### 2-Hilbert Spaces

This blog has been on hiatus for a while, as I’ve been doing various other things, including spending some time in Hamburg getting set up for the move there. Another of these things has been working with Jamie Vicary on our project on the groupoidified Quantum Harmonic Oscillator (QHO for short). We’ve now put the first of two papers on the arXiv – this one is a relatively nonrigorous look at how this relates to categorification of the Heisenberg Algebra. Since John Baez is a high-speed blogging machine, he’s already beaten me to an overview of what the paper says, and there’s been some interesting discussion already. So I’ll try to say some different things about what it means, and let you take a look over there, or read the paper, for details.

I’ve given some talks about this project, but as we’ve been writing it up, it’s expanded considerably, including a lot of category-theoretic details which are going to be in the second paper in this series. But the basic point of this current paper is essentially visual and, in my opinion, fairly simple. The groupoidification of the QHO has a nice visual description, since it is all about the combinatorics of finite sets. This was described originally by Baez and Dolan, and in more detail in my very first paper. The other visual part here is the relation to Khovanov’s categorification of the Heisenberg algebra using a graphical calculus. (I wrote about this back when I first became aware of it.)

As a Representation

The scenario here actually has some common features with my last post. First, we have a monoidal category with duals, let’s say $C$ presented in terms of some generators and relations. Then, we find some concrete model of this abstractly-presented monoidal category with duals in a specific setting, namely $Span(Gpd)$.

Calling this “concrete” just refers to the fact that the objects in $Span(Gpd)$ have some particular structure in terms of underlying sets and so on. By a “model” I just mean a functor $C \rightarrow Span(Gpd)$ (“model” and “representation” mean essentially the same thing in this context). In fact, for this to make sense, I think of $C$ as a 2-category with one object. Then a model is just some particular choices: a groupoid to represent the unique object, spans of groupoids to represent the generating morphisms, spans of spans to represent the generating 2-morphisms, all chosen so that the defining relations hold.

In my previous post, $C$ was a category of cobordisms, but in this case, it’s essentially Khovanov’s monoidal category $H'$ whose objects are (oriented) dots and whose morphisms are certain classes of diagrams. The nice fact about the particular model we get is that the reasons these relations hold are easy to see in terms of a the combinatorics of sets. This is why our title describes what we got as “a combinatorial representation” Khovanov’s category $H'$ of diagrams, for which the ring of isomorphism classes of objects is the integral form of the algebra. This uses that $Span(Gpd)$ is not just a monoidal category: it can be a monoidal 2-category. What’s more, the monoidal category $H'$ “is” also a 2-category – with one object. The objects of $H'$ are really the morphisms of this 2-category.

So $H'$ is in some sense a universal theory (because it’s defined freely in terms of generators and relations) of what a categorification of the Heisenberg algebra must look like. Baez-Dolan groupoidification of the QHO then turns out to be a representation or model of it. In fact, the model is faithful, so that we can even say that it provides a combinatorial interpretation of that category.

The Combinatorial Model

Between the links above, you can find a good summary of the situation, so I’ll be a bit cursory. The model is described in terms of structures on finite sets. This is why our title calls this a “combinatorial representation” of Khovanov’s categorification.

This means that the one object of $H$ (as a 2-category) is taken to the groupoid $FinSet_0$ of finite sets and bijections (which we just called $S$ in the paper for brevity). This is the “Fock space” object. For simplicity, we can take an equivalent groupoid, which has just one $n$-element set for each $n$.

Now, a groupoid represents a system, whose possible configurations are the objects and whose symmetries are the morphisms. In this case, the possible configurations are the different numbers of “quanta”, and the symmetries (all set-bijections) show that all the quanta are interchangeable. I imagine a box containing some number of ping-pong balls.

A span of groupoids represents a process. It has a groupoid whose objects are histories (and morphisms are symmetries of histories). This groupoid has a pair of maps: to the system the process starts in, and to the system it ends in. In our model, the most important processes (which generate everything else) are the creation and annihilation operators, $a^{\dagger}$ and $a$ – and their categorified equivalents, $A$ and $A^{\dagger}$. The spans that represent them are very simple: they are processes which put a new ball into the box, or take one out, respectively. (Algebraically, they’re just a way to organize all the inclusions of symmetric groups $S_n \subset S_{n+1}$.)

The “canonical commutation relation“, which we write without subtraction thus:

$A A^{\dagger} = A^{\dagger} A + 1$

is already understood in the Baez-Dolan story: it says that there is one more way to remove a ball from a box after putting a new one into it (one more history for the process $A A^{\dagger}$) than to remove a ball and then add a new one (histories for $a^{\dagger} a$). This is fairly obvious: in the first instance, you have one more to choose from when removing the ball.

But the original Baez-Dolan story has no interesting 2-morphisms (the actual diagrams which are the 1-morphisms in $H$), whereas these are absolutely the whole point of a categorification in the sense Khovanov gets one, since the 1-morphisms of $H'$ determine what the isomorphism classes of objects even are.

So this means that we need to figure out what the 2-morphisms in $Span(Gpd)$ need to be – first in general, and second in our particular representation of $H$.

In general, a 2-morphism in $Span(Gpd)$ is a span of span-maps. You’ll find other people who take it to be a span-map. This would be a functor between the groupoids of histories: roughly, a map which assigns a history in the source span to a history in the target span (and likewise for symmetries), in a way that respects how they’re histories. But we don’t want just a map: we want a process which has histories of its own. We want to describe a “movie of processes” which change one process into another. These can have many histories of their own.

In fact, they’re not too complicated. Here’s one of Khovanov’s relation in $H'$ which forms part of how the commutation relation is expressed (shuffled to get rid of negatives, which we constantly need to do in the combinatorial model since we have no negative sets):

We read an upward arrow as “add a ball to the box”, and a downward arrow as “remove a ball”, and read right-to-left.  Both processes begin and end with“add then remove”. The right-hand side just leaves this process alone: it’s the identity.

The left-hand side shows a process-movie whose histories have two different cases. Suppose we begin with a history for which we add $x$ and then remove $y$. The first case is that $x = y$: we remove the same ball we put in. This amounts to doing nothing, so the first part of the movie eliminates all the adding and removing. The second part puts the add-remove pair back in.

The second case ensures that $x \neq y$, since it takes the initial history to the history (of a different process!) in which we remove $y$ and then add $x$ (impossible if $y = x$, since we can’t remove this ball before adding it). This in turn is taken to the history (of the original process!) where we add $x$ and then remove $y$; so this relates every history to itself, except for the case that $x = y$. Overall the sum of these relations give the identity on histories, which is the right hand side.

This picture includes several of the new 2-morphisms that we need to add to the Baez-Dolan picture: swapping the order of two generators, and adding or removing a pair of add/remove operations. Finding spans of spans which accomplish this (and showing they satisfy the right relations) is all that’s needed to finish up the combinatorial model.  So, for instance, the span of spans which adds a “remove-then-add” pair is this one:

If this isn’t clear, well, it’s explained in more detail in the paper.  (Do notice, though, that this is a diagram in groupoids: we need to specify that there are identity 2-cells in the span, rather than some other 2-cells.)

So this is basically how the combinatorial model works.

But in fact this description is (as often happens) chronologically backwards: what actually happened was that we had worked out what the 2-morphisms should be for different reasons. While trying to to understand what kind of structure this produced, we realized (thanks to Marco Mackaay) that the result was related to $H$, which in turn shed more light on the 2-morphisms we’d found.

So far so good. But what makes it possible to represent the kind of monoidal category we’re talking about in this setting is adjointness. This is another way of saying what I meant up at the top by saying we start with a monoidal category with duals.  This means morphisms each have a partner – a dual, or adjoint – going in the opposite direction.  The representations of the raising and lowering operators of the Heisenberg algebra on the Hilbert space for the QHO are linear adjoints. Their categorifications also need to be adjoints in the sense of adjoint 1-morphisms in a 2-category.

This is an abstraction of what it means for two functors $F$ and $G$ to be adjoint. In particular, it means there have to be certain 2-cells such as the unit $\eta : Id \Rightarrow G \circ F$ and counit $\epsilon : F \circ G \Rightarrow Id$ satisfying some nice relations. In fact, this only makes $F$ a left adjoint and $G$ a right adjoint – in this situation, we also have another pair which makes $F$ a right adjoint and $G$ a left one. That is, they should be “ambidextrous adjoints”, or “ambiadjoints” for short. This is crucial if they’re going to represent any graphical calculus of the kind that’s involved here (see the first part of this paper by Aaron Lauda, for instance).

