Klein 2-Geometry X

April 15, 2008

Posted by David Corfield

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Our early work on Klein 2-geometry led us to think about the action of discrete 2-groups. Later we got thinking about actions on vector bundles, seen as categories with abelian vertex groups $ℝ^{n}$ .

Perhaps the odd thing was why, if we knew we’d need to consider symmetries of groupoids, we never looked more closely at symmetries of a group, i.e., $AUT (G)$ for a group $G$ .

One important issue is that $AUT (G)$ doesn’t act transitively on $G$ , which takes us away from the Kleinian outlook. On the other hand, we could look at its action on the lattice of subgroups of $G$ , where there will be transitive actions on orbits in the lattice. No doubt we ought to be careful, however, categorifying transitivity of action.

We should also think about the lattice of subgroups for powers of $G$ . And perhaps also the action of $AUT (G)$ on the category of representations of $G$ .

Posted at April 15, 2008 11:23 AM UTC

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Re: Klein 2-Geometry X

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If you think of

AUT (G)

as the symmetries of the stack

[* / G]

then

AUT (G)

does act transitively on the stack. It may be worth it to think of isometries of compact Riemannian orbifolds. If any natural geometric object has 2-groups as symmetries, surely these must be natural examples.

Posted by: Eugene Lerman on April 16, 2008 2:20 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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Yes, orbifolds seem like excellent candidates. And we have a nice account of Orbifolds as Groupoids by Moerdijk. Then, as you say, we can look at mappings preserving geometric structures on the orbifolds.

Would you happen to know if people look at the 2-group of symmetries of an orbifold? If an orbifold is a groupoid, it would seem to be the obvious thing to do.

By the way, a chapter on orbifolds from The Geometry and Topology of Three-Manifolds by Thurston is availble here. I see a manifold with boundary may be thought of as an orbifold – two copies of the manifold quotiented by an action of $ℤ_{2}$ .

Posted by: David Corfield on April 18, 2008 10:56 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

I see the answer to my question

Would you happen to know if people look at the 2-group of symmetries of an orbifold?

is “Yes, me!”.

From the abstract of your Orbifolds as a localization of the 2-category of groupoids

In our construction the spaces of 1- and 2-arrows admit natural topologies, the space of morphisms (1-arrows) between two orbifolds is naturally a groupoid and the symmetries of an orbifold form a strict 2-group.

Now, I find I’ve referred to this paper of yours before.

Posted by: David Corfield on April 18, 2008 11:22 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

In many ways the paper you refer to is a mistake and it is embarrassing to explain why. One problem is that it’s really hard in that construction to get your hands on 2-arrows. Another is that the 2-category constructed is some sort of a quotient of the category of Deligne-Mumford stacks over manifolds and not a particularly nice one. It looks like the DM stacks are a good way to think about orbifolds (if not the way) but don’t look for a paper explaining why.

Posted by: Eugene Lerman on April 19, 2008 3:17 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

I am not entirely sold on Moerdijk’s approach to orbifolds. For one thing his category of orbifolds is a 1-category. So hom’s are sets and have no chance of being groupoids/orbifolds themselves. Another problem is that morphisms in his category are not local, as pointed out by Heriques and Metzler. For instance sections of vector bundles over orbifolds in his setup will not form a sheaf.

Thurston thinks of orbifolds as metric spaces with particularly nice singularities. So no higher category theory there.

Posted by: Eugene Lerman on April 19, 2008 3:10 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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For one thing his category of orbifolds is a 1-category.

I don’t have the details of his presentation present. But what would prevent us from picking our favoite closed category of smooth spaces and then take a smooth orbifold to be nothing but a groupoid internal to smooth spaces. These live in the obvious 2-category.

For instance sections of vector bundles over orbifolds in his setup will not form a sheaf.

Could you expand on that? What’s a sheaf on an orbifold for you?

I’d proceed like this: with the orbifold $X$ taken to be a groupoid internal to my favorite closed category of smooth spaces, a smooth vector bundle over it is a locally smoothly trivializable functor $V : X \to Vect .$

A section $σ$ of that is a morphism from the trivial vector bundle $1 : X \to pt \overset{ℂ}{\to} Vect$ $σ : 1 \to V .$

You want to say that sections form something like a sheaf on an orbifold.

For that we first of all would have to say which epimorphisms $π : Y \to X$ of smooth groupoids are regular epimorphisms.

The sheaf condition for local sections with respect to Y $σ : 1_{Y} \to π^{*} V$ must be the descent condition which says that sections of $V$ over $X$ are the same as those sections of $π^{*} V$ over $Y$ for which $π_{1}^{*} σ = π_{2}^{*} σ$ over $Y \times_{X} Y$ .

