-sphere. Each of these is an \infty-subgroup of the other (induced by inclusion and Hopf fibration)? Are either of π (S 3)//π (S 2)\pi_{\infty}(S^3)//\pi_{\infty}(S^2) or π (S 2)//π (S 3)\pi_{\infty}(S^2)//\pi_{\infty}(S^3) interesting? I would imagine that the former is an \infty-groupoid equivalent to the fundamental \infty-groupoid of the circle.

Or do we have to be more careful working in Top?

Posted by: David Corfield on March 5, 2009 12:14 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

Well, whatever π (S 3)//π (S 2)\pi_\infty(S^3)// \pi_\infty(S^2) is, it’s a transitive π (S 3)\pi_\infty(S^3)-space. Do you have an action of π (S 3)\pi_\infty(S^3) on S 1S^1 in mind?

Classical homotopy theorists would write π (S 3)//π (S 2)\pi_\infty(S^3)// \pi_\infty(S^2) as a bar construction B(*,ΩS 2,ΩS 3)B(*,\Omega S^2,\Omega S^3). I feel like there may be more to say here, but I haven’t thought of it yet.

Posted by: Mike Shulman on March 5, 2009 5:11 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

Okay, got it.

Via Galois theory, when AA is connected, spaces with an action of π (A)\pi_\infty(A) can be identified with fibrations over AA. The π (A)\pi_\infty(A)-space corresponding to a fibration is, guess what, its fiber.

Now given any map f:ACf:A\to C, we can pull back fibrations over CC to fibrations over AA; this corresponds to restricting an action of π (C)\pi_\infty(C) to an action of π (A)\pi_\infty(A). We can also compose with ff to turn a fibration over AA into a fibration over CC. (Well, it won’t necessarily be a fibration, but we can just replace it by one.) This corresponds to the (,1)(\infty,1)-categorical left adjoint to restriction along π (f)\pi_\infty(f), which we might call (Kan) “extension” or “induction;” a homotopy theorist would write it as a bar construction B(ΩC,ΩA,)B(\Omega C, \Omega A, -). (There’s also a right adjoint, but we won’t need it.) A careful study of the relationships between these equivalences and transition functors can be found in my paper Parametrized spaces model locally constant homotopy sheaves (arXiv).

Now, the quotient π (C)//π (A)\pi_\infty(C)//\pi_\infty(A), written by the homotopy-theorist as B(,ΩA,ΩC)B(\star, \Omega A, \Omega C), is constructed by starting with π (C)\pi_\infty(C) as a π (C)\pi_\infty(C)-space, restricting it to a π (A)\pi_\infty(A)-space, then quotienting by the π (A)\pi_\infty(A)-action, i.e. extending forward along the projection π (A)\pi_\infty(A)\to \star to the trivial \infty-group.

Translating over into the world of fibrations, the fibration corresponding to π (C)\pi_\infty(C) as a π (C)\pi_\infty(C)-space is the path fibration PCCP C \to C, where PCP C is the space of paths starting at the basepoint c 0Cc_0\in C, and the map PCCP C \to C projects to the endpoint of the path. The fiber of this map, over the basepoint, is the loop space ΩC\Omega C of paths starting and ending at the basepoint, so that makes sense. We now pull back this fibration along f:ACf:A\to C, and what we get is the space of pairs (a,γ)(a,\gamma) where aAa\in A and γ\gamma is a path in CC from f(a)f(a) to the basepoint c 0c_0; in other words, it’s just the homotopy fiber of ff over the basepoint, equipped with its obvious projection to AA. Next we compose with the projection AA\to \star from AA to a point, which doesn’t change the space but just forgets the projection to AA. Now we have a fibration over \star, i.e. just a space. We then have to transform this fibration over \star into a space with an action by π ()=\pi_\infty(\star)=\star by taking its fiber; but of course this does nothing.