So one of the theorems in the longer paper will show concretely that any 1-morphism in $Span(Gpd)$ has an ambiadjoint – which happens to look like the same span, but thought of as going in the reverse direction. This is somewhat like how the adjoint of a real linear map, expressed as a matrix relative to well-chosen bases, is just the transpose of the same matrix. In particular, $A$ and $A^{\dagger}$ are adjoints in just this way. The span-of-span-maps I showed above is exactly the unit for one side of this ambi-adjunction – but it is just a special case of something that will work for any span and its adjoint.

Finally, there’s something a little funny here. Since the morphisms of $Span(Gpd)$ aren’t functors or maps, this combinatorial model is not exactly what people often mean by a “categorified representation”. That would be an action on a category in terms of functors and natural transformations. We do talk about how to get one of these on a 2-vector space out of our groupoidal representation toward the end.

In particular, this amounts to a functor into $2Vect$ – the objects of $2Vect$ being categories of a particular kind, and the morphisms being functors that preserve all the structure of those categories. As it turns out, the thing about this setting which is good for this purpose is that all those functors have ambiadjoints. The “2-linearization” that takes $Span(Gpd)$ into $2Vect$ is a 2-functor, and this means that all the 2-cells and equations that make two morphisms ambiadjoints carry over. In $2Vect$, it’s very easy for this to happen, since all those ambiadjoints are already present. So getting representations of categorified algebras that are made using these monoidal categories of diagrams on 2-vector spaces is fairly natural – and it agrees with the usual intuition about what “representation” means.

Anything I start to say about this is in danger of ballooning, but since we’re already some 40 pages into the second paper, I’ll save the elaboration for that…

I’ve written here before about building topological quantum field theories using groupoidification, but I haven’t yet gotten around to discussing a refinement of this idea, which is in the most recent version of my paper on the subject.  I also gave a talk about this last year in Erlangen. The main point of the paper is to pull apart some constructions which are already fairly well known into two parts, as part of setting up a category which is nice for supporting models of fairly general physical systems, using an extension of the  concept of groupoidification. So here’s a somewhat lengthy post which tries to unpack this stuff a bit.

Factoring TQFT

The older version of this paper talked about the untwisted version of the Dijkgraaf-Witten (DW for short) model, which is a certain kind of TQFT based on a gauge theory with a finite gauge group.  (Freed and Quinn put it as: “Chern-Simons theory with finite gauge group”).  The new version gets the general – that is, the twisted – form in the same way: factoring the theory into two parts. So, the DW model, which was originally described by Dijkgraaf and Witten in terms of a state-sum, is a functor

$Z : 3Cob \rightarrow Vect$

The “twisting” is the point of their paper, “Topological Gauge Theories and Group Cohomology”.  The twisting has to do with the action for some physical theory. Now, for a gauge theory involving flat connections, the kind of gauge-theory actions which involve the curvature of a connection make no sense: the curvature is zero.  So one wants an action which reflects purely global features of connections.  The cohomology of the gauge group is where this comes from.

Now, the machinery I describe is based on a point of view which has been described in a famous paper by Freed, Hopkins, Lurie and Teleman (FHLT for short – see further discussion here) in terms in which the two stages are called the “classical field theory” (which has values in groupoids), and the “quantization functor”, which takes one into Hilbert spaces.

Actually, we really want to have an “extended” TQFT: a TQFT gives a Hilbert space for each 2D manifold (“space”), and a linear map for a 3D cobordism (“spacetime”) between them. An extended TQFT will assign (higher) algebraic data to lower-dimension boundaries still.  My paper talks only about the case where we’ve extended down to codimension 2, whereas FHLT talk about extending “down to a point”. The point of this first stopping point is to unpack explicitly and computationally what the factorization into two parts looks like at the first level beyond the usual TQFT.

In the terminology I use, the classical field theory is:

$A^{\omega} : nCob_2 \rightarrow Span_2(Gpd)^{U(1)}$

This depends on a cohomology class $[\omega] \in H^3(G,U(1))$. The “quantization functor” (which in this case I call “2-linearization”):

$\Lambda^{U(1)} : Span_2(Gpd)^{U(1)} \rightarrow 2Vect$

The middle stage involves the monoidal 2-category I call $Span_2(Gpd)^{U(1)}$.  (In FHLT, they use different terminology, for instance “families” rather than “spans”, but the principle is the same.)

Freed and Quinn looked at the quantization of the “extended” DW model, and got a nice geometric picture. In it, the action is understood as a section of some particular line-bundle over a moduli space. This geometric picture is very elegant once you see how it works, which I found was a little easier in light of a factorization through $Span_2(Gpd)$.

This factorization isolates the geometry of this particular situation in the “classical field theory” – and reveals which of the features of their setup (the line bundle over a moduli space) are really part of some more universal construction.

In particular, this means laying out an explicit definition of both $Span_2(Gpd)^{U(1)}$ and $\Lambda^{U(1)}$.

2-Linearization Recalled

While I’ve talked about it before, it’s worth a brief recap of how 2-linearization works with a view to what happens when you twist it via groupoid cohomology. Here we have a 2-category $Span(Gpd)$, whose objects are groupoids ($A$, $B$, etc.), whose morphisms are spans of groupoids:

$A \stackrel{s}{\leftarrow} X \stackrel{t}{\rightarrow} B$

and whose 2-morphisms are spans of span-maps (taken up to isomorphism), which look like so:

(And, by the by: how annoying that WordPress doesn’t appear to support xypic figures…)

These form a (symmetric monoidal) 2-category, where composition of spans works by taking weak pullbacks.  Physically, the idea is that a groupoid has objects which are configurations (in the cause of gauge theory, connections on a manifold), and morphisms which are symmetries (gauge transformations, in this case).  Then a span is a groupoid of histories (connections on a cobordism, thought of as spacetime), and the maps $s,t$ pick out its starting and ending configuration.  That is, $A = A_G(S)$ is the groupoid of flat $G$-connections on a manifold $S$, and $X = A_G(\Sigma)$ is the groupoid of flat $G$-connections on some cobordism $\Sigma$, of which $S$ is part of the boundary.  So any such connection can be restricted to the boundary, and this restriction is $s$.

Now 2-linearization is a 2-functor:

$\Lambda : Span_2(Gpd)^{U(1)} \rightarrow 2Vect$

It gives a 2-vector space (a nice kind of category) for each groupoid $G$.  Specifically, the category of its representations, $Rep(G)$.  Then a span turns into a functor which comes from “pulling” back along $s$ (the restricted representation where $X$ acts by first applying $s$ then the representation), then “pushing” forward along $t$ (to the induced representation).

What happens to the 2-morphisms is conceptually more complicated, but it depends on the fact that “pulling” and “pushing” are two-sided adjoints. Concretely, it ends up being described as a kind of “sum over histories” (where “histories” are the objects of $Y$), which turns out to be exactly the path integral that occurs in the TQFT.

Or at least, it’s the path integral when the action is trivial! That is, if $S=0$, so that what’s integrated over paths (“histories”) is just $e^{iS}=1$. So one question is: is there a way to factor things in this way if there’s a nontrivial action?

Cohomological Twisting

The answer is by twisting via cohomology. First, let’s remember what that means…

We’re talking about groupoid cohomology for some groupoid $G$ (which you can take to be a group, if you like).  “Cochains” will measure how much some nice algebraic fact, such as being a homomorphism, or being associative, “fails to occur”.  “Twisting by a cocycle” is a controlled way to force some such failure to happen.

So, an $n$-cocycle is some function of $n$ composable morphisms of $G$ (or, if there’s only one object, “group elements”, which amounts to the same thing).  It takes values in some group of coefficients, which for us is always $U(1)$

The trivial case where $n=0$ is actually slightly subtle: a 0-cocycle is an invariant function on the objects of a groupoid. (That is, it takes the same value on any two objects related by an (iso)morphism. (Think of the object as a sequence of zero composable morphisms: it tells you where to start, but nothing else.)

