Whatever ultimately becomes of some aspects of the Standard Model – the Higgs boson, for example – here is a report (based on an experiment described here) that some of the fundamentals hold up well to experimental test. Specifically, the Spin-Statistics Theorem – the relationship between quantum numbers of elementary particles and the representation theory of the Poincare group. It would have been very surprising if things had been otherwise, but as usual, the more you rely on an idea, the more important it is to be sure it fits the facts. The association between physics and representation theory is one of those things.

So the fact that it all seems to work correctly is a bit of a relief for me. See below.

Since the paperwork is now well on its way, I may as well now mention here that I’ve taken a job as a postdoctoral researcher at CAMGSD, a centre at IST in Lisbon, starting in September. In a week or so I will be heading off to visit there – there are quite a few people there doing things I find quite interesting, so it should be an interesting trip. After that, I’ll be heading down to the south of the country for the Oporto meeting on Geometry, Topology and Physics, which is held this year in Faro. This year the subject is “categorification”, so my talk will be mainly about my paper on ETQFT. There are a bunch of interesting speakers – two I happen to know personally are Aaron Lauda and Joel Kamnitzer, but many others look quite promising.

In particular, one of the main invited speakers is Mikhail Khovanov, whose name is famously (for some values of “famous”) attached to Khovanov Homology, which is a categorification of the Jones Polynomial. Instead of a polynomial, it associates a graded complex of vector spaces to a knot. (Dror Bar-Natan wrote an intro, with many pictures and computations). Khovanov’s more recent work, with Aaron Lauda, has been on categorifying quantum groups (starting with this).

Now, as for me, since my talk in Faro will only be about 20 minutes, I’m glad of the opportunity to give some more background during the visit at IST. In particular, a bunch of the background to the ETQFT paper really depends on this paper on 2-linearization. I’ve given some previous talks on the subject, but this time I’m going to try to get a little further into how this fits into a more general picture. To repeat a bit of what’s in this post, 2-linearization describes a (weak) 2-functor:

$\Lambda : Span(Gpd) \rightarrow 2Vect$

where $Span(Gpd)$ has groupoids as its objects, spans of groupoid homomorphisms as its arrows, and spans-of-span-maps as 2-morphisms. $2Vect$ is the 2-category of 2-vector spaces, which I’ve explained before. This 2-functor is supposed to be a sort of “linearization”, which is a very simple functor

$L : Span(FinSet) \rightarrow Vect$

It takes a set $X$ to the free vector space $L(X) = \mathbb{C}^X$, and a span $X \stackrel{s}{\leftarrow} S \stackrel{t}{\rightarrow} Y$ to a linear map $L(S) : L(X) \rightarrow L(Y)$. This can be described in two stages, starting with a vector in $L(S)$, namely, a function $\psi : X \rightarrow \mathbb{C}$. The two stages are:

• First, “pull” $\psi$ up along $s$ to $\mathbb{C}^S$ (note: I’m conflating the set $S$ with the span $(S,s,t)$), to get the function $s^*\psi = \psi \circ s : S \rightarrow \mathbb{C}$.
• Then “push” this along $t$ to get $t_*(s^*\psi)$. The “push” operation $f_*$ along any map $f : X \rightarrow Y$ is determined by the fact that it takes the basis vector $\delta_x \in \mathbb{C}^X$ to the basis vector $\delta_{f(x)} \in \mathbb{C}^Y$ (these are the delta functions which are 1 on the given element and 0 elsewhere)

It’s helpful to note that, for a given map $f : X \rightarrow Y$, are linear adjoints (using the standard inner product where the delta functions are orthonormal). Combining them together – it’s easy to see – gives a linear map which can be described in the basis of delta functions by a matrix. The $(x,y)$-entry of the matrix counts the elements of $S$ which map to $(x,y)$ under $(s,t) : S \rightarrow X \times Y$. We interpret this by saying the matrix “counts histories” connecting $x$ to $y$.

In groupoidification, a-la Baez and Dolan (see the various references beyond the link), one replaces $FinSet$ with $FinGpd$, the 2-category of (essentially) finite groupoids, but we still have a functor into $Vect$. In fact, into $FinHilb$: the vector space $D(G)$ is the free one on isomorphism classes in $G$, but the linear maps (and the inner product) are tweaked using the groupoid cardinality, which can be any positive rational number. Then we say the matrix does a “sum over histories” of certain weights. In this paper, I extend this to “$U(1)$-groupoids”, which are labelled by phases – which represent the exponentiated action in quantum mechanics – and end up with complex matrices. So far so good.