In conclusion, for any map f:ACf:A\to C of connected based spaces, the (,1)(\infty,1)-categorical quotient π (C)//π (A)\pi_\infty(C)//\pi_\infty(A) is just the homotopy fiber of ff. And if ff is itself a fibration, this is equivalent to its ordinary fiber. The \infty-group π (C)\pi_\infty(C) acts on this in the standard way, by transporting along paths in CC; the action is transitive because AA is connected.

So, if we start with the Hopf fibration S 3S 2S^3\to S^2, then the quotient π (S 2)//π (S 3)\pi_\infty(S^2)//\pi_\infty(S^3) is just the fiber S 1S^1. I read what you wrote as saying that π (S 3)//π (S 2)\pi_\infty(S^3)//\pi_\infty(S^2) should be S 1S^1, but maybe this is actually what you meant; if so, you’re right! The other one π (S 3)//π (S 2)\pi_\infty(S^3)//\pi_\infty(S^2) is the homotopy fiber of the inclusion S 2S 3S^2\to S^3, whatever that is.

In the other direction, suppose I start with a connected based space CC and a space XX having a transitive action of π (C)\pi_\infty(C), and I want to find a map f:ACf:A\to C of connected based spaces such that Xπ (C)//π (A)X\simeq \pi_\infty(C)//\pi_\infty(A), or equivalently that XX is the homotopy fiber of ff. To use the above approach, I should start by translating XX into a fibration over CC—but then I’m already done! This is a fibration with fiber XX, and its total space is connected since the action is transitive.

We can sort of give an explicit description of that total space: if XX has an action of ΩC\Omega C, we can make the bar construction B(,ΩC,X)B(\star,\Omega C, X). This isn’t exactly a fibration over CC, but it comes with a map to B(,ΩC,)=BΩCB(\star, \Omega C, \star) = B\Omega C, which is equivalent to CC (since CC is connected), and its fiber is indeed XX. Presumably we could trace through the equivalences and figure out why this looks like the “stabilizer” of a point in XX.

Posted by: Mike Shulman on March 5, 2009 6:22 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

It may seem weird that we get the quotient π (C)//π (A)\pi_\infty(C)//\pi_\infty(A) by taking a fiber of f:ACf:A\to C, since fibers are dual to quotients. At least, it seems weird to me! But we can reassure ourselves by dropping back to low dimensions.

Let GG be an ordinary discrete 1-group and let i:HGi:H\hookrightarrow G be a subgroup. Now the claim is that we can recover G/HG/H as the essential fiber of the map Bi:BHBGB i:B H \to B G of groupoids. Of course, now I’m using BGB G for the groupoid with one object \star and BG(,)=GB G (\star,\star) = G.

Well, in general, for a map f:ACf:A\to C of groupoids, an object of the essential fiber over c 0Cc_0\in C is an object aAa\in A and an isomorphism f(a)c 0f(a)\cong c_0, and a morphism in the essential fiber is a morphism aaa\to a' in AA making the obvious triangle commute in CC. For Bi:BHBGB i:B H \to B G, this means that an object of the essential fiber over BG\star\in B G is the unique object BH\star \in B H together with an isomorphism Bi()B i(\star) \cong \star—in other words, an element of GG. And a morphism in the essential fiber from g 1g_1 to g 2g_2 will be a morphism \star\to\star in BHB H—that is, an element of HH—such that g 1h=g 2g_1 h = g_2 in GG. Thus, the essential fiber is equivalent to the discrete set G/HG/H, which is what we wanted.

This still feels like magic to me.

Posted by: Mike Shulman on March 5, 2009 7:07 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

Let GG be an ordinary discrete 1-group and let i:HGi : H \hookrightarrow G be a subgroup. Now the claim is that we can recover G/HG/H as the essential fiber of the map Bi:BHBG\mathbf{B} i : \mathbf{B}H \to \mathbf{B}G of groupoids.