The case $n=1$ is maybe a little more obvious. A 1-cochain $f \in Z^1_{gpd}(G,U(1))$ can measure how a function $h$ on objects might fail to be a 0-cocycle. It is a $U(1)$-valued function of morphisms (or, if you like, group elements).  The natural condition to ask for is that it be a homomorphism:

$f(g_1 \circ g_2) = f(g_1) f(g_2)$

This condition means that a cochain $f$ is a cocycle. They form an abelian group, because functions satisfying the cocycle condition are closed under pointwise multiplication in $U(1)$. It will automatically by satisfied for a coboundary (i.e. if $f$ comes from a function $h$ on objects as $f(g) = \delta h (g) = h(t(g)) - h(s(g))$). But not every cocycle is a coboundary: the first cohomology $H^1(G,U(1))$ is the quotient of cocycles by coboundaries. This pattern repeats.

It’s handy to think of this condition in terms of a triangle with edges $g_1$, $g_2$, and $g_1 \circ g_2$.  It says that if we go from the source to the target of the sequence $(g_1, g_2)$ with or without composing, and accumulate $f$-values, our $f$ gives the same result.  Generally, a cocycle is a cochain satisfying a “coboundary” condition, which can be described in terms of an $n$-simplex, like this triangle. What about a 2-cocycle? This describes how composition might fail to be respected.

So, for instance, a twisted representation $R$ of a group is not a representation in the strict sense. That would be a map into $End(V)$, such that $R(g_1) \circ R(g_2) = R(g_1 \circ g_2)$.  That is, the group composition rule gets taken directly to the corresponding rule for composition of endomorphisms of the vector space $V$.  A twisted representation $\rho$ only satisfies this up to a phase:

$\rho(g_1) \circ \rho(g_2) = \theta(g_1,g_2) \rho(g_1 \circ g_2)$

where $\theta : G^2 \rightarrow U(1)$ is a function that captures the way this “representation” fails to respect composition.  Still, we want some nice properties: $\theta$ is a “cocycle” exactly when this twisting still makes $\rho$ respect the associative law:

$\rho(g_1) \rho( g_2 \circ g_3) = \rho( g_1 \circ g_2) \circ \rho( g_3)$

Working out what this says in terms of $\theta$, the cocycle condition says that for any composable triple $(g_1, g_2, g_3)$ we have:

$\theta( g_1, g_2 \circ g_3) \theta (g_2,g_3) = \theta(g_1,g_2) \theta(g_1 \circ g_2, g_3)$

So $H^2_{grp}(G,U(1))$ – the second group-cohomology group of $G$ – consists of exactly these $\theta$ which satisfy this condition, which ensures we have associativity.

Given one of these $\theta$ maps, we get a category $Rep^{\theta}(G)$ of all the $\theta$-twisted representations of $G$. It behaves just like an ordinary representation category… because in fact it is one! It’s the category of representations of a twisted version of the group algebra of $G$, called $C^{\theta}(G)$. The point is, we can use $\theta$ to twist the convolution product for functions on $G$, and this is still an associative algebra just because $\theta$ satisfies the cocycle condition.

The pattern continues: a 3-cocycle captures how some function of 2 variable may fail to be associative: it specifies an associator map (a function of three variables), which has to satisfy some conditions for any four composable morphisms. A 4-cocycle captures how a map might fail to satisfy this condition, and so on. At each stage, the cocycle condition is automatically satisfied by coboundaries. Cohomology classes are elements of the quotient of cocycles by coboundaries.

So the idea of “twisted 2-linearization” is that we use this sort of data to change 2-linearization.

Twisted 2-Linearization

The idea behind the 2-category $Span(Gpd)^{U(1)}$ is that it contains $Span(Gpd)$, but that objects and morphisms also carry information about how to “twist” when applying the 2-linearization $\Lambda$.  So in particular, what we have is a (symmetric monoidal) 2-category where:

• Objects consist of $(A, \theta)$, where $A$ is a groupoid and $\theta \in Z^2(A,U(1))$
• Morphisms from $A$ to $B$ consist of a span $(X,s,t)$ from $A$ to $B$, together with $\alpha \in Z^1(X,U(1))$
• 2-Morphisms from $X_1$ to $X_2$ consist of a span $(Y,\sigma,\tau)$ from $X$, together with $\beta \in Z^0(Y,U(1))$

The cocycles have to satisfy some compatibility conditions (essentially, pullbacks of the cocycles from the source and target of a span should land in the same cohomology class).  One way to see the point of this requirement is to make twisted 2-linearization well-defined.

One can extend the monoidal structure and composition rules to objects with cocycles without too much trouble so that $Span(Gpd)$ is a subcategory of $Span(Gpd)^{U(1)}$. The 2-linearization functor extends to $\Lambda^{U(1)} : Span(Gpd)^{U(1)} \rightarrow 2Vect$:

• On Objects: $\Lambda^{U(1)} (A, \theta) = Rep^{\theta}(A)$, the category of $\theta$-twisted representation of $A$
• On Morphisms: $\Lambda^{U(1)} ( (X,s,t) , \alpha )$ comes by pulling back a twisted representation in $Rep^{\theta_A}(A)$ to one in $Rep^{s^{\ast}\theta_A}(X)$, pulling it through the algebra map “multiplication by $\alpha$“, and pushing forward to $Rep^{\theta_B}(B)$
• On 2-Morphisms: For a span of span maps, one uses the usual formula (see the paper for details), but a sum over the objects $y \in Y$ picks up a weight of $\beta(y)$ at each object

When the cocycles are trivial (evaluate to 1 always), we get back the 2-linearization we had before. Now the main point here is that the “sum over histories” that appears in the 2-morphisms now carries a weight.

So the twisted form of 2-linearization uses the same “pull-push” ideas as 2-linearization, but applied now to twisted representations. This twisting (at the object level) uses a 2-cocycle. At the morphism level, we have a “twist” between “pull” and “push” in constructing . What the “twist” actually means depends on which cohomology degree we’re in – in other words, whether it’s applied to objects, morphisms, or 2-morphisms.

The “twisting” by a 0-cocycle just means having a weight for each object – in other words, for each “history”, or connection on spacetime, in a big sum over histories. Physically, the 0-cocycle is playing the role of the Lagrangian functional for the DW model. Part of the point in the FHLT program can be expressed by saying that what Freed and Quinn are doing is showing how the other cocycles are also the Lagrangian – as it’s seen at higher codimension in the more “local” theory.

For a TQFT, the 1-cocycles associated to morphisms describe how to glue together values for the Lagrangian that are associated to histories that live on different parts of spacetime: the action isn’t just a number. It is a number only “locally”, and when we compose 2-morphisms, the 0-cocycle on the composite picks up a factor from the 1-morphism (or 0-morphism, for a horizontal composite) where they’re composed.

This has to do with the fact that connections on bits of spacetime can be glued by particular gauge transformations – that is, morphisms of the groupoid of connections. Just as the gauge transformations tell how to glue connections, the cocycles associated to them tell how to glue the actions. This is how the cohomological twisting captures the geometric insight that the action is a section of a line bundle – not just a function, which is a section of a trivial bundle – over the moduli space of histories.

So this explains how these cocycles can all be seen as parts of the Lagrangian when we quantize: they explain how to glue actions together before using them in a sum-over histories. Gluing them this way is essential to make sure that $\Lambda^{U(1)}$ is actually a functor. But if we’re really going to see all the cocycles as aspects of “the action”, then what is the action really? Where do they come from, that they’re all slices of this bigger thing?

Twisting as Lagrangian

Now the DW model is a 3D theory, whose action is specified by a group-cohomology class $[\omega] \in H^3_{grp}(G,U(1))$. But this is the same thing as a class in the cohomology of the classifying space: $[\omega] \in H^3(BG,U(1))$. This takes a little unpacking, but certainly it’s helpful to understand that what cohomology classes actually classify are… gerbes. So another way to put a key idea of the FHLT paper, as Urs Schreiber put it to me a while ago, is that “the action is a gerbe on the classifying space for fields“.

What does this mean?

This map is given as a path integral over all connections on the space(-time) $S$, which is actually just a sum, since the gauge group is finite and so all the connections are flat.  The point is that they’re described by assigning group elements to loops in $S$:

$A : \pi_1(M) \rightarrow G$

But this amounts to the same thing as a map into the classifying space of $G$:

$f_A : M \rightarrow BG$

This is essentially the definition of $BG$, and it implies various things, such as the fact that $BG$ is a space whose fundamental group is $G$, and has all other homotopy groups trivial. That is, $BG$ is the Eilenberg-MacLane space $K(G,1)$. But the point is that the groupoid of connections and gauge transformations on $S$ just corresponds to the mapping space $Maps(S,BG)$. So the groupoid cohomology classes we get amount to the same thing as cohomology classes on this space. If we’re given $[\omega] \in H^3(BG,U(1))$, then we can get at these by “transgression” – which is very nicely explained in a paper by Simon Willerton.