Here one has to be careful with interpreting the possibly weak pullback $Y \times_{X} Y$ of groupoids.

Is this the point where there is a problem, to your mind? I haven’t tried to fully go through this, but I’d think that that sections on $Y$ which glue are indeed in bijection to sections on $X$ .

Either I am missing something or you are thinking of a different setup.

Posted by: Urs Schreiber on April 19, 2008 4:52 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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But what would prevent us from picking our favoite closed category of smooth spaces and then take a smooth orbifold to be nothing but a groupoid internal to smooth spaces. These live in the obvious 2-category.

Several things. The main problem is that your 2-category won’t have enough 1-arrows: there are plenty of smooth equivalences of groupoids that don’t even have continuous inverses. For this reason you need to localize.

>> How you localize is a subtle issue.

In Moerdijk’s set up $Hom (S^{1}, Γ ⇉ *)$ is the set of equivalence classes of principal $Γ$ -bundles over $S^{1}$ ( $Γ$ is a finite group). What you want instead is the category of principal $Γ$ -bundles.

What’s a sheaf on an orbifold for you?

Since a manifold is an orbifold, the space of maps from a manifold to an orbifold should be a sheaf of categories on the manifold.

Either I am missing something or you are thinking of a different setup.

I am mostly thinking of orbifolds as Deligne-Mumford stacks with all the grief that this entails.

Posted by: Eugene Lerman on April 19, 2008 8:28 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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How you localize is a subtle issue.

I would tend to localize by saying quasi-equivalences are the fully faithful surjections. (Around here we call the result of that localization “smooth anafunctors”.) What goes wrong if I do that?

What’s a sheaf on an orbifold for you?

Since a manifold is an orbifold, the space of maps from a manifold to an orbifold should be a sheaf of categories on the manifold.

This is reading off from an orbifold the given stack on manifolds. But what is a sheaf on an orbifold for you?

Posted by: Urs Schreiber on April 19, 2008 9:56 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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Three quick points:

I can’t emphasize Pronk’s 1996 paper enough - in it is a localisation of the (2-)category of algebraic groupoids to a bicategory, and this bicategory is shown to be equivalent to the bicategory of algebraic stacks, and the same result for etale groupoids and topological stacks. (I think these would all be DM stacks from the context).

A sheaf on a (toological, say) groupoid is a sheaf on the space of objects and an action of the groupoid on that sheaf. This is easiest to see if we consider the etale space of he sheaf. Vector bundles on groupoids are defined in exactly the same way. If you like, Urs, this can be recast in a groupoidification style - the object map is a(n etale space of a) sheaf or a vector bundle. Such sheaves generalise $G$ -sets for $G$ a group.

The last is one closer to some of my own work at this point: how can we sensibly make ${Hom}_{W^{- 1}} (X, Y)$ a topological groupoid when $X, Y$ are topological groupoids, and the maps are in the localised category.

[Aside: The fully faithful, essentially surjective maps are the ones to invert, but “essentially surjective” needs to use surjective submersions or local-section-admitting maps (smooth or continuous cases resp.) - in other words some sort of regular epimorphism)]

To get the objects of the hom-groupoid above (at least in the topological case), we could naively form $∐ Obj (W (A, X)) \times Obj (Hom (A, Y))$ where $W$ is here thought of as a sub-bicategory. We can topologise $Obj (Hom (A, Y))$ as a subspace of $C (A_{0}, Y_{0}) \times C (A_{1}, Y_{1})$ - doing this for smooth spaces I suppose uses the completeness properties of the category of such. This makes $Obj ({Hom}_{W^{- 1}} (X, Y))$ a space. (We might have to restrict to a convenient category: $Top = CompGenHaus$ , say)

Trickier, and the more important part is the space of morphisms. In Pronk’s setup, these are described by equivalence classes of diagrams which are too complicated for me to draw here. Essentially the representatives are maps of spans, where the map is itself a span (and recall this map is intended to be an equivalence in a bicategory, so it is symmetric). The equivalence relation is best thought of as being football-shaped (Australian or American), being a span between spans between spans. Apologies for all the words, but the only place I know this description appears is in some talk slides Dorette Pronk kindly scanned in for me - the original paper is much more confusing as to the shape of these things.

The point is, finding a topology on this collection looks to be a bit hard. The best thing I can think of (just now it was) is some sort of coend…

The motivating example should probably be, as was mentioned, ${Hom}_{W^{- 1}} (X, B G)$ , $B G$ the one-object groupoid associated to the group $G$ . We can do better than letting this be a category enriched over spaces - objects the set of $G$ -bundles and hom spaces the obvious things, because continuously varying transition functions should vary the bundle continuously.