Yes, this is a nice fact. Let me just remark that we have discussed this here from time to time in slightly different but equivalent language:

As described at homotopy limit and at generalized universal bundle one way to compute the essential fiber of a morphism g:XBGg : X \to \mathbf{B}G by an ordinary pullback is to pull back the universal GG-bundle E ptG\mathbf{E}_{pt} G in its \infty-groupoid incarnation.

Dually, for SS any pointed object and ρ:BHS\rho : \mathbf{B}H \to S a morphism of pointed objects, the homotopy fiber is the pullback of E ptS\mathbf{E}_{pt} S and computes the action groupoid pt S//Hpt_S//H of HH acting on the point of SS, as described there.

The situation you just mentioned is the combination of these two cases: that where we have a morphism BHBG\mathbf{B}H \to \mathbf{B}G of groups:

here the homotopy fiber is both:

a) the GG-bundle associated to the universal HH-bundle by this morphism in its \infty-groupoid incarnaton;

- the action groupoid G//HG//H.

Posted by: Urs Schreiber on March 5, 2009 8:53 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

Very neat! I didn’t see those connections, but of course you’re right. It still feels like magic to me that we compute a quotient by computing a fiber, but at least now I see that it’s an instance of a familiar general phenomenon.

Posted by: Mike Shulman on March 5, 2009 11:40 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

It still feels like magic to me that we compute a quotient by computing a fiber, but at least now I see that it’s an instance of a familiar general phenomenon.

Yes, indeed. Just for the record, some may recall that a while ago I had been wondering about this in the context of not \infty-groupoids but their infinitesimal version, L L_\infty-algebroids.

I wanted to get a good definition of \infty-action of one L L_\infty-algebroid gg on another hh.

The direct way to do this would be to define something like an L L_\infty-algebroid end(h)end(h) of hh and then look at L L_\infty-morphisms gend(h)g \to end(h).

But it turns out that defining end(h)end(h) in the right way is at best painful. Well, I wouldn’t be shocked if somebody presents me a nice solution, but in all the discussion we had about this here no really good general such seems to have surfaced.

So at some point I decided to go the other route and instead of defining the action directly, define the fibration that it should induce, i.e. the “action L L_\infty-algebroid” and let the action L L_\infty-algebroid define the action instead of the other way round.

This is described in p. 9, section Actions and their homotopy quotients of On \infty-Lie, more details from p. 21 on (from a talk I once gave).

Later it turned out that the same definition, but in in rather different language, was given by Jonathan Block. I talk about that in the entry Block on L L_\infty-module categories.

(With Jim Stasheff and Hisham Sati we have an almost finished article where we apply this kind of thing to L L_\infty-connections to reproduce various formulas known in String theory. Much to the infuration of at least Jim I kept postponing finishing this article because I felt I needed to better understand the homotopy-theoretic abstract nonsense in which it lives. It seems I am getting closer…)

Posted by: Urs Schreiber on March 6, 2009 1:33 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

Oh yes, the wrong way round.

You’d kinda want π (S 3)//π (S 2)\pi_{\infty}(S^3)//\pi_{\infty}(S^2) to be an ‘inverse’ to S 1S^1.

I think I need to spend some time reading all your interesting comments.

Posted by: David Corfield on March 5, 2009 7:46 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

Sorry to keep adding more while probably no one else has yet digested what I’m saying, but this is too nice not to share. I just realized that the equivalence between GG-actions and spaces over BGB G is not limited to classical homotopy theory; it’s true in any exact (,1)(\infty,1)-category.

In an exact 1-category KK, regular epis are automatically descent morphisms, meaning that XK/XX\mapsto K/X is a stack for them. That means that for an equivalence relation RA×AR\to A\times A, objects over the quotient A/RA/R are equivalent to objects over AA with an “action” by RR, which amounts to being constant along equivalence classes.