The essential idea is that a 3-cocycle $\omega$ (representing the class $[\omega]$) amounts to a nice 3-form on $BG$ which we can integrate over a 3-dimentional submanifold to get a number. For a $d$-dimensional $S$, we get such a 3-manifold from a $(3-d)$-dimensional submanifold of $Maps(S,BG)$: each point gives a copy of $S$ in $BG$. Then we get a $(3-d)$-cocycle on $Maps(S,BG)$ whose values come from integrating $\omega$ over this image. Here’s a picture I used to illustrate this in my talk:

Now, it turns out that this gives 2-cocycles for 1-manifolds (the objects of $3Cob_2$, 1-cocycles on 2D cobordisms between them, and 0-cocycles on 3D cobordisms between these cobordisms. The cocycles are for the groupoid of connections and gauge transformations in each case. In fact, because of Stokes’ theorem in $BG$, these have to satisfy all the conditions that make them into objects, morphisms, and 2-morphisms of $Span^{U(1)}(Gpd)$. This is the geometric content of the Lagrangian: all the cocycles are really “reflections” of $\omega$ as seen by transgression: pulling back along the evaluation map $ev$ from the picture. Then the way you use it in the quantization is described exactly by $\Lambda^{U(1)}$.

What I like about this is that $\Lambda^{U(1)}$ is a fairly universal sort of thing – so while this example gets its cocycles from the nice geometry of $BG$ which Freed and Quinn talk about, the insight that an action is a section of a (twisted) line bundle, that actions can be glued together in particular ways, and so on… These presumably can be moved to other contexts.

In the first week of November, I was in Montreal for the biannual meeting of the Philosophy of Science Association, at the invitation of Hans Halvorson and Steve Awodey.  This was for a special session called “Category Theoretical Reflections on the Foundations of Physics”, which also had talks by Bob Coecke (from Oxford), Klaas Landsman (from Radboud University in Nijmegen), and Gonzalo Reyes (from the University of Montreal).  Slides from the talks in this session have been collected here by Steve Awodey.  The meeting was pretty big, and there were a lot of talks on a lot of different topics, some more technical, and some less.  There were enough sessions relating to physics that I had a full schedule just attending those, although for example there were sessions on biology and cognition which I might otherwise have been interested in sitting in on, with titles like “Biology: Evolution, Genomes and Biochemistry”, “Exploring the Complementarity between Economics and Recent Evolutionary Theory”, “Cognitive Sciences and Neuroscience”, and “Methodological Issues in Cognitive Neuroscience”.  And, of course, more fundamental philosophy of science topics like “Fictions and Scientific Realism” and “Kinds: Chemical, Biological and Social”, as well as socially-oriented ones such as “Philosophy of Commercialized Science” and “Improving Peer Review in the Sciences”.  However, interesting as these are, one can’t do everything.

In some ways, this was a really great confluence of interests for me – physics and category theory, as seen through a philosophical lens.  I don’t know exactly how this session came about, but Hans Halvorson is a philosopher of science who started out in physics (and has now, for example, learned enough category theory to teach the course in it offered at Princeton), and Steve Awodey is a philosopher of mathematics who is interested in category theory in its own right.  They managed to get this session brought in to present some of the various ideas about the overlap between category theory and physics to an audience mostly consisting of philosophers, which seems like a good idea.  It was also interesting for me to get a view into how philosophers approach these subjects – what kind of questions they ask, how they argue, and so on.  As with any well-developed subject, there’s a certain amount of jargon and received ideas that people can refer to – for example, I learned the word and current usage (though not the basic concept) of supervenience, which came up, oh, maybe 5-10 times each day.

There are now a reasonable number of people bringing categorical tools to bear on physics – especially quantum physics.  What people who think about the philosophy of science can bring to this research is the usual: careful, clear thinking about the fundamental concepts involved in a way that tries not to get distracted by the technicalities and keep the focus on what is important to the question at hand in a deep way.  In this case, the question at hand is physics.  Philosophy doesn’t always accomplish this, of course, and sometimes get sidetracked by what some might call “pseudoquestions” – the kind of questions that tend to arise when you use some folk-theory or simple intuitive understanding of some subtler concept that is much better expressed in mathematics.  This is why anyone who’s really interested in the philosophy of science needs to learn a lot about science in its own terms.  On the whole, this is what they actually do.

And, of course, both mathematicians and physicists try to do this kind of thinking themselves, but in those fields it’s easy – and important! – to spend a lot of time thinking about some technical question, or doing extensive computations, or working out the fiddly details of a proof, and so forth.  This is the real substance of the work in those fields – but sometimes the bigger “why” questions, that address what it means or how to interpret the results, get glossed over, or answered on the basis of some superficial analogy.  Mind you – one often can’t really assess how a line of research is working out until you’ve been doing the technical stuff for a while.  Then the problem is that people who do such thinking professionally – philosophers – are at a loss to understand the material because it’s recent and technical.  This is maybe why technical proficiency in science has tended to run ahead of real understanding – people still debate what quantum mechanics “means”, even though we can use it competently enough to build computers, nuclear reactors, interferometers, and so forth.

Anyway – as for the substance of the talks…  In our session, since every speaker was a mathematician in some form, they tended to be more technical.  You can check out the slides linked to above for more details, but basically, four views of how to draw on category theory to talk about physics were represented.  I’ve actually discussed each of them in previous posts, but in summary:

• Bob Coecke, on “Quantum Picturalism”, was addressing the monoidal dagger-category point of view, which looks at describing quantum mechanical operations (generally understood to be happening in a category of Hilbert spaces) purely in terms of the structure of that category, which one can see as a language for handling a particular kind of logic.  Monoidal categories, as Peter Selinger as painstakingly documented, can be described using various graphical calculi (essentially, certain categories whose morphisms are variously-decorated “strands”, considered invariant under various kinds of topological moves, are the free monoidal categories with various structures – so anything you can prove using these diagrams is automatically true for any example of such categories).  Selinger has also shown that, for the physically interesting case of dagger-compact closed monoidal categories, a theorem is true in general if and only if it’s true for (finite dimensional) Hilbert spaces, which may account for why Hilbert spaces play such a big role in quantum mechanics.  This program is based on describing as much of quantum mechanics as possible in terms of this kind of diagrammatic language.  This stuff has, in some ways, been explored more through the lens of computer science than physics per se – certainly Selinger is coming from that background.  There’s also more on this connection in the “Rosetta Stone” paper by John Baez and Mike Stay,
• My talk (actually third, but I put it here for logical flow) fits this framework, more or less.  I was in some sense there representing a viewpoint whose current form is due to Baez and Dolan, namely “groupoidification”.  The point is to treat the category $Span(Gpd)$ as a “categorification” of (finite dimensional) Hilbert spaces in the sense that there is a representation map $D : Span(Gpd) \rightarrow Hilb$ so that phenomena living in $Hilb$ can be explained as the image of phenomena in $Span(Gpd)$.  Having done that, there is also a representation of $Span(Gpd)$ into 2-Hilbert spaces, which shows up more detail (much more, at the object level, since Tannaka-Krein reconstruction means that the monoidal 2-Hilbert space of representations of a groupoid is, at least in nice cases, enough to completely reconstruct it).  This gives structures in $2Hilb$ which “conceptually” categorify the structures in $Hilb$, and are also directly connected to specific Hilbert spaces and maps, even though taking equivalence classes in $2Hilb$ definitely doesn’t produce these.  A “state” in a 2-Hilbert space is an irreducible representation, though – so there’s a conceptual difference between what “state” means in categorified and standard settings.  (There’s a bit more discussion in my notes for the talk than in the slides above.)
• Klaas Landsman was talking about what he calls “Bohrification“, which, on the technical side, makes use of Topos theory.  The philosophical point comes from Niels Bohr’s “doctrine of classical concepts” – that one should understand quantum systems using concepts from the classical world.  In practice, this means taking a (noncommutative) von Neumann algebra $A$ which describes the observables a quantum system and looking at it via its commutative subalgebras.  These are organized into a lattice – in fact, a site.  The idea is that the spectrum of $A$ lives in the topos associated to this site: it’s a presheaf that, over each commutative subalgebra $C \subset A$, just gives the spectrum of $C$.  This is philosophically nice in that the “Bohrified” propositions actually behave in a logically sensible way.  The topos approach comes from Chris Isham, developed further with Andreas Doring. (Note the series of four papers by both from 2007.  Their approach is in some sense dual to that of Lansman, Heunen and Spitters, in the sense that they look at the same site, but look at dual toposes – one of sheaves, the other of cosheaves.  The key bit of jargon in Isham and Doring’s approach is “daseinization”, which is a reference to Heidegger’s “Being and Time”.  For some reason this makes me imagine Bohr and Heidegger in a room, one standing on the ceiling, one on the floor, disputing which is which.)
• Gonzalo Reyes talked about synthetic differential geometry (SDG) as a setting for building general relativity.  SDG is a way of doing differential geometry in a category where infinitesimals are actually available, that is, there is a nontrivial set $D = \{ x \in \mathbb{R} | x^2 = 0 \}$.  This simplifies discussions of vector fields (tangent vectors will just be infinitesimal vectors in spacetime).  A vector field is really a first order DE (and an integral curve tangent to it is a solution), so it’s useful to have, in SDG, the fact that any differentiable curve is, literally, infinitesimally a line.  Then the point is that while the gravitational “field” is a second-order DE, so not a field in this sense, the arguments for GR can be reproduced nicely in SDG by talking about infinitesimally-close families of curves following geodesics.  Gonzalo’s slides are brief by necessity, but happily, more details of this are in his paper on the subject.