This is just to get a closed bicategory. Is there any hope of getting something like cartesian closed?

Posted by: David Roberts on April 20, 2008 1:14 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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I by and large agree with the points made by David Roberts. However I would put the emphasis a bit differently. For me the moral of Pronk’s theorem is that if you try to localize etale Lie groupoids at equivalences as a bicategory, you end up with DM stacks over the category of manifolds [They are really “DM” stacks, since diagonal is never representable, but this is a small technical annoyance].

In many cases the (2-)category of stacks in nicer to work with. For instance, try integrating a vector field in Pronk’s bicategory! You don’t have a prayer of getting a flow as a an action of the reals. You will get a weak action of the reals, and you’ll never see what’s going on for the forest of spans.

Posted by: Eugene Lerman on April 20, 2008 3:00 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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One important issue is that $AUT (G)$ doesn’t act transitively on $G$

I am confused about what transitivity should mean here. The action is certainly transitive on the space of objects! :-)

Remind me, what’s the grand question we’d want to ask given a 2-group $G$ acting on a groupoid $X$ ? Isn’t it whether $G$ has a sub-2-group $H$ such that $X = G / H$ for some notion of quotient of 2-groups?

Posted by: Urs Schreiber on April 16, 2008 2:51 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

Surely you don’t mean =. That is a naughty symbol when looking at 2-things!!! And then if it is equivalence, one might have a suitable notion.

Posted by: Tim Porter on April 16, 2008 4:30 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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Sure, I mean equivalence. Thanks.

Posted by: Urs Schreiber on April 16, 2008 6:22 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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I guess I was concerned that if we opted for transitivity in terms of whether for any two objects there’s an object of the acting 2-group which sends the first to an object isomorphic to the second, then any action of a 2-group on a connected groupoid would be transitive.

I was also wondering whether we might need to consider not just shapes composed of objects being fixed, but also shapes involving arrows. And this might require some knowledge of how the 2-group permutes the arrows of the groupoid.

As you say, after Tim’s point, what we want to find are situations where $G$ has a sub-2-group $H$ such that a groupoid $X$ is equivalent to $G / H$ for some notion of quotient of 2-groups.

I’m just not too sure about the range of thing $H$ can be the stabilizer of.

Posted by: David Corfield on April 18, 2008 10:31 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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then any action of a 2-group on a connected groupoid would be transitive.

I wouldn’t find that disturbing. If we are thinking of the groupoid as an orbifold, as you indicate, that’s what one would expect, heuristically: each connected component of the groupoid is really “one point with internal symmetry”.

What we should really do is try to characterize the actions of a 2-group $G$ on “cosets” $G / H$ and then derive the right conditions from that.

I haven’t thought it through, but it seems to me that the result of this would also be that $G$ acts transitively on isomorphism classes of $G / H$ . No?

Posted by: Urs Schreiber on April 18, 2008 12:37 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

Would it be right to think that quotienting two smooth 2-groups we won’t reach a space of points of differing internal symmetries?

Or to put it another way, thinking of an orbifold with some points of nontrivial isotropy, the 2-group of symmetries won’t be transitive.

If Klein 2-geometry needs homogeneity, maybe its natural home is the study of principal bundles.

Posted by: David Corfield on April 18, 2008 2:31 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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There are orbifolds which are “homogeneous” in the sense that all points have the same symmetry group. They are gerbs. The cheapest non-trivial example I know of is represented by the groupoid

S^{1} \times S^{3} \to S^{3}

where the circle acts ineffectively, say

λ \cdot (z_{1}, z_{2}) = (λ^{2} z_{1}, λ^{2} z_{2})

. I think

SU (2)

acts transitively on that, no? I am not sure what the full 2-group is….

Posted by: Eugene Lerman on April 18, 2008 9:16 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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I’ll need a little educating about your cheapest non-trivial example.

Am I right to think that the action is rotating the fibres of the Hopf fibration, so that for one journey of $λ$ around $S^{1}$ , the fibres are rotated by $4 π$ ?

If we chose to multiply for other values of $n$ in $λ^{n}$ , would that still give an orbifold?

Posted by: David Corfield on April 19, 2008 10:17 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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Am I right to think that the action is rotating the fibres of the Hopf fibration, so that for one journey of λ around $S^{1}$ , the fibres are rotated by 4π?

If we chose to multiply for other values of n in $λ^{n}$ , would that still give an orbifold?

1. Yes.

2. Yes. As long as the action groupoid is equivalent to an etale groupoid, it is an orbifold. Or, if you prefer, as long as the 1-stack represented by you groupoid is a Deligne-Mumford stack, you have an orbifold.