Now, the same should be true in an exact (,1)(\infty,1)-category, for the quotient of a groupoid. But recall what I said here about the construction of BGB G from a group object GG in any exact (,1)(\infty,1)-category. That means that 1BG1\to B G is a descent morphism, so that objects over BGB G can be identified with objects over 11 (that is, just objects) having an action by GG. But this is precisely what we wanted.

That means that my argument above that identifies the quotient G//HG//H with the fiber of BHBGB H \to B G should also work in any exact (,1)(\infty,1)-category. In particular, it should work in the (,1)(\infty,1)-category of stacks of \infty-groupoids on the category of manifolds.

Posted by: Mike Shulman on March 5, 2009 8:27 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

That means that my argument above that identifies the quotient G//HG//H with the fiber of BHBG\mathbf{B}H \to \mathbf{B}G should also work in any exact (,1)(\infty,1)-category. In particular, it should work in the (,1)(\infty,1)-category of stacks of \infty-groupoids on the category of manifolds.

Compare section 2.2 of

J. F. Jardine, Cocycle categories

which has a proof of this kind of statement for \infty-stacks with \infty-stacks modeled by simplicial presheaves.

Posted by: Urs Schreiber on March 5, 2009 9:03 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

Excellent! I also like seeing it as a consequence of exactness, though.

Posted by: Mike Shulman on March 6, 2009 1:39 AM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

What’s a nice nontrivial example of an \infty-group?

For a topological space XX, let ΠX\Pi X be its fundamental \infty-groupoid. For any point xXx \in X we have the \infty-group Π(X)(x,x)\Pi (X) (x,x) of automorphisms of xx.

This Π(X)(x,x)\Pi(X)(x,x) is an \infty-group which subsumes all the ordinary homotopy groups π n(X,x)\pi_n(X,x) of XX at xx, together with all their actions and interrelations on each other.

This is the canonical example but also a bit tautological: equivalently one could say: an \infty-groups is a connected topological space.

More generally: for any \infty-groupoid GG and every object aGa \in G the thing G(a,a)G(a,a) is an \infty-group.

That, too, is a bit tautological, of course.

Posted by: Urs Schreiber on March 5, 2009 2:31 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

Old-fashioned homotopy theorists would call Π(X)(x,x)\Pi(X)(x,x) the loop space ΩX\Omega X of the based space (X,x)(X,x). And in fact, every example is of this form: given any \infty-group GG there is a space BGB G such that GΩBGG\simeq \Omega B G. This is a very classical theorem in homotopy theory, although classical homotopy theorists would say “grouplike A A_\infty-space” rather than “\infty-group.”

It’s also a special case of the statement that the (,1)(\infty,1)-category of spaces (= \infty-groupoids) is exact: an \infty-group GG is a group object in spaces, hence G1G \;\rightrightarrows\; 1 is a groupoid object, and so (by exactness) it is the kernel of some map 1BG1\to B G, i.e. we have a pullback G 1 1 BG\array{G & \overset{}{\to} & 1\\ \downarrow && \downarrow\\ 1& \underset{}{\to} & B G} which says exactly that GΩBGG\simeq \Omega B G.

Posted by: Mike Shulman on March 5, 2009 4:35 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

So is the lesson that infinitely categorified Klein geometry (at least the (\infty, 1) variety) is just homotopy theory for a pair of connected spaces with base points and a mapping between them?

Double cosets are next on the agenda.

Posted by: David Corfield on March 5, 2009 4:50 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

Yes, I think so. That shouldn’t really be a surprise; most (,1)(\infty,1)-category theory is just a new language for homotopy theory. Although, like any new language, it does suggest new ways of looking at old things. I doubt that homotopy theorists ever thought of a map between connected based spaces as analogous to a Klein geometry.

Posted by: Mike Shulman on March 5, 2009 5:27 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

In light of what I said above, it might be more accurate to say that Klein (,1)(\infty,1)-geometry is the homotopy theory of pairs of connected based spaces and fibrations between them—the corresponding homogeneous space then being the fiber of the fibration.