The other sessions I went to were mostly given by philosophers, rather than physicists or mathematicians, though with exceptions.  I’ll briefly present my own biased and personal highlights of what I attended.  They included sessions titled:

Quantum Physics“: Edward Slowik talked about the “prehistory of quantum gravity”, basically revisiting the debate between Newton and Leibniz on absolute versus relational space, suggesting that Leibniz’ view of space as a classification of the relation of his “monads” is more in line with relational theories such as spin foams etc.  M. Silberstein and W. Stuckey – gave a talk about their “relational blockworld” (described here) which talks about QFT as an approximation to a certain discrete theory, built on a graph, where the nodes of the graph are spacetime events, and using an action functional on the graph.

Meinard Kuhlmann gave an interesting talk about “trope bundles” and AQFTTrope ontology is an approach to “entities” that doesn’t assume there’s a split between “substrates” (which have no properties themselves), and “properties” which they carry around.  (A view of ontology that goes back at least to Aristotle’s “substance” and “accident” distinction, and maybe further for all I know).  Instead, this is a “one-category” ontology – the basic things in this ontology are “tropes”, which he defined as “individual property instances” (i.e. as opposed to abstract properties that happen to have instances).  “Things” then, are just collections of tropes.  To talk about the “identity” of a thing means to pick out certain of the tropes as the core ones that define that thing, and others as peripheral.  This struck me initially as a sort of misleading distinction we impose (say, “a sphere” has a core trope of its radial symmetry, and incidental tropes like its colour – but surely the way of picking the object out of the world is human-imposed), until he gave the example from AQFT.  To make a long story short, in this setup, the key entites are something like elementary particles, and the core tropes are those properties that define an irreducible representation of a $C^{\star}$-algebra (things like mass, spin, charge, etc.), whereas the non-core tropes are those that identify a state vector within such a representation: the attributes of the particle that change over time.

I’m not totally convinced by the “trope” part of this (surely there are lots of choices of the properties which determine a representation, but I don’t see the need to give those properties the burden of being the only ontologically primaries), but I also happen to like the conclusions because in the 2Hilbert picture, irreducible representations are states in a 2-Hilbert space, which are best thought of as morphisms, and the state vectors in their components are best thought of in terms of 2-morphisms.  An interpretation of that setup says that the 1-morphism states define which system one’s talking about, and the 2-morphism states describe what it’s doing.

New Directions Concerning Quantum Indistinguishability“: I only caught a couple of the talks in this session, notably missing Nick Huggett’s “Expanding the Horizons of Quantum Statistical Mechanics”.  There were talks by John Earman (“The Concept of Indistinguishable Particles in Quantum
Mechanics”), and by Adam Caulton (based on work with Jeremy Butterfield) on “On the Physical Content of the Indistinguishability Postulate”.  These are all about the idea of indistinguishable particles, and the statistics thereof.  Conventionally, in QM you only talk about bosons and fermions – one way to say what this means is that the permutation group $S_n$ naturally acts on a system of $n$ particles, and it acts either trivially (not altering the state vector at all), or by sign (each swap of two particles multiplies the state vector by a minus sign).  This amounts to saying that only one-dimensional representations of $S_n$ occur.  It is usually justified by the “spin-statistics theorem“, relating it to the fact that particles have either integer or half-integer spins (classifying representations of the rotation group).  But there are other representations of $S_n$, labelled by Young diagrams, though they are more than one-dimensional.  This gives rise to “paraparticle” statistics.  On the other hand, permuting particles in two dimensions is not homotopically trivial, so one ought to use the braid group $B_n$, rather than $S_n$, and this gives rise again to different statistics, called “anyonic” statistics.

One recurring idea is that, to deal with paraparticle statistics, one needs to change the formalism of QM a bit, and expand the idea of a “state vector” (or rather, ray) to a “generalized ray” which has more dimensions – corresponding to the dimension of the representation of $S_n$ one wants the particles to have.  Anyons can be dealt with a little more conventionally, since a 2D system may already have them.  Adam Caulton’s talk described how this can be seen as a topological phenomenon or a dynamical one – making an analogy with the Bohm-Aharonov effect, where the holonomy of an EM field around a solenoid can be described either dynamically with an interacting Lagrangian on flat space, or topologically with a free Lagrangian in space where the solenoid has been removed.

Quantum Mechanics“: A talk by Elias Okon and Craig Callender about QM and the Equivalence Principle, based on this.  There has been some discussion recently as to whether quantum mechanics is compatible with the principle that relates gravitational and inertial mass.  They point out that there are several versions of this principle, and that although QM is incompatible with some versions, these aren’t the versions that actually produce general relativity.  (For example, objects with large and small masses fall differently in quantum physics, because though the mean travel time is the same, the variance is different.  But this is not a problem for GR, which only demands that all matter responds dynamically to the same metric.)  Also, talks by Peter Lewis on problems with the so-called “transactional interpretation” of QM, and Bryan Roberts on time-reversal.

Why I Care About What I Don’t Yet Know“:  A funny name for a session about time-asymmetry, which is the essentially philosophical problem of why, if the laws of physics are time-symmetric (which they approximately are for most purposes), what we actually experience isn’t.  Personally, the best philosophical account of this I’ve read is Huw Price’s “Time’s Arrow“, though Reichenbach’s “The Direction of Time” has good stuff in it also, and there’s also Zeh’s more technical “The Physical Basis of the Direction of Time“. In the session, Chris Suhler and Craig Callender gave an account of how, given causal asymmetry, our subjective asymmetry of values for the future and the past can arise (the intuitively obvious point being that if we can influence the future and not the past, we tend to value it more).  Mathias Frisch talked about radiation asymmetry (the fact that it’s equally possible in EM to have waves converging on a source than spreading out from it, yet we don’t see this).  Owen Maroney argued that “There’s No Route from Thermodynamics to the Information Asymmetry” by describing in principle how to construct a time-reversed (probabilisitic) computer.  David Wallace spoke on “The Logic of the Past Hypothesis”, the idea inspired by Boltzmann that we see time-asymmetry because there is a point in what we call the “past” where entropy was very low, and so we perceive the direction away from that state as “forward” it time because the world tends to move toward equilibrium (though he pointed out that for dynamical reasons, the world can easily stay far away from equilibrium for a long time).  He went on to discuss the logic of this argument, and the idea of a “simple” (i.e. easy-to-describe) distribution, and the conjecture that the evolution of these will generally be describable in terms of an evolution that uses “coarse graining” (i.e. that repeatedly throws away microscopic information).