Posted by: Eugene Lerman on April 19, 2008 2:12 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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Would it be right to think that quotienting two smooth 2-groups we won’t reach a space of points of differing internal symmetries?

I think that is correct. 2-groups have all their isotropy groups isomorphic, so the quotient of one of these by the other shouldn’t give us non-isomorphic groups of internal symmetries. This is not to say that the groupoid which is the quotient is transitive! Take for example the fundamental groupoid of $BG ∐ BG$ for $G$ a discrete group.

Posted by: David Roberts on April 20, 2008 3:17 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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so the quotient of one of these by the other shouldn’t […]

We should really start thinking about how that “quotient” is really to be defined. And we should be thinking “2-coset” i.e. “co-2-set” i.e. “co-groupoid”.

At least for strict 2-groups, that definition should be obvious, but let’s make it explicit:

for strict 2-groups, take the (ordinary) group of morphisms of $G$ and form the (ordinary) coset $Mor (C) : = Mor (G) / Mor (H)$ by the group of morphisms of $H \subset G$ .

To recover the groupoid structure on the resulting coset $Mor (C)$ , we need that groupoid to have as objects the coset of the group of objects of $G$ by the group of morphisms of $H$ : $Obj (C) : = Obj (G) / Mor (H) .$ Then the source-target map is the obvious $Mor (C) : = Mor (G) / Mor (H) \overset{(s, t) / Mor (H)}{\to} Obj (G) / Mor (H) = : Obj (c) .$ Same for composition.

I think. Feel free to disagree.

This is not to say that the groupoid which is the quotient is transitive! Take for example […]

For sure, just take for example 1-groups as special kinds of 2-groups: we know that the coset $G / H$ of two 1-groups is not a transitive (=connected) groupoid (but a discrete groupoid, even)!

Posted by: Urs Schreiber on April 20, 2008 4:01 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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Are you saying $Mor (H)$ is a subgroup of $Obj (G)$ when you form $Obj (C)$ ?

Could you illustrate the construction with $AUT (S_{4})$ and $AUT (S_{3})$ ? Neither $S_{3}$ nor $S_{4}$ has outer automorphisms. So

$∣ Mor (AUT (S_{4})) ∣ = 24^{2} and ∣ Mor (AUT (S_{3})) ∣ = 6^{2} .$

So then

$∣ Mor (C)) ∣ = 16 .$

But how do you quotient $Obj (AUT (S_{4}))$ , of size 24, by $Mor (AUT (S_{3}))$ of size 36?

Perhaps you mean to look at orbits of $Obj (G)$ under the action of $Mor (H)$ .

Posted by: David Corfield on April 21, 2008 10:41 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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Perhaps you mean to look at orbits of $Obj (G)$ under the action of $Mor (H)$ .

Yes, thanks, I didn’t say that well.

Then entire “co-set” groupoid is, as a set of morphisms, $Mor (G) / Mor (H)$ and its objects are the classes which contain the original identity morphisms of $G$ .

The best way to think about this is to think of both $G$ and $H$ as one-object 2-groupoids and draw the action of $H$ on $G$ as horizontal composition of bigons in these 2-groupoids.

Then you want to identify $G$ -bigons modulo acting with $H$ -bigons from the right. And drawing that picture makes it clear how to identitfy objects and composition in the quotient, I think. That’s what i tried to describe.

Posted by: Urs Schreiber on April 21, 2008 1:11 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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From Katherine Norrie’s paper Actions and automorphisms of crossed modules, Bull. Soc. Math. France. 118 (1990) 129-146, we have that a subcrossed module of $(G, H, d)$ is a subgroup of $G$ and a subgroup of $H$ , restricting the boundary map, and with the induced action.

In the case of the Poincaré 2-group, $(SO (n, 1), ℝ^{n + 1}, d)$ , $d$ trivial, we need only make sure about the action. We can now choose any subgroup of $SO (n, 1)$ and any subspace of $R^{n}$ on which there is an induced action.

Posted by: David Corfield on April 21, 2008 2:53 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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I must disagree. This construction doesn’t seem to make use of the fact $H$ is a 2-group - only the action of the underlying groupoid of $H$ .

I would have thought something more along the lines of $Obj (G) / Obj (H)$ , but I haven’t checked to see if it works done like this.