The 2-linearization process is really “just” a categorification of what happens for sets, where we treat “groupoid” as the right categorification of “set”, and “Kapranov-Voevodsky 2-vector space” as the right categorification of “vector space”. (To treat “category” as the right categorification of “set”, one would have to use Elgueta’s “generalized 2-vector space“, which is probably morally the right thing to do, but here I won’t.) To a groupoid $X$, we assign the category of functors into $Vect$ – that is, $Rep(X)$ (in smooth cases, we might want to restrict what kind of representations we mean – see below).

To pull such a functor along a groupoid homomorphism $f : X \rightarrow Y$ is again done by precomposition: $f^*F = F \circ f$. The push map in 2-linearization is the Kan extension of the functor $\Psi$ along $f$. This is the universal way to push a functor forward, and is the (categorical!) adjoint to the pull map. (Kan extensions are supposed to come equipped with some natural transformations: these are the ones associated to the adjunction). Then composing “pull” and “push”, one categorifies “sum over histories”.

So here’s one thing this process is related to: in the case where our groupoids have just one object (i.e. are groups), and the homomorphism $f : X \rightarrow Y$ is an inclusion (conventionally written $H < G$), this goes by a familiar name in representation theory: restriction and induction. So, given a representation $\rho$ of $G$ (that is, a functor from $Y$ into $Vect$), there is an induced representation $res_H^G \rho = f^*\rho$, which is just the same representation space, acted on only by elements of $H$ (that is, $X$). This is the easy one. The harder one is the induced representation of $G$ from a representation $\tau$ of $H$ (i.e. $\tau : X \rightarrow Vect$, which is to say $ind^G_H \tau = f_* \tau : Y \rightarrow Vect$. The fact that these operations are adjoints goes in representation theory by the name “Frobenius reciprocity”.

These two operations were studied by George Mackey (in particular, though I’ve been implicitly talking about discrete groups, Mackey’s better known for looking at the case of unitary representations of compact Lie groups). The notion of a Mackey functor is supposed to abstract the formal properties of these operations. (A Mackey functor is really a pair of functors, one covariant and one contravariant – giving restriction and “transfer”/induction maps for – which have formal properties similar to the functor from groups into their representation rings – which it’s helpful to think of as the categories of representations, decategorificatied. In nice cases, a Mackey functor from a category $C$ is the same as a functor out of $Span(C)$).

Anyway, by way of returning to groupoids: the induced representation for groups is found by $\mathbb{C}[G] \otimes_{\mathbb{C}[H]} V$, where $V$ is the representation space of $\tau$. (For compact Lie groups, replace the group algebra $\mathbb{C}[G]$ with $L^2(G)$, and likewise for $H$). A similar formula shows up in the groupoid case, but with a contribution from each object (see the paper on 2-linearization for more details). This is also the formula for the Kan extension.

“Now wait a minute”, the categorically aware may ask, “do you mean the left Kan extension, or the right Kan extension?” That’s a good question! For one thing, they have different formulas: one involving limits, and the other involving colimits. Instead of answering it, I’ll talk about something not entirely unrelated – and a little more context for 2-linearization.

The setup here is actually a rather special case of Grothendieck’s six-operation framework, in the algebro-geometric context, for sheaves on (algebraic) spaces (there’s an overview in this talk by Joseph Lipman, the best I’ve been able to find online). Now, , these operations as extended to derived categories of sheaves (see this intro by R.P. Thomas). The derived category $D(X)$ is described concretely in terms of chain complexes of sheaves in $Sh(X)$, taken “up to homotopy” – it is a sort of categorification of cohomology. But of course, this contains $Sh(X)$ as trivial complexes (i.e. concentrated at level zero). The fact that our sheaves come from functors into $Vect$, which form a 2-vector space, so that functors between these are exact, means that there’s no nontrivial homology – so in our special case, the machinery of derived categories is more than we need.