The Emergence of Spacetime in Quantum Theories of Gravity“:  This session addressed the idea that spacetime (or in some cases, just space) might not be fundamental, but could emerge from a more basic theory.  Christian Wüthrich spoke about “A-Priori versus A-Posteriori” versions of this idea, mostly focusing on ideas such as LQG and causal sets, which start with discrete structures, and get manifolds as approximations to them.  Nick Huggett gave an overview of noncommutative geometry for the philosophically minded audience, explaining how an algebra of observables can be treated like space by means of all the concepts from geometry which can be imported into the theory of $C^{\star}$-algebras, where space would be an approximate description of the algebra by letting the noncommutativity drop out of sight in some limit (which would be described as a “large scale” limit).  Sean Carroll discussed the possibility that “Space is Not Fundamental – But Time Might Be”, pointing out that even in classical mechanics, space is not a fundamental notion (since it’s possible to reformulate even Hamiltonian classical mechanics without making essential distinctions between position and momentum coordinates), and suggesting that space arises from the dynamics of an actual physical system – a Hamiltonian, in this example – by the principle “Position Is The Thing In Which Interactions Are Local”.  Finally, Sean Maudlin gave an argument for the fundamentality of time by showing how to reconstruct topology in space from a “linear structure” on points saying what a (directed!) path among the points is.

Last week I spoke in Montreal at a session of the Philosophy of Science Association meeting.  Here are some notes for it.  Later on I’ll do a post about the other talks at the meeting.

Right now, though, the meeting slowed me down from describing a recent talk in the seminar here at IST.  This was Gonçalo Rodrigues’ talk on categorifying measure theory.  It was based on this paper here, which is pretty long and goes into some (but not all) of the details.  Apparently an updated version that fills in some of what’s not there is in the works.

In any case, Gonçalo takes as the starting point for categorifying ideas in analysis the paper “Measurable Categories” by David Yetter, which is the same point where I started on this topic, although he then concludes that there are problems with that approach.  Part of the reason for saying this has to do with the fact that the category of Hilbert spaces has many bad properties – or rather, fails to have many of the good ones that it should to play the role one might expect in categorifying ideas from analysis.

Yetter’s idea can be described, very roughly, as follows: we would like to categorify the concept of a function-space on a measure space $(X,\mu)$.  That is, spaces like $L^2(X,\mu)$ or $L^{\infty}(X,\mu)$.  The reason for this is that the 2-vector-spaces of Kapranov and Voevodsky are very elegant, but intrinsically finite-dimensional, categorifications of “vector space”.  An infinite-dimensional version would be important for representation theory, particularly of noncompact Lie groups or 2-groups, but even just infinite ones, since there are relatively few endomorphisms of KV 2-vector spaces.  Yetter’s paper constructs analogs to the space of measurable functions $\mathcal{M}(X)$, where “functions” take values in Hilbert spaces.

A measurable field of Hilbert spaces is, roughly, a family of Hilbert spaces indexed by points of $X$, together with a nice space of “measurable sections”.  This is supposed to be an infinite-dimensional, measure-theoretic counterpart to an object in a KV 2-vector space, which always looks like $\mathbf{Vect}^k$ for some natural number $k$, which is now being replaced by $(X,\mu)$.  One of the key tools in Yetter’s paper is the direct integral of a field of Hilbert spaces, which is similarly the counterpart to the direct sum $\bigoplus$ in the discrete world.  It just gives the space of measurable sections (taken up to almost-everywhere equivalence, as usual).  This was the main focus of Gonçalo’s talk.

The direct integral has one major problem, compared to the (finite) direct sum it is supposed to generalize – namely, the direct sum is a categorical coproduct, in $\mathbf{Vect}$ or any other KV 2-vector space.  Actually, it is both a product and a coproduct ($\mathbf{Vect}$ is abelian), so it is defined by a nice universal property.  The direct integral, on the other hand, is not.  It doesn’t have any similarly nice universal property.  (In the infinite-dimensional case, colimits and limits would be expected to become different in any case, but the direct integral is neither).  This means that many proofs in analysis will be hard to reproduce in the categorified setting – universal properties mean one doesn’t have to do nearly as much work to do this, among their other good qualities.  This is related to the issue that the category $\mathbf{Hilb}$ does not have all limits and colimits

Gonçalo’s paper and talk outline a program where one can categorify a lot of the proofs in analysis, by using a slightly different framework which uses a bigger category than $\mathbf{Hilb}$, namely $Ban_C$, whose objects are Banach spaces and whose maps are (linear) contractions.  A Banach Category is a category enriched in $Ban_C$.  Now, Banach spaces have a norm, but not necessarily an inner product, and this small weakening makes them much worse than Hilbert spaces as objects.  Many intuitions from Hilbert spaces, like the one that says any subspace has a complement, just fail: the corresponding notion for Banach spaces is the quasicomplement ($X$ and $Y$ are quasicomplements if they intersect only at zero, and their sum is dense in the whole space), and it’s quite possible to have subspaces which don’t have one.  Other unpleasant properties abound.

Yet $Ban_C$ is a much nicer category than $Hilb$.  (So we follow the general dictum that it’s better to have a nice category with bad objects than a bad category with nice objects – the same motivation behind “smooth spaces” instead of manifolds, and the like.)  It’s complete and cocomplete (i.e. has all limits and colimits), as well as monoidal closed – for Banach spaces $A$ and $B$, the space $Hom(A,B)$ is also in $Ban_C$.  None of these facts holds for $Hilb$.  On the other hand, the space of bounded maps between Hilbert spaces is a Banach space (with the operator norm), but not necessarily a Hilbert space.  So even $Hilb$ is already a Banach category.

It also turns out that, unlike in $Hilb$, limits and colimits (where those exist in $Hilb$) are not necessarily isomorphic.  In particular, in $Ban_C$, the coproduct and product $A + B$ and $A \times B$ both have the same underlying vector space $A \oplus B$, but the norms are different.  For Hilbert spaces, the inner product comes from the Pythagorean formula in either case, but for Banach spaces, the coproduct gets the sum of the two norms, and the product gets the supremum.  It turns out that coproducts are the more important concept, and this is where the direct integral comes in.

First, we can talk about Banach 2-spaces (the analogs of 2-vector spaces): these are just Banach categories which are cocomplete (have all weighted colimits).  Maps between them are cocontinuous functors – that is, colimit-preserving ones.  (Though properly, functors between Banach categories ought to be contractions on Hom-spaces).  Then there are categorified analogs of all sorts of Banach space structure in a familiar way – the direct sum (coproduct) is the analog of vector addition, the category $Ban_C$ is the analog of the base field (say, $\mathbb{R}$), and so on.

This all gives the setting for categorified measure theory.  Part of the point of choosing $Ban_C$ is that you can now reason out at least some of how it works by analogy.  To start with, one needs to fix a Boolean algebra $\Omega$ – this is to be the $\sigma$-algebra of measurable sets for some measure space, though it’s important that it needn’t have any actual points (this is a notion of measure space akin to the notion of locale in the topological world).  This part of the theory isn’t categorified (arguably a limitation of this approach, but not one that’s any different from Yetter’s).  Instead, we categorify the definition of measure itself.

A measure is a function $\mu : \Omega \mapsto \mathbb{R}$ – it assigns a number to each measurable set.  The pair $(\Omega,\mu)$ is a measure algebra, and relates to a measure space the way a locale relates to a topological space.  So a categorified measure $\nu$ should be a functor from $\Omega$ (seen now as a category) into $Ban_C$.  (We can generalize this: the measure could be valued in some vector space over $\mathbb{R}$, and a categorified measure could be a functor into some other Banach 2-space.)  Since we’re thinking of $\Omega$ as a lattice of subsets, it makes some sense to call $\nu$ a presheaf, or rather co-presheaf.  What’s more, just as a measure is additive ($\mu(A + B) = \mu(A) + \mu(B)$, for disjoint sets, where $+$ is the union), so also the categorical measure $\nu$ should be (finitely) additive up to isomorphism.  So we’re assigning Banach spaces to all the measurable sets.  This is a “co”-presheaf – which is to say, a covariant functor, so the spaces “nest”: when for measurable sets, we have $A \subset B$, then $\nu(A) \leq \nu(B)$ also.

An intuition for how this works comes from a special case (not at all exhaustive), where we start with an actual, uncategorified, measure space $(X,\mu)$.  Then one categorified measure will arise by taking $\nu(E) = L_1(E,\mu)$: the Banach space associated to a measurable set $E$ is the space of integrable functions.  We can take any “scalar” multiple of this, too: given a fixed Banach space $B$, let $\nu(E) = L_1(E,\mu) \otimes B$.  But there are lots of examples that aren’t like this.

