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 $G$ and abelian group $H$, and which is related to the semidirect product $G \ltimes H$.

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 $n$-groups – that is, when we categorify.

To begin with, we can think of a representation $(V, \rho)$ of a group $G$ as a functor. The group $G$ 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:

$\rho : G \rightarrow Vect$

since this assigns some one vector space (the representation space, $\rho(\star) = V$) to the one object of $G$, and a linear map $\rho(g): V \rightarrow V$ to each morphism of $G$ (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 $F : G \rightarrow Vect$ can be organized into a structure $hom(G,Vect)$, and this is most naturally seen as a category, not just a set. The category of representations of $G$ is usually called $Rep(G)$, but seen as a category of functors, it is a general case of a category $hom(C,D)$ of functors from category $C$ to category $D$. 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 $F : C \rightarrow D$ is a map between categories, assigning
• to each $C$-object a corresponding $D$-object
• to each $C$-morphism a corresponding $D$-morphism

in a way compatible with composition and identities

• a natural transformation $n$ between functors $F,F' : C \rightarrow D$ assigns
• to each $C$-object a $D$-morphism

making a naturality square commute for any morphism $g : x \rightarrow y$ in $C$:

(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 $V$.)

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

• a 2-functor $F : C \rightarrow D$ is a map between 2-categories, assigning
• to each $C$-object a corresponding $D$-object
• to each $C$-morphism a corresponding $D$-morphism
• to each $C$-2-morphism a corresponding $D$-2-morphism

in a way compatible with composition and identities

• a natural transformation $n$ between 2-functors $F,F' : C \rightarrow D$ assigns
• to each $C$-object a $D$-morphism
• to each $C$-morphism a $D$-2-morphism

making a generalized naturality square commute for any 2-morphism $h : f \rightarrow g$ in $C$ (where $f,g : x \rightarrow y$):

• a modification (what I might have named a “2-natural transformation” or similar) $m$ between natural transformations $n,n' : F \rightarrow n'$ assigns
• to each $C$-object a $D$-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 $C = G$ is a 2-group – a 2-category with one object and all $j$-morphisms invertible, and $D = 2Vect$, 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 “$k$-natural transformation” assigns a $k$-morphism in $D$ to an object in the $n$-category $C$, and a $(k+j)$-morphism in $D$ to each $j$-morphism in $C$. This morphism fits into a diagram filling a commutative diagram which was the coherence law for the top dimensional transformation for $(n-1)$-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 $k$-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 $n$-morphisms in an $n$-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 $n$-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 $n$-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 $n$-categories describe what functor $n$-categories are like. When written down and decoded, these definitions can be turned into a concrete definition of representations and the various $k$-intertwiners involved in the representation theory of $n$-groups.

However, next time I’ll take a look at some of what is known in the slightly more down to earth world where $n=2$.