Here’s a test case we can calculate explicitly: $H$ a normal sub-2-group of $G$ . This should be given by a map of crossed modules $(H_{1} \to H_{2}) \to (G_{1} \to G_{2})$ such that $H_{i} \to G_{i}$ is the inclusion of a normal subgroup and the whole thing forms a crossed square. The resulting 2-crossed module as described in our paper, $H_{1} \to H_{0} \times G_{1} \to G_{0}$ is (weakly) equivalent to the crossed module $(H_{0} \times G_{1}) / H_{1} \to G_{0},$ and this will be the homogeneous groupoid we are after (albeit one that is a 2-group).

Conceivably, one might argue that all we need for a normal sub-2-group is that the group $π_{3}$ we get from the crossed square is trivial (In the notation of the appendix in the paper, this is $\ker u \cap \ker f$ ), but from some other work I have been doing I don’t feel this more general notion is appropriate.

Emboldened by the power of the Loday mapping cone, we could try to form the mapping cone of a general map of crossed modules such that it is a componentwise inclusion of a subgroup. We get a complex of groups again, but this time it is not a normal complex. It is possible to form a quotient $(H_{0} \times G_{1}) / H_{1} \to G_{0}$ and get a sort of ‘crossed homogenous set’. This is a bit like when we get exact sequences of groups that tail off into exact sequences of pointed sets, but the other way around. The challenge is to try to identify the groupoid corresponding to such a thing.

Posted by: David Roberts on April 21, 2008 2:56 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

I mean I disagree with Urs’ construction, not David’s! Didn’t refresh often enough for David’s reply to arrive on my screen before my own.

Posted by: David Roberts on April 21, 2008 3:01 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

With regard to these examples, may I point out the link between crossed squares and 2-crossed modules found by Daniel Conduché. The `mapping cone’ of Loday applied to any crossed square naturally has a 2-crossed module structure using the h-map of the crossed square. For higher dimensions it is possibly relevant to look at my old paper in Topology and the alternative treatment by Bullejos, Cegarra and Duskin at about the same time. That was the algebraic proof of Loday’s result on homotopy n-types.

Posted by: Tim Porter on April 21, 2008 5:14 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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…the link between crossed squares and 2-crossed modules found by Daniel Conduché.

My apologies - I didn’t mean to claim that Urs and I found this. I’ve been citing Conduche on this for a year now, and I choose to blame the lateness of the hour in Adelaide at which I posted for my slip.

Posted by: David Roberts on April 21, 2008 11:48 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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This construction doesn’t seem to make use of the fact $H$ is a 2-group

But it does! That’s what makes the quotient a groupoid.

I should have drawn the pictures in the first place. Let me do it now:

this is a 2-cell in $B G$ :

$\begin{matrix} ↗ ↘^{g} \\ • & ⇓^{g_{1}} & • \\ ↘ ↗_{g'} \end{matrix}$

and we want to identify two of them when they differ by right composition $\begin{matrix} ↗ ↘^{g} & ↗ ↘^{h} \\ • & ⇓^{g_{1}} & • & ⇓^{h_{1}} & • \\ ↘ ↗_{g'} & ↘ ↗_{h'} \end{matrix}$

with a 2-morphism in $B H$ .

The objects of the “co-set” groupoid are classes of things that look like

$\begin{matrix} ↗ ↘^{h} \\ • & \overset{g}{\to} & • & ⇓^{h_{1}} & • \\ ↘ ↗_{h'} \end{matrix} .$

This means that $\begin{matrix} • & \overset{g}{\to} & • & \overset{h}{\to} & • \end{matrix}$ and $\begin{matrix} • & \overset{g}{\to} & • & \overset{h'}{\to} & • \end{matrix}$ are the same object in $G / H$ and that $\begin{matrix} ↗ ↘^{h} \\ • & \overset{g}{\to} & • & ⇓^{h_{1}} & • \\ ↘ ↗_{h'} \end{matrix}$ is identified with the identity morphism on that object, for all $h_{1}$ .

A morphism from the class of $• \overset{g}{\to} •$ to the class of $• \overset{g'}{\to} •$ is represented by a morphism $\begin{matrix} ↗ ↘^{g} \\ • & ⇓^{g_{1}} & • \\ ↘ ↗_{g'} \end{matrix}$ which we take to be the same morphism as $\begin{matrix} ↗ ↘^{g} & ↗ ↘^{h} \\ • & ⇓^{g_{1}} & • & ⇓^{h_{1}} & • \\ ↘ ↗_{g'} & ↘ ↗_{h'} \end{matrix}$ for each $h_{1}$ .

For composing two morphisms in the quotient you pick any two representatives whose source and target match.

That this is (I think) well defined crucially makes use of the exchange law in $G$ , and hence of the 2-group property also of $H$ .

So, the set of morphisms of the “coset” groupoid is $Mor (G / H) : = Mor (G) / Mor (H) .$ And the groupoid structure on that set comes from the 2-group structure on $G$ and $H$ .

