Recently I finished up my series of talks on 2-Hilbert spaces with a description of the basics of 2-group representation theory, and a little about the special case of the Poincaré 2-group. The main sources were a paper by Crane and Yetter describing 2-group representations in general, and another by Crane and Sheppeard. The Poincaré 2-group, so far as I know, was first explicitly mentioned by John Baez in the context of higher gauge theory. It’s an example of a kind of 2-group which can be cooked up from any group and abelian group , and which is related to the semidirect product .

One reason people are starting to take an interest in the representation theory of the Poincaré 2-group is that representations of the Poincaré group (among others) and intertwiners between them play a role in spin foam models for field theories such as BF theory, various models of quantum gravity, and so on. Some of these, turn up naturally when looking at TQFT’s, and generalizations of these, which is how I got here. Extending this to 2-groups gives a richer structure to work with. (Whether the extra richness is useful is another matter).

Before getting into more detail, I first would like to take a look at representation theory for groups from a categorical point of view, and then see what happens when we move to -groups – that is, when we categorify.

To begin with, we can think of a representation of a group as a functor. The group can be thought of as a category with one object and all morphisms invertible – so that the group elements are morphisms, and the group operation is composition. In this case, a representation of the group is just any functor:

since this assigns some one vector space (the representation space, ) to the one object of , and a linear map to each morphism of (i.e. to each group element) in a way consistent with composition. The nice thing about this point of view is that knowing a little category theory is enough to suggest one of the fundamental ideas of representation theory, namely intertwining operators (“intertwiners”). These are natural transformations between functors. *This is the idea to categorify*.

The point is that functors can be organized into a structure , and this is most naturally seen as a category, not just a set. The category of representations of is usually called , but seen as a category of functors, it is a general case of a category $hom(C,D)$ of functors from category to category . Let’s look at how this is structured, then consider what happens with higher dimensional categories. There seems to be a general pattern which one can just begin to see with 1-categories:

- a functor is a map between categories, assigning
- to each -object a corresponding -object
- to each -morphism a corresponding -morphism

in a way compatible with composition and identities

- a natural transformation between functors assigns
- to each -object a -morphism

making a naturality square commute for any morphism in :

(In the case where the functors are representations of a group, this is an intertwiner – a linear map which commutes with the action of the group on .)

The pattern is a little more obvious for 2-categories:

- a 2-functor is a map between 2-categories, assigning
- to each -object a corresponding -object
- to each -morphism a corresponding -morphism
- to each -2-morphism a corresponding -2-morphism

in a way compatible with composition and identities

- a natural transformation between 2-functors assigns
- to each -object a -morphism
- to each -morphism a -2-morphism

making a generalized naturality square commute for any 2-morphism in (where ):

- a
*modification*(what I might have named a “2-natural transformation” or similar) between natural transformations assigns- to each -object a -2-morphism

making a similar diagram commute (OK, well, it appears on p11 of John Baez’ Introduction to n-Categories, but I don’t have a web-ified version of it – I haven’t learned how to turn LaTeX diagrams into handy web format).

…

In the case where is a 2-group – a 2-category with one object and all -morphisms invertible, and , then we have here the (quite abstract!) definition of a representation, an 1-intertwiner between representations, and a 2-intertwiner between 1-intertwiners.

It’s not too hard to see the pattern suggested here – a “-natural transformation” assigns a -morphism in to an object in the -category , and a -morphism in to each -morphism in . This morphism fits into a diagram filling a commutative diagram which was the coherence law for the top dimensional transformation for -categories. (I might point out that if I were to come up with terminology for these things from scratch, I’d try to build in some flexibility from the start. Instead of “functor”, “natural transformation”, and “modification”, I’d have used terms more analogous to the terminology for morphisms. Probably I’d have used, respectively, “1-functor”, “2-functor”, “3-functor”, and so on. This is already a problem, since these terms are in use with a different meaning! Instead, I’ve used “natural -transformation”.) It’s less easy to say what, explicitly, the various coherence laws should be at each stage, except that there should be an equation between the composites of (a) the -morphisms in an -natural transformation with (b) the two possible images of any chosen lower dimensional morphisms.