Of course, things get more complicated when you want to add in a smooth structure. To do that in homotopy theory, you’d probably have to think about, say, simplicial (pre)sheaves on the category of manifolds.

Posted by: Mike Shulman on March 5, 2009 8:15 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

David wrote:

infinitely categorified Klein geometry […] is just homotopy theory

Mike replied

That shouldn’t really be a surprise; most (,1)(\infty,1)-category theory is just a new language for homotopy theory

Maybe a remark on this point:

Saying that

Working in the (,1)(\infty,1)-category of (,0)(\infty,0)-categories (= \infty-groupoids) is nothing but working with topological spaces.

is the analog of saying

Working in the 11-category of 0-categories (= sets) is nothing but doing set theory.

In both cases one can ask: So what’s the point? Why do we talk about (\infty-)categories and not just about sets and spaces.

The answer is both cases is: because the category theory allows us to consider parameterized sets and parameterized spaces: we have the Yoneda lemma and (pre)sheaves of sets and (pre)sheaves spaces, hence (Grothendieck) topoi and (Rezk-Lurie) \infty-topoi.

This allows us to do things that set theorists and homotopy theorists can not do with plain sets and plain topological spaces: add extra structure, for instance smooth structure, by considering presheaves over suitable test objects.

So a central point is that there are (presheaf) \infty-topoi. Whence the title of Lurie’s book, I suppose.

(E.g. his remarks at the beginning of section 5.1).

Posted by: Urs Schreiber on March 6, 2009 11:06 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

Yes!

Of course, set theorists and homotopy theorists do study parametrized things and add extra structure. Set theorists call it “forcing.” Homotopy theorists do “equivariant homotopy theory” and “parametrized homotopy theory” and talk about simplicial (pre)sheaves. The merits of the categorical approach are that we have a unified framework in which to do all of these things, and that when we phrase classical things correctly in the categorical language, we don’t have to re-prove them in all the other situations. And probably others.

Posted by: Mike Shulman on March 7, 2009 4:08 AM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

Ahem

Just note that the link at that comment to the ‘notes’ has moved to here, due to departmental server migration.

Posted by: David Roberts on March 11, 2009 1:38 AM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

As an added inducement to get you onboard, Mike, with a categorified Erlanger Programme we can categorify Tarski’s What are Logical Notions?, using what Todd was telling us about here and here.

Posted by: David Corfield on March 3, 2009 4:57 PM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

I’m glad we’re taking up this theme again. I’m pretty tired, so all I have to say tonight is: “a moment of insight near the Nine Zigzag Bridge in Shanghai” sounds pretty cool.

Cooler than if you also know I was in a Starbucks packed with tourists trying to recover from the heat.

Posted by: John Baez on March 3, 2009 6:45 AM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

What sort of object do we expect the stabiliser to be? Certainly the stabiliser of a point is a sub-2-group (i.e. a faithful functor to the group), but can we allow more general things? Here is a silly idea that comes from just copying my own private definition of the stabiliser of a point, replacing the point with something more general.

Let GG be a 2-group, i:YXi:Y \to X a sub-thingie of the groupoid XX, which has an action a:X×GXa:X \times G \to X. I guess we would want ii to be faithful.

Consider the weak pullback i(a| Y)i\downarrow (a\big|_Y) of ii along the composite a| Y:Y×Gi×idX×GaX. a\big|_Y:Y \times G \stackrel{i\times id}{\to} X \times G \stackrel{a}{\to} X.

This admits the structure of a double groupoid where one of the directions is as weak as the 2-group (so strict 2-group gives a double groupoid, and a weak 2-group gives a pseudo double groupoid). There is a functor jj from i(a| Y)i\downarrow (a\big|_Y) to GG given by the composite of the canonical arrow to Y×GY\times G and projection to GG. I think that this double groupoid has connections (at least in the strict case) so is the ‘same as’ a 2-groupoid (and bigroupoid in the non-strict case, once we know that this would even work. Has it been done anywhere in the literature?) In the case that Y=*Y=\ast, the double groupoid reduces to a 2-group.