Posted by: Urs Schreiber on April 21, 2008 5:44 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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Of course we also have the systematic general definition of the action 2-groupoid:

Let $i : B H ↪ B G$ be the 2-functor including $H$ in $G$ , and let $ρ : B G ↪ Grpd$ be the fundamental 2-representation of $G$ on itself.

(This 2-functor $ρ$ sends the single object of $B G$ to the groupoid $G$ , sends a morphism in $B G$ to right-multiplication in $G$ and a 2-morphism to a natural transformation between two right multiplications).

Then we know from David C.’s thinking about higher topos theory, that the action 2-groupoid $G / / H$ of the action of $H$ on $G$ is the pullback

$\begin{matrix} G / / H & \to & E G & \to & T_{pt} Grpd \\ ↓ & ↓ & ↓ \\ B H & \overset{i}{\to} & B G & \overset{ρ}{\to} & Grpd \end{matrix} .$ Here the step in the middle is the action 2-groupoid of $G$ acting on itself, which is the 2-groupoid $E G : = {INN}_{0} (G) : = T_{•} B G$ from David R.’s and my article.

If we divide out 2-isomorphism in the 2-groupoid $G / / H$ here we obtain the $G / H$ 1-groupoid that we are after.

And I dare to say that this reproduces the prescription which I have given.

Posted by: Urs Schreiber on April 21, 2008 6:39 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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Another fun remark:

what I just said can, using the remark on 2-bundles at the end of David R.’s and my article be read as the construction of the (2-groupoid incarnation) of the $G$ -2-bundle on $B ∣ H ∣$ obtained by pullback along the inclusion $B ∣ H ∣ \to B ∣ G ∣$ :

$\begin{matrix} G & \to & G & \to & s^{- 1} pt \\ ↓ & ↓ & ↓ \\ G / / H & \to & E G & \to & T_{pt} Grpd \\ ↓ & ↓ & ↓ \\ B H & \overset{i}{↪} & B G & \to & Grpd \end{matrix}$

on the right: the universal principal $Grpd$ 2-groupoid bundle

in the middle: the universal $G$ -2-bundle (described in David R.’s and my article)

on the left: the $G$ -2-bundle over $B ∣ H ∣$

Here $∣ \cdot ∣$ denotes taking 2-nerves and forming geometric realization, and $B X$ is the one-object 2-groupoid corresponding to a monoidal 1-groupoid $X$ .

This is such that $∣ B X ∣ = B ∣ X ∣ .$

Posted by: Urs Schreiber on April 21, 2008 7:01 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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I’m formulating a quotient groupoid based on some of my own comments above, but there is just not enough time to get it finished and available to the general public today. Happily, things turned out much nicer than I though. Generally it looks equivalent to Urs’ construction, but has more arrows, coming from $H$ , instead of quotienting the objects of $G$ .

Watch this space!

Posted by: David Roberts on April 22, 2008 8:37 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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Normal sub-2-groups and their cokernels are defined in general in

P. Carrasco, A.R. Garzón and E.M. Vitale, On categorical crossed modules

(definition 3.7 and above) : they are categorical crossed modules

T : G \to H

where

T

is faithful (thanks to the faithfulness of

T

, some conditions in the definition of categorical crossed modules are automatically satisfied, so the definition of normal sub-2-group is not too complicated). Example 2.6 (vi) seems to be the special case you are considering. Every kernel is a normal sub-2-group and, by proposition 3.6, every normal sub-2-group is the kernel of its cokernel.

Posted by: Mathieu Dupont on April 22, 2008 1:19 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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Normal sub-2-groups and their cokernels are defined in general in […]

I have read this, but thanks for the reminder. However, the real task is to define a general sub-2-group $K_{(2)} < G_{(2)}$ and the quotient “homogeneous” groupoid $K_{(2)} \ G_{(2)}$ , with its action etc.

Posted by: David Roberts on April 23, 2008 6:28 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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Do we have any objections to Norrie’s definition of sub-crossed modules on p. 131 of the paper mentioned here?

Also in that paper, after discussing the the automorphism group of a group $N$ , she says on p. 130

…in some other familiar categories, the set of structure preserving selfmaps of a given object will not fulfil the role just delineated for the automorphism group $Aut N$ . In the category of groups, the automorphism group plays a dual role of capturing the notions of action and of structure preserving selfmaps. In other categories these notions do not necessarily coincide, and for our present purposes it is the notion of action that is significant. We shall define actor crossed modules, and show how they provide an analogue of automorphism groups of groups.

What are those ‘regular derivations’ (p. 132) all about?