This framework has been extended to groupoids – so the sheaves are on the space of objects, and are equivariant – as described in a paper by Moerdijk called “Etale Groupoids, Derived Categories, and Operations” (the situation of sheaves that are equivariant under a group action is described in more detail by Bernstein and Lunts in the Springer lecture notes “Equivariant Sheaves and Functors”). Sheaves on groupoids are essentially just equivariant sheaves on the space of objects. Now, given a morphism $f : X \ra Y$, there are four induced operations:

• $f^* , f^! : D(Y) \rightarrow D(X)$
• $f_*, f^! : D(X) \rightarrow D(Y)$ (in general right adjoint to $f^*$ and $f^!$)

(The other operations of the “six” are $hom$ and $\otimes$). The basic point here is that we can “pull” and “push” sheaves along the map $f$ in various ways. For our purposes, it’s enough to consider $f^*$ and $f_*$. The sheaves we want come from functors into $Vect$ (we actually have a vector space at each point in the space of objects). These are equivariant “bundles”, albeit not necessarily locally trivial. The fact that we can think of these as sheaves – of sections – tends to stay in the background most of the time, but in particular, being functors automatically makes the resulting sheaves equivariant. In the discrete case, we can just think of these as sheaves of vector spaces: just take $F(U)$ to be the direct sum of all the vector spaces at each object in any subset $U$ – all subsets are open in the discrete topology… For the smooth situation, it’s better not to do this, and think of the space of sections as a module over the ring of suitable functions.

Now to return to your very good question about “left or right Kan extension”… the answer is both. since for $Vect$-valued functors (where $Vect$ is the category of finite dimensional vector spaces), we have natural isomorphisms $f^* \cong f^!$ and $f_* \cong f_!$: these functors are \textit{ambiadjoint} (ie. both left and right adjoint). We use this to define the effect of $\Lambda$ on 2-morphisms in $Span_2(Gpd)$.

This isomorphism is closely related to the fact that finite-dimensional vector spaces are canonically isomorphic to their double-dual: $V \cong V^{**}$. That’s because the functors $f^*$ and $f_*$ are 2-linear maps. These are naturally isomorphic to maps represented as matrices of vector spaces. Taking an adjoint – aside from transposing the matrix, naturally replaces the matrices with their duals. Doing this twice, we get the isomorphisms above. So the functors are both left and right adjoint to each other, and thus in particular we have what is both left and right Kan extension. (This is also connected with the fact that, in $Vect$, the direct sum is both product and coproduct – i.e. limit and colimit.)

It’s worth pointing out, then, that we wouldn’t generally expect this to happen for infinite-dimensional vector spaces, since these are generally not canonically isomorphic to their double-duals. Instead, for this case we would need to be looking at functors valued in $Hilb$, since Hilbert spaces do have that property. That’s why, in the case of smooth groupoids (say, Lie groupoids), we end up talking about “(measurable) equivariant Hilbert bundles”. (In particular, the ring of functions over which our sheaves are modules is: the measurable ones. Why this is the right choice would be a bit of a digression, but roughly it’s analogous to the fact that $L^2(X)$ is a space of measurable functions. This is the limitation on which representations we want that I alluded to above.).

Now, $\Lambda$ is supposed to be a 2-functor. In general, given a category $C$ with all pullbacks, $Span_2(C)$ is the universal 2-category faithfully containing $C$ such that every morphism has an ambiadjoint. So the fact that the “pull” and “push” operations are ambiadjoint lets this 2-functor respect that property. It’s the unit and counits of the adjunctions which produce the effect of $\Lambda$ on 2-morphisms: given a span of span-maps, we take the two maps in the middle, consider the adjoint pairs of functors that come from them, and get a natural transformation which is just the composite of the counit of one adjunction and the unit of the other.

Here’s where we understand how this fits into the groupoidification program – because the effect of $\Lambda$ on 2-morphisms exactly reproduces the “degroupoidification” functor of Baez and Dolan, from spans of groupoids into $Vect$, when we think of such a span as a 2-morphism in $Hom(1,1)$ – that is, a span of maps of spans from the terminal groupoid to itself. In other words, degroupoidification is an example something we can do between ANY pair of groupoids – but in the special case where the representation theory all becomes trivial. (This by no means makes it uninteresting: in fact, it’s a perfect setting to understand almost everything else about the subject).

Now, to actually get all the coefficients to work out to give the groupoid cardinality, one has to be a bit delicate – the exact isomorphism between the construction of the left and right adjoint has some flexibility when we’re working over the field of complex numbers. But there’s a general choice – the Nakayama isomorphism – which works even when we’re replace $Vect$ by $R$-modules for some ring $R$. To make sure, for general $R$, that we have a true isomorphism, the map needs some constants. These happen to be, in our case, exactly the groupoid cardinalities to make the above statement true!

To me, this last part is a rather magical aspect of the whole thing, since the motivation I learned for groupoid cardinalities is quite remote from this – it’s just a valuation on groupoids which gets along with products and coproducts, and also with group actions (so that $|X/G| = |X|/|G|$, even when the action isn’t free). So one thing I’d like to know, but currently don’t is: how is it that this is “secretly” the same thing as the Nakayama isomorphism?