There is a lot of useful information out there about various forms of -categories, such as the Illustrated Guidebook by Cheng and Lauda, and Tom Leinster’s “Higher Operads, Higher Categories” (also in print). They’re a little less packed with information on functors, natural transformations, and their higher generalizations. I don’t know a reference that explains the generalization thoroughly, though. If anyone does know a good source on this, I’d like to hear about it. Probably this is somewhere in the work of Street, Kelly, maybe Batanin (whose definition of -category is the one implicitly used here) or others, but I’m not familiar enough with the literature to know where this is done.

These generalizations of functors and natural transformations to higher -categories describe what functor -categories are like. When written down and decoded, these definitions can be turned into a concrete definition of representations and the various -intertwiners involved in the representation theory of -groups.

However, next time I’ll take a look at some of what is known in the slightly more down to earth world where .

January 24, 2008 at 5:26 pm

So what exactly

isthe Poincaré 2-group? The Poincaré 1-group has to do with the symmetries of Minkowski space… does the Poincaré 2-group have something to do with the symmetries of a different space?January 24, 2008 at 10:23 pm

Jamie: Interesting question. Moving from a group to a 2-group doesn’t obviously change which is the space whose symmetries are being described, although one should think about “2-spaces”, which Toby Bartels has explained in his thesis (these are categories internal in the relevant category of spaces – say smooth manifolds). While 2-groups can act on 2-spaces, I’m not quite sure if the concept “2-group of symmetries of a 2-space” works quite as well as one would like. For one thing, the 2-maps from one 2-space to another are not just functors, at least the way Toby handles them (they’re anafunctors – *locally* functors in some sense). There is also the difference between 2-groups which are “automorphic”, like the Poincaré 2-group (all 2-morphisms are 2-AUTOmorphisms), and those which aren’t. You might have to be careful what you mean by “symmetry” to ensure the natural transformations between 2-maps from a 2-space to itself have this property.

Assuming the various possible issues don’t cause too many problems, then since a space can be seen as a 2-space with only identity morphisms, I would guess the Poincaré 2-group would be the 2-group of symmetries of Minkowski space thought of this way. I haven’t given this enough thought to be sure that’s right, though.

More plainly, the Poincaré 2-group is a way of restoring the distinction between rotations/boosts on the one hand, and translations on the other which is lost when you look at the Poincaré group as one big group, rather than as a construct (usually the semidirect product) built from a pair of groups. Then the group of translations is seen as automorphisms of the identity Lorentz transformation, and the Lorentz group acts on translations in the obvious way.

January 24, 2008 at 11:27 pm

Is this the same connection between 2-groups and semidirect products as is illustrated in the last section of the 2nd edition of

CWM?January 25, 2008 at 2:15 am

John: Unfortunately, I only have the first edition, so I haven’t read that section. Explicitly, the connection is that given a 2-group whose group of objects is , and where the automorphism group of the identity is , then the collection of all morphisms of gets a group structure, and is . That’s the only connection between the two ideas I know of, so I assume it’s the one in

CWM.January 25, 2008 at 4:29 am

Exactly. So is that the Poincaré 2-group? Use Minkowski spacetime points as objects and the Lorentz group as the automorphisms of the identity object? (or the other way around.. I always have to look up which direction the semidirect product goes)

January 25, 2008 at 11:27 am

It’s the other way around (the 2-automorphism group has to be abelian), but yes, that’s it.

January 30, 2008 at 5:19 am

Hi! I invented the Poincare 2-group! 🙂

Whenever you have a group G acting as automorphisms of an abelian group H, you get a 2-group with G as the group of objects and H as the group of endomorphisms of the identity object. Every strict 2-group where all morphisms are endomorphisms is of this form.