All this is fine, but the point here is to define integration.  The usual way to go about this when you learn analysis is to start with characteristic functions of measurable sets, then define a sequence through simple functions, measurable functions, and so forth.  Eventually one can define $L^p$ spaces based on the convergence of various integrals.  Something similar happens here.

The analog of a function here is a sheaf: a (compatible) assignment of Banach spaces to measurable sets.  (Technically, to get to sheaves, we need an idea of “cover” by measurable sets, but it’s pretty much the obvious one, modulo the subtlety that we should only allow countable covers.) The idea will be to start with characteristic sheaves for measurable sets, then take some kind of completion of the category of all of these as a definition of “measurable sheaf”.  Then the point will be that we can extend the measure from characteristic sheaves to all measurable sheaves using a limit (actually, a colimit), analogous to the way we define a Lebesgue integral as a limit of simple functions approximating a measurable one.

A characteristic sheaf $\chi(E)$ for a measurable set $E \in \Omega$ might be easiest to visualize in terms of a characteristic bundle, which just puts a copy of the base field (we’ve been assuming it’s $\mathbb{R}$) at each point of $E$, and the zero vector space everywhere else.  (This is a bundle in the measurable sense, not the topological one – assuming $X$ has a topology other than $\Omega$ itself.)  Very intuitively, to turn this into a sheaf, one can just use brute force and take a set $A$ the product of all the spaces lying in $A$.  A little less crudely, one should take a space of sections with decent properties – so that $\chi(E)$ assigns to $A$ a space of functions on $E \cap A$.  In particular, the functor $\chi : \Omega \rightarrow L_{\infty}(\Omega)$ which picks out all the (measurable) bounded sections is a universal way to do this.

Now the point is that the algebra of measurable sets, $\Omega$, thought of as a category, embeds into the category of presheaves on it by $\chi : \Omega \rightarrow \mathbf{PShv}(\Omega)$, taking a set to its characteristic sheaf.  Given a measure valued in some Banach category, $\nu : \Omega \rightarrow \mathcal{B}$, we can find the left Kan extension $\int_X d\nu : \mathbf{PShv}(\Omega) \rightarrow \mathcal{B}$, such that $\nu = \int_X d\nu \circ \chi$.  The Kan extension is a universal way to extend $\nu$ to all of $\mathbf{PShv}(\Omega)$ so that this is true, and it can be calculated as a colimit.

The essential fact here is that the characteristic sheaves are dense in $\mathbf{PShv}(\Omega)$: any presheaf can be found as a colimit of the characteristic ones.  This is analogous to how any function can be approximated by linear combinations of characteristic functions.  This means that the integral defined above will actually give interesting results for all the sheaves one might expect.

I’m glossing over some points here, of course – for example, the distinction between sheaves and presheaves, the role of sheafification, etc.  If you want to get a more accurate picture, check out the paper I linked to up above.

All of this granted, however, many of the classical theorems of measure theory have analogs that are proved in essentially the same way as the standard versions.  One can see the presheaf category as a categorified analog of $L_1(X,\nu)$, and get the Fubini theorem, for instance: there is a canonical equivalence (no longer isomorphism) between (a suitable) tensor product of $\mathbf{PShv}(X)$ and $\mathbf{PShv}(Y)$ on one hand, and on the other $\mathbf{PShv}(X \times Y)$.  Doing integration, one can then do all the usual things – exchange order of integration between $X$ and $Y$, say – in analogous conditions.  The use of universal properties to define integrals etc. means that one doesn’t need to fuss about too much with coherence laws, and so the proofs of the categorified facts are much the same as the original proofs.

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?

So I recently received word that this paper had been accepted for publication by Applied Categorical Structures. Since I’ll shortly be putting out another which uses its main construction to build Extended Topological Quantum Field Theories, it’s nice and appropriate to say something about that. But actually, just at the moment, I want to take a slightly different approach.

Toward the end of February, I went up to Waterloo to the Perimeter Institute, where my friend Derek Wise was visiting with Andy Randono – apparently they’re working on a project together that has something to do with Cartan Geometry, which is a subject that plays a big role in Derek’s thesis.

However, Derek was speaking in their seminar about Extended TQFT (his slides are now up on his website, and there’s also a video of the talk available). Actually, a lot of what he was talking about was work of mine, since we’re working on a project together to constructs ETQFT’s from Lie groups (most likely compact ones at first, since all the usual analytical problems with noncompact groups turn up here). However, I really enjoyed seeing Derek talk about it, because he has a sharper grasp than I do of how this subject appears to physicists, and the way he presented this stuff is very different from the way I usually talk about it (you can see me in the video trying to help deal with a question at the end from Rafael Sorkin and Laurent Freidel, and taking a while to correctly understand what it was, partly because of this jargon gap – I hope to get better).

So, for example, describing a TQFT in the Atiyah/Segal axiomatic formulation is fairly natural to someone who works with category theory, but Derek motivated it as a way of taking a “deeper look at the partition function” for a certain field theory. The idea is that a partition function $Z$ for a quantum field theory associates a number to a space $M$, satisfying certain rules. It is usually described by some kind of integral. Typically in QFT, these are rather tricky integrals – a topological QFT has the nice feature that, since it has no local degrees of freedom, these integrals are much more tractable. Of course, this is a mathematically nice feature that comes at the expense of physical relevance, but such is life.

Anyway, the idea is that the partition function $Z$ for an $n$-dimensional TQFT can be thought of as assigning, not just numbers to $n$-dimensional manifolds $M$, but something more which reduces to this in a special case. Specifically, $Z$ assigns a Hilbert space to any codimension-1 submanifold of $M$, in a particular way which Derek passed over by saying it “satisfies some compatibility conditions”. For an audience of mathematicians, you can gloss over this just as quickly by saying the assignments are “functorial”, or even with more detail saying the conditions make $Z$ a symmetric monoidal functor.

Part of the point is that these conditions are about as obvious on physical grounds as they are if you’re a category theorist. For example, the fact that composition is preserved by the functor $Z$ can be interpreted physically as saying that the number $Z(M)$ given by the partition function isn’t affected by how we chop up the manifold $M$ to analyse it. The fact that $Z$ is a monoidal functor ends up meaning that the “unit” for manifolds under unions (namely, the empty manifold with no points, which you can add to things without affecting them) gets assigned the Hilbert space $\mathbb{C}$, which is the unit for Hilbert spaces with respect to the tensor product $\otimes$. The fact that this is so means we can treat a manifold with no boundary as going from one (empty) boundary to another (empty) boundary – it therefore gets assigned a linear map from $\mathbb{C}$ to $\mathbb{C}$ – a number. Seeing how this linear map comes from composing pieces of the manifold is what “a deeper look at the partition function” means.

ETQFT does essentially the same thing, at one level deeper. The point is that a TQFT breaks apart a manifold by treating it as a series of pieces – manifolds with boundary, glued together at their boundaries. An ETQFT does the same to these pieces, treating them as composed of pieces – manifolds with corners – which are glued orthogonally to the gluing just mentioned. That is, there are two kinds of composition, so we’re in some sort of 2-category (bi-, or double- depending on how you formulate things). The essential point is that now, to manifolds without boundary, which are of codimension 1, we assign Hilbert spaces – and to top-dimensional manifolds WITH boundary, we assign maps of Hilbert spaces.

An ETQFT attempts to give a “deeper-still look at the partition function” by seeing how the Hilbert space arises from composition of pieces in this new direction, along boundaries of codimension 2. The way Derek describes this for physicists is to say that the ETQFT describes how that Hilbert space is “built from local data”, which he described in the usual physics language of path integrals. First of all, the conventional thing in physics is to take $Z(\Sigma)$ for a (codimension-1) manifold $\Sigma$ to be $L^2(\mathcal{A}_0(\Sigma)/\mathcal{G}(\Sigma))$ – the space of square-integrable functions on the quotient of the space $\mathcal{A}_0(\Sigma)$ of flat $G$-connections on $M$ by the action of the group of gauge transformations $\mathcal{G}(\Sigma)$.

