The String Coffee Table

[Update: I had a typo in the original version of the following, which was kindly pointed out to me by Mark Hovey.]

As is well known, a representation in $C$ of a group $G$ is nothing but a functor

(Here I denote by $\Sigma(G)$ the suspension of $G$. This is the category with a single object and a morphism for each element of $G$.)

Better yet, the representation category is nothing but the functor category

Therefore it is obvious that a representation for a 2-group $G$ should be a 2-functor

where now $C$ is some 2-category. We have an entire 2-category of 2-representations, namely the 2-functor 2-category

Nothing more obvious than this.

Unpacking this definition requires work. And there is far more freedom of choice of representation than in the in the ordinary setup. For one, there are “more” sensible target 2-categories $C$ that one would be interested in, because there are many flavors of categorified vector spaces. Furthermore, there is a large freedom in choosing the level of coherent weakening one is working with.

As far as I am aware, the first study of representations of (strict) 2-groups is the PhD thesis

Magnus Forrester-Barker
Representations of Crossed Modules and $\mathrm{Cat}^1$-Groups
$\to$ pdf.

Here strict 2-groups (or the crossed modules of groups they are equivalent to) are represented on Baez-Crans 2-vector spaces. (The author thinks of these equivalently in terms of 2-term chain complexes.)

In general, a 2-vector space should be a 2-module over a 2-(semi)ring. Baez-Crans 2-vector spaces are modules over the 2-ring $K[0]$ which has elements of a field $K$ as objects and only identity morphisms between these.

Another interesting 2-semi ring is $\mathrm{Vect}_K$, the category of vector spaces over $K$. The 2-vector spaces invented by Kapranov and Voevodsky are modules over these. These are the ones appearing in BDR’s approach to elliptic cohomology ($\to$) and this is what Ganter and Kapranov are interested in.

Representation of 2-groups on the 2-category $\text{KV2Vect}$ of Kapranov-Voevodsky 2-vector spaces have been investigated before in

Josep Elgueta
Representation theory of 2-groups on Kapranov and Voevodsky’s 2-vector spaces
math.CT/0408120.

Ganter and Kapranov actually restrict attention to a special case of this. They restrict attention to 2-groups coming from crossed modules of the form

(Note that this is my paraphrasing of what they do, supposed to embed this into the context of the two papers mentioned above. Ganter/Kapranov don’t mention the term 2-group at all. Compare their paragraph 4.1 on p. 9.)

Hence for them, a 2-representation of an ordinary group $G$ is a lax 2-functor

This means that $\rho$ is a gadget which picks a single object $X \in \mathrm{Obj}(C)$, assigns to each group element $g \in G$ an endo-1-morphisms $X\overset{\rho(g)}{\to } X$ , and assigns to pairs of group elements $(g,h)$ a coherent 2-isomorphism

$\phi$ being coherent means that it satisfies an obvious associativity condition. This will turn out, as usual, to be related to certain cocycle data.

While being “only” a very special case of the most general notion of 2-representation, this already has very interesting examples. For instance “1-dimensional” 2-representations of this kind are nothing but projective ordinary representations of given central charge.

So far, this is nothing but a specialization of the above mentioned general idea of a 2-representation. The crucial new idea introduced by Ganter and Kapranov is that of a categorified concept of the notion of trace of a representation - hence a categorification of the concept of character.

The strong motivation for this, mentioned in their introduction and explained in more detail in section 8.1, is the following.

As I will explain in a minute, the trace operation invented by Ganter/Kapranov maps 2-representations of the group $G$ to representations of its inertia groupoid $\Lambda(G)$.

Now broaden the perspective by thinking of representations as special cases of equivariant bundles. A representation of an ordinary group $G$ is obviously the same as a $G$-equivariant bundle over a single point. From this it follows that the group completed decategorification of the category $\mathrm{Rep}(G)$ is nothing but the $G$-equivariant $K$-theory of a point. So there is nothing more natural than generalizing to $K$-theory over general orbifolds. But - and that is finally the point - Moerdijk tells us that the orbifold K-theory maps to the homology of the inertia groupoid $\Lambda(M)$ of the groupoid $M$ representing the orbifold (see p. 17 of his review).