Posted by: David Corfield on April 23, 2008 1:37 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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What are those ‘regular derivations’ (p. 132) all about?

I find it helpful to rephrase such structural questions about crossed modules in terms of strict one-object 2-groupoids, which they are equivalent to.

I write $B G : = B (T \overset{\partial}{\to} G)$ for the strict one-object 2-groupoid corresponding to Norrie’s crossed module $(T, G, \partial)$ .

Then you can see that the definition of a “derivation” of $(T, G, \partial)$ on p. 132 is equivalent to precisely those pseudonatural transformations starting at the identity on $B G$ $\begin{matrix} ↗ ↘^{Id} \\ B G & ⇓^{χ} & B G \\ ↘ ↗_{{Ad}_{χ (-)}} \end{matrix}$ whose component map is trivial on objects.

Thes form a monoid under horizontal composition of pseudonatural transformation. The iduced composition on the component functions is the one Norrie defines on p. 132.

The “regular derivations” are those such pseudonatural transformations which have a horizontal inverse.

Notice that pseudonatural transformation on the identity which have trivial component on objects are the complementary concept to the inner automorphisms in the article with David R.: there we looked at those which have components only on the objects.

In my discussion here I addressed these two kinds of inner automorphisms of 2-groups a “horizontal” and “vertical”:

(beware: back then I wrote $G_{(2)}$ for wht above I am denoting $B G$ .)

For $n$ -groups, there is a tower of $n$ different notions of inner automorphisms, differening in the $k$ above which they are required to have vanishing component $k$ -morphisms.

A discussion of that is here. The Lie $n$ -algebraic analog became section 3.2 of the set of notes Structure of Lie $n$ -algebras, which was a precursor of our text on $L_{∞}$ -connections (but this particular discussion didn’t make it into that.)

Posted by: Urs Schreiber on April 24, 2008 1:43 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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Do we have any objections to Norrie’s definition of sub-crossed modules […]

It’s fine. I wouldn’t use it as a definition of sub-2-group, though, since it was generally agreed way back in the past that a sub-2-group needs to be defined in terms of homotopy invariants (e.g. John’s August 14 posts in Klein 2-Geometry IV (which was about the date the seed for this Cafe was planted!))

The promised quotient groupoid has arrived, and since I’m not flash at itex/MathML, it is visible here.

A summary in words is as follows: Instead of forming the quotient of the objects of our 2-group by the objects of the sub-2-group, we add them as extra morphisms to the existing ones. We then divide out the action of the morphisms of the sub-2-group on these new morphisms. That this works was completely surprising to me, contradicting and overriding what I said in my last paragraph above.

Posted by: David Roberts on April 24, 2008 5:47 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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For arrows in the category of abelian groups (or, more generally, arrows in an abelian category $C$ ), the construction $L \ (K ⋉ H) \to G$ coincides with the cokernel defined by Grandis in section 4 of A note on exactness and stability in homotopical algebra. In the 2-category of arrows in $C$ , commutative squares between them and homotopies of chain complexes between them, this cokernal has the universal property of a “regular standard homotopy cokernel” (section 1.2 in Grandis’ paper) and also the universal property of the cokernel given in A Picard-Brauer exact sequence of categorical groups (E.M. Vitale, JPAA 175, 383-408). In particular, this construction applies to the 2-category of Baez-Crans 2-vector spaces.

If $C$ is the category of abelian groups, if you “realize” this situation as symmetric 2-groups, symmetric monoidal functors and monoidal natural transformations, you get the cokernel of a morphism of symmetric 2-groups described in “A Picard-Brauer exact sequence of categorical groups”. So I think that your construction $L \ (K ⋉ H) \to G$ corresponds to the cokernel described in the paper by Carrasco, Garzón and Vitale, but I haven’t checked the details.

This construction is not so surprising if you think in this way : to quotient a set, you add “equalities” between the objects of the set; to quotient a category, you add arrows between the objects of the category (and you keep the same objects). This parallel is blurred by the fact that quotients of sets are usually defined in terms of equivalence classes. But in Errett Bishop’s book “Foundations of Constructive Analysis”, for example, quotients of sets by equivalence relations are defined exactly in this way : you keep the same elements, but replace the original equality by the equivalence relation.

Posted by: Mathieu Dupont on April 24, 2008 8:53 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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Thanks for the references to the ‘abelian’ versions! It’s always good to have that backwards compatibility ;)

to quotient a category, you add arrows between the objects of the category (and you keep the same objects)

that much was clear - it is finding the right analogy for the arrows that is tricky. Using the Loday mapping cone we get something that is secretly a 2-groupoid, and I’ve just demoted the 2-arrows of that 2-groupoid to equalities - one might call this a ‘locally decategorified’ 2-groupoid.