How much jj could be thought of as a sub-2-group of GG is debatable, but some homotopy-theoretic considerations may be able to guide these ill-formed thoughts into something sensible.

Posted by: David Roberts on March 4, 2009 1:32 AM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

I don’t understand. The pullback you mention presumably takes place in GpdGpd, so it will be a groupoid. Where does the double groupoid come from?

Posted by: Mike Shulman on March 5, 2009 3:00 AM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

Apologies for the delay, public holiday here in Australia, plus the one day a week I’m not at uni. All of what is below may be less than optimal to the problem at hand, but the structure seems interesting.

Where does the double groupoid come from?

The pullback admits the structure of a double groupoid as follows:

Recall: we have a faithful functor i:YXi:Y \to X between strict GG-groupoids for GG a 2-group (GG strict for now, as well as ii, so it is really equivariant). The weak pullback has as objects

(y;γ;y,g)Y 0×X 1×Y 0×G 0(y;\gamma;y',g) \in Y_0\times X_1 \times Y_0 \times G_0

such that γ:i(y)i(y)g\gamma:i(y) \to i(y')\cdot g. Morphisms between (y 1;γ 1;y 1,g 1)(y_1;\gamma_1;y_1',g_1) and (y 2;γ 2;y 2,g 2)(y_2;\gamma_2;y_2',g_2) are triples (a;a,h)Y 1×Y 1×G 1 (a;a',h) \in Y_1 \times Y_1 \times G_1 such that γ 2i(a)=i(a)hγ 1\gamma_2 \circ i(a) = i(a')\cdot h \circ \gamma_1 It helps now to draw the commuting square I could not (Oh when will I learn to do diagrams in here?), with the aa’s vertical.

The objects of the pullback groupoid become horizontal arrows, we let the objects of the double groupoid be just the objects of YY and the arrows of the pullback groupoid become the squares in the double groupoid.

Then vertical composition of squares is just composition in the pullback groupoid (componentwise composition of (a;a,h)(a;a',h)’s). The horizontal composition is akin to semidirect product multiplication.

The horizontal composition of (y;γ;y,g)(y;\gamma;y',g) and (y;δ;y,k)(y';\delta;y'',k) is (y;δgγ;y,kg). (y;\delta\cdot g \circ \gamma;y'',kg). That this is associative uses strictness of GG, ii and the action. I haven’t thought seriously about precisely to say what thing I get for weak GG etc. A pseudo double groupoid maybe.

It may be worth noting that the vertical category in this double category is just YY.

There is a functor to GG, which is considered as a double category as follows: There is only one object, the vertical arrows are trivial, the horizontal arrows are the objects of GG, and the squares are the arrows of GG. Clearly this is the 2-groupoid with one object corresponding to GG considered as a double groupoid in the usual way.

My claim above that this has connections is obviously wrong, as it isn’t edge symmetric.

Now to go back and read all of the above comments…

Posted by: David Roberts on March 11, 2009 1:37 AM | Permalink | Reply to this

Re: The Stabilizer of a Subcategory

Interesting. That double groupoid is sort of a “GG-equivariant kernel” of i:YXi:Y\to X. In particular, although it doesn’t have connections, it does have companions and conjoints for vertical arrows (which are, of course, the same in a double groupoid). And if Y=XY=X, its quotient (aka its reflection into groupoids, left adjoint to forming the double groupoid of commutative squares in a groupoid) is the weak quotient X//GX//G.

Just musing. I’m not sure how this fits into the picture.

Posted by: Mike Shulman on March 11, 2009 3:46 AM | Permalink | Reply to this

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