While it may take a little to digest this analogy completely, the simple upshot is this: Ganter and Kapranov expect that their theory of 2-representations is nothing but the restriction to point-like orbifolds of a categorification of equivariant K-theory, hence of equivariant ellitptic cohomology.

This explains why the results they obtain, parts of which I will discuss now, reproduce phenomena known from what they call equivariant string theory in the introduction.

One main result is this.

Recall that for an ordinary group $G$ a class function is a map

to some field $K$ (i.e. a 1-dimensional representation) which is invariant under conjugation

Now, it turns out that in $G$-equivariant versions of higher cohomology theories (such as elliptic cohomology), one encounters generalizations of these, the $n$-class functions

which map $n$-tuples of pairwise commuting elements of $G$ to $K$, such that they are invariant under simultaneous conjugation of all $n$ elements.

But, of course for $n=1$ class functions come from traces of representations $\rho : \Sigma(G) \to \mathrm{Vect}$

Hence one is led to wonder if $n$-class functions are related to categorified traces. Indeed, Ganter and Kapranov show that their categorified trace gives rise to 2-class functions.

The definition of a categorified trace can nicely be motivated using Kapranov-Voevodsky 2-vector spaces. Here a linear map of 2-vector spaces (to be precise, we are implicitly using a semi-coordinatized version of these 2-vector spaces, but never mind) is a $n \times n$ matrix $A$ whose entries are vector spaces, $A = (A_{ij})$. Its 2-trace should hence be the vector space

2-morphisms in this category are simply matrices whose entries are linear maps, going componentwise between the entries of the source and target 1-morphisms. Hence it is obvious that the above 2-trace is nothing but the space of 2-morphisms from the identity 1-morphisms on $n$ to $A$:

This, then, suggests that we define generally for every 1-morphism $x \overset{F}{\to} x$ in some 2-category $C$

That’s definition 3.1 on p. 6 of. Apparently, the same concept is currently being studied by Bruce Bartlett and Simon Willerton.

If that’s the “categorical trace” then it’s obvious what the “categorical character” of a 2-representation is going to be (def. 4.6 on p. 11). Given a 2-representation $\rho$, its categorical character is simply

(Recall that $\rho(g)$ is the 1-morphisms which represents the action of the group element $g$.) Note that this takes values in $\mathrm{Set}$! The categorical trace is a set of 2-morphisms. Of course, if our target 2-category is suitably enriched over some category $V$, the categorical trace takes values in that $V$.

We expect the categorical trace to be not just a map, but in fact a functor. Indeed, it is. As is easy to see (prop. 4.8) the images $\mathrm{Tr}(\rho(g))$ and $\mathrm{Tr}(\rho(fgf^{-1}))$ are isomorphic. Hence, if we let $\Lambda(G)$ be the category whose objects are the elements of $G$ and whose morphisms are conjugations in $G$

then the categorical trace is a functor

One interesting example given is example 3.5 : the categorical trace of a 2-representations in the derived 2-category of bimodules gives the Hochschild cohomology of bimodules.

But a functor on a groupoid like $\Lambda(G)$ is the obvious generalization of a representation of a group. Hence $\mathrm{Tr}(\rho)$ is a groupoid representation. Here $\Lambda(G)$ is called the inertia groupoid of $G$ and in fact the above functor is just the 1-morphism part of the map

that I discussed above as a crucial motivation for the study of categorified traces.

So how do we get a 2-class function from this categorical trace? This way (def. 4.10):

Consider two commuting elements $g,h \in G$. They give rise to an automorphism

in the inertia groupoid $\Lambda(G)$. This means that

is an automorphism of the set $\mathrm{Tr}(\rho(g))$. If we work in the enriched case and our set is really a vector space, then $\mathrm{Tr}(\rho(g \overset{h}{\to} g))$ is an automorphism of that vector space. Hence we can take the ordinary trace of this linear map and finally obtain simply a number

One finds that this number is invariant under simultaneous conjugatins of $g$ and $h$

Hence, in form of a slogan we have

The ordinary trace of the categorical trace of a 2-representation of $G$ is a 2-class function for $G$.

Ganter and Kapranov discuss a couple of interesting examples. In particular, there is a close relation to E-theory (which Aaron Bergman a while ago mentioned here). There are also examples related to derived categories of coherent sheaves, which look particularly relevant to the idea of equivariant string theory, but I don’t have the energy left to talk about these now.