Posted by: David Roberts on April 25, 2008 12:49 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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That this works was completely surprising to me, contradicting and overriding what I said in my last paragraph…

So did you end up agreeing with Urs?

Now we could do (or a least I could) with some concrete calculations. So with crossed module denoted $(G, H, \partial)$ , we can consider:

$H$ is a $G$ -module, $\partial = 0$ . We might look at $(SO (2), ℝ^{2}, 0)$ in $(SO (3), ℝ^{3}, 0)$ .
$H$ is a normal subgroup of $G$ , $\partial$ is inclusion, action by conjugation. We can choose $K$ any subgroup of $G$ , and then $L$ a normal subgroup of $K$ contained in $H$ . $H \cap K$ would work.
An epimorphism $H \to G$ with central kernel. What’s a nice example of a central extension and sub-central extension?
$G$ is $Aut (H)$ , $\partial$ sends to $h$ to conjugation by $h$ . We might look at $AUT (S_{3})$ in $AUT (S_{4})$ .

Posted by: David Corfield on April 24, 2008 9:18 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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So taking the first case, we have $SO (3)$ worth of objects, arranged in $S^{2}$ worth of components. Between any two objects in the same component there are $ℝ$ worth of morphisms. $G_{(2)}$ acts on this by permuting around the objects, which remain in their connected components. Its objects also act on the set of morphisms in the quotient groupoid, and its own morphisms add on their projection to $ℝ$ .

With the second example, connected components of the quotient groupoid would correspond to cosets of $H K$ .

Posted by: David Corfield on April 24, 2008 6:22 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

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I haven’t sat down and calculated Urs’ version to my own satisfaction, but I am less doubting than at first. My surprise was the fact $L$ is normal in $K ⋉ H$ , given that it isn’t normal in $H$ .

For the central extension case, how about $U (1) \to U (m) \to SU (m)$ sitting diagonally inside $U (1) \to U (2 m) \to SU (2 m) .$ or their orthogonal counterparts. I’m sure $U (2 m) / U (m)$ has some interesting geometry. One could also try $1 \to SU (n)$ inside $U (n + 1) \to SU (n + 1)$ . Note that this last isn’t injective on the induced map of cokernels - this condition isn’t necessary for the quotient object to be a 1-groupoid, but I’m checking to see if having said injectivity makes things nice for (2-)transitivity.

Posted by: David Roberts on April 25, 2008 12:39 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

I’m intrigued to find out the different flavours of Klein 2-geometry. I don’t know to what extent the four types of crossed module I gave exhaust them. It seems fairly clear that case 1 is pointing us to the geometry of vector bundles. But what about the others?

Can we have blends of cases, like maybe of 1 and 3 in the case of projective representations of a group?

Posted by: David Corfield on April 25, 2008 12:57 PM | Permalink | Reply to this

Re: Klein 2-Geometry X

Presumably the ‘blending’ would need a theory of 2-group extensions. John’s worked on such things. How do we lure him into the thread?

Posted by: David Corfield on April 28, 2008 9:35 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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I’m very sorry — I just got back from Shanghai and I’m desperately writing a grant proposal that’s due at 8:59 pm on Thursday. So, please try this paper:

P. Carrasco, A. M. Cegarra and A. R. Grandjeá, (Co)homology of crossed modules.

A crossed module is just another way of thinking about a 2-group, and the most vanilla flavor of group cohomology ( $H^{2}$ , to be precise) classifies central extensions. So, we should not be surprised that in Section 6 of this paper, they classify central extensions of crossed modules using their $H^{2}$ for crossed modules.

(To classifying the more general noncentral extensions, we will need ‘Schreier theory’ for 2-groups, also known as ‘nonabelian’ cohomology. I guess Larry Breen discusses this in his paper in the Grothendieck Festschrift — but since I don’t really read French, I’ve never absorbed that paper.)

Posted by: John Baez on April 30, 2008 9:29 AM | Permalink | Reply to this

Re: Klein 2-Geometry X

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(2-)transitivity

This seems to be the next big step: what does it mean for a 2-group action on a groupoid $X$ to be appropriately transitive? The obvious thing that comes to mind is that given two objects $x, y$ of $X$ , there is an object $g$ in the 2-group $G$ and a morphism $x \cdot g \to y$ . Then we only have to worry about morphisms (which is the crux of the matter).

The condition above can be described better as $1 \times G \to X \times G \to X$ is essentially surjective for every $1 \to X$ . This is different to the version I described in my notes, (April 24 version) namely $X \times G \to X \times X$ being essentially surjective for

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April 15, 2008