Given a manifold $M$ with boundary components $\Sigma$ and $\Sigma '$, the standard quantum field theory formalism to describe the map $Z(M) : Z(\Sigma) \rightarrow Z(\Sigma ')$ given by a TQFT is to describe how it interacts with particular state-vectors in the Hilbert spaces for the source and target boundary components of $M$. So then:

$\langle \psi | Z(M) | \phi \rangle = \int_{\mathcal{A}_0(M)/\mathcal{G}} \mathcal{D}A \overline{\psi(A|_{\Sigma '})} e^{i S([A])} \phi(A|_{\Sigma})$

The point being, a flat connection $A$ has some action on it, which depends only on its gauge equivalence class $[A]$ (“the Lagrangian has gauge symmetry”), and it restricts to give flat connections on $\Sigma$ and $\Sigma '$, on which the $L^2$-functions $\psi$ and $\phi$ act, to give something we can integrate. The measure $\mathcal{D}[A]$ is a crucial entity here, and in general can be a real puzzle, but at least for discrete groups, it’s just a weighted counting measure which effectively gives us the groupoid cardinality of the quotient space. As for the action $S$, the simplest possible case just says the action of any flat connection is zero – hence this expression is just finding the (groupoid) cardinality, or more generally measuring the (stacky) volume, of the configuration space for flat connections. There are other possible actions, though.

Derek gives an explanation of how to interpret this in terms of the “pull-push” construction, which I’ve talked about elsewhere here, including in the above paper, so right now, I’ll just pass to the next layer of the ETQFT layer cake – codimension-2. Here, there is a similar formula, which also has an interpretation in terms of a “pull-push” construction, but which can be written as a categorified path integral.

So now the $\Sigma$ has boundary, and connects “inner” codimension-2 boundary component $B_1$ to “outer” boundary component $B_2$. Then, say, $B_1$ gets assigned the category of all gauge-equivariant “bundles” of Hilbert spaces on $\mathcal{A}_0(B_1)$, rather than the space of gauge-invariant functions. (Derek carefully avoided using the term “category”, to stay physically motivated – and the term “bundle” is accurate in the case of a discrete gauge group $G$, but in general one has to appeal to the theory of measurable fields of Hilbert spaces, since they needn’t be locally trivial). Then given particular Hilbert bundles $\mathcal{H}$ and $\mathcal{K}$ on the spaces $\mathcal{A}_0(B_1)$ and $\mathcal{A}_0(B_2)$ respectively, we can define what $Z(\Sigma)$ is by:

$\langle \mathcal{K} | Z(M) | \mathcal{H} \rangle = \int_{\mathcal{A}_0(M)/\mathcal{G}} \mathcal{D}A \mathcal{K}(A|_{B_2}) \otimes T_A \otimes \mathcal{H}(A|_{B_1})$

The interpretation is much like the previous formula: now we’re direct-integrating Hilbert spaces, instead of integrating complex functions – and we get a Hilbert space instead of a complex number, but this is in some sense superficial. Something any physicist would notice right away (or anyone comparing this to the previous formula) is that the exponential of the action $S([A])$ seems to have gone missing, to be replaced by some Hilbert space $T_A$. If we’re using the trivial action $S \cong 0$, this is fine, but otherwise, how exactly $S$ affects the direct integral would take some explaining. For now, let’s just say that we should think of $S([A])$ as being folded into either the inner product on $T_A$, or into the measure $\mathcal{D}A$: it shows up in its effect on the inner product on the Hilbert space that this direct integral produces.

Let me jump to the end of Derek’s talk here, to get at some conceptual aspect of what’s happening here. The axiomatic way of talking about ETQFT, namely Ruth Lawrence’s way, is to say we assign a 2-Hilbert space to the codimension-2 manifolds. But “2-Hilbert space” is an off-putting bit of jargon, so instead the suggestion is to replace it with “von Neumann algebra”.

The point is that 2-Hilbert spaces are thought (according to a paper by Baez, Baratin, Friedel and Wise) to be just categories of representations of vN algebras. Being a 2-Hilbert space means, for instance, that they’re additive (by direct sum), $\mathbb{C}$-linear (there is a vector space of intertwiners between any two representations), have duals, and so on. Moreover, they’re monoidal 2-Hilbert spaces, since there is a tensor product. Their idea is that the two ideas correspond exactly. In any case, the way the ETQFT construction in question works actually passes through a von Neumann algebra. This comes from the groupoid algebra that’s associated to a certain group action. Namely, the action of the gauge group on the space of flat $G$-connections on the manifold $M$.

Then the way we can look more closely at the “structure of the partition function” is by seeing the Hilbert space associated to a codimension-1 manifold as actually being a kind of morphism of von Neumann algebras. In particular, it’s a Hilbert bimodule, which is acted on by the source algebra (say $A$) on the left, and the target algebra ($B$) on the right. This is intimately connected with the stuff I was writing about recently about Morita equivalence, and so to the 2-Hilbert space view. In particular, a Hilbert bimodule $H$ gives an adjoint pair of linear functors (or “2-linear maps”) between the representation categories of algebras.

So shortly I’ll make a post about some papers coming out, and get back to this point…

Last week there was an interesting series of talks by Ivan Dynov about the classification of von Neumann algebras, and I’d like to comment on that, but first, since it’s been a while since I posted, I’ll catch up on some end-of-term backlog and post about some points I brought up a couple of weeks ago in a talk I gave in the Geometry seminar at Western. This was about getting Extended TQFT’s from groups, which I’ve posted about plenty previously . Mostly I talked about the construction that arises from “2-linearization” of spans of groupoids (see e.g. the sequence of posts starting here).

The first intuition comes from linearizing spans of (say finite) sets. Given a map of sets $f : A \rightarrow B$, you get a pair of maps $f^* : \mathbb{C}^B \rightarrow \mathbb{C}^A$ and $f_* : \mathbb{C}^A \rightarrow \mathbb{C}^B$ between the vector spaces on $A$ and $B$. (Moving from the set to the vector space stands in for moving to quantum mechanics, where a state is a linear combination of the “pure” ones – elements of the set.) The first map is just “precompose with $f$“, and the other involves summing over the preimage (it takes the basis vector $a \in A$ to the basis vector $f(a) \in B$. These two maps are (linear) adjoints, if you use the canonical inner products where $A$ and $B$ are orthonormal bases. So then a span $X \stackrel{s}{\leftarrow} S \stackrel{t}{\rightarrow} Y$ gives rise to a linear map $t_* \circ s^* : \mathbb{C}^X \rightarrow \mathbb{C}^Y$ (and an adjoint linear map going the other way).

There’s more motivation for passing to 2-Hilbert spaces when your “pure states” live in an interesting stack (which can be thought of, up to equivalence, as a groupoid hence a category) rather than an ordinary space, but it isn’t hard to do. Replacing $\mathbb{C}$ with the category $\mathbf{FinHilb}_\mathbb{C}$, and the sum with the direct sum of (finite dimensional) Hilbert spaces gives an analogous story for (finite dimensional) 2-Hilbert spaces, and 2-linear maps.

I was hoping to get further into the issues that are involved in making the 2-linearization process work with Lie groups, rather than finite groups. Among other things, this generalization ends up requiring us to work with infinite dimensional 2-Hilbert spaces (in particular, replacing $\mathbf{FinHilb}$ with $\mathbf{Hilb}$). Other issues are basically measure-theoretic, since in various parts of the construction one uses direct sums. For Lie groups, these need to be direct integrals. There are also places where counting measure is used in the case of a discrete group $G$. So part of the point is to describe how to replace these with integrals. The analysis involved with 2-Hilbert spaces isn’t so different for than that required for (1-)Hilbert spaces.

Category theory and measure theory (analysis in general, really), have not historically got along well, though there are exceptions. When I was giving a similar talk at Dalhousie, I was referred to some papers by Mike Wendt, “The Category of Disintegration“, and “Measurable Hilbert Sheaves“, which is based on category-theoriecally dealing with ideas of von Neumann and Dixmier (a similar remark applies Yetter’s paper “Measurable Categories“), so I’ve been reading these recently. What, in the measurable category, is described in terms of measurable bundles of Hilbert spaces, can be turned into a description in terms of Hilbert sheaves when the category knows about measures. But categories of measure spaces are generally not as nice, categorically, as the category of sets which gives the structure in the discrete case. Just for example, the product measure space $X \times Y$ isn’t a categorical product – just a monoidal one, in a category Wendt calls $\mathbf{Disint}$.

This category has (finite) measure spaces as objects, and as morphisms has disintegrations. A disintegration from $(X