In particular, whenever you have a group G and a representation of it on a vector space H, you get an example of this situation. So, any group representation gives a 2-group. I find this very cute, but I don’t feel I understand its profound significance (if there is one).

When I realized this fact (which surely had been known for a long time), I decided to take the Lorentz group and its representation on R^4, turn this stuff into a 2-group, and call it the Poincare 2-group.

The use of this thing, if any, is still quite mysterious to me. Derek Wise, Laurent Freidel, Aristide Baratin and I are about 2/3 done with a paper on representations of 2-groups, which began as a paper representations of the Poincare 2-group. Jeff was also involved in working on this paper. Baratin and Freidel have done some calculations relating the representation theory of the Poincare 2-group to some 4d TQFT-like models, but I don’t think anyone really understands what’s going on yet.

By the way, for any n you can build an (n+1)-group by starting with a group G acting as automorphisms of a group H, and taking G to be the objects and H to be the endo-n-morphisms of the identity object. Well, that’s

almosttrue… actually the case n = 0 is a weird degenerate case which doesn’t quite work the same way. This case is the usual semidirect product!January 30, 2008 at 1:24 pm

And there was me thinking that a 2-group was a 2-category with one object, one morphism, and all 2-morphisms invertible. (I guess that would be a category enriched in the 2-category of groups, rather than internal to it… or something like that.) It’s nice how the translation and boost symmetries get separated by this construction. I shall have to wait for this new paper to see why it’s fundamental to quantum gravity! 🙂

February 15, 2008 at 10:13 am

Jamie wrote:

“And there was me thinking that a 2-group was a 2-category with one object, one morphism, and all 2-morphisms invertible.”

Indeed, every such 2-category yields a 2-group, namely the “shifted” version of the ordinary (abelian) group of endomorphisms of that single morphism.

More generally, though, 2-groups come from 2-groupoids with a single object.

Even though it may seem backwards, that’s actually a good way to define $n$-groups: they are the Hom (n-1)-categories of one-object n-groupoids.

Depending on how strict or weak you chose your n-groupoid here, you get correspondingly a more or less strict or weak notion of n-group.

As John Armstrong mentioned, it makes sense to think of an n-group as a one-object n-groupoid, because we should think of it as (a sub n-group, possibly, of) the n-group of automorphisms (or autoequivalences, if we are working weakly) of some object. That’s the single object we keep seeing.

It might be an interesting exercise to try to reverse-engineer a given n-group to realize it as the automorphism n-group of some concrete object in some given n-category.

February 15, 2008 at 10:26 am

Speaking of unwritten papers:

I am still trying to find somebody interested in un-unwriting this unwritten article:

The canonical 2-representation(pdf, 11 unwritten pages)Abstract:Every finite strict 2-group has a canonical 2-representation on Vect-module categories. This easily generalizes to strict Lie 2-groups and possibly to Fréchet Lie 2-groups.Examples discussed are the 2-representations which yield the associated 2-bundles known as line 2-bundle (abelian gerbes) and String 2-bundles.

If I were to un-unwrite this myself now, I’d stop talking about Fréchet and start talking instead internal to sheaves on smooth test domains.

As Andrew Stacey pointed out to me: it is actually known that doing loop groups not as Freéchet manifolds but as things internal to sheaves on smooth test domains is possible and, more shockingly, yields the familiar representation theory.

May 23, 2008 at 2:56 pm

The inventor of the Poincaré 2-group said:

Can’t you do this trick for ANY group $G$ and abelian group $H$? I don’t see why $G$ needs to act as automorphisms of $H$.

May 23, 2008 at 3:10 pm

Of course you can, because you can always just choose the trivial representation. This is quite a nice theorem! Can anybody tell me where it’s proved? Maybe I’ll have a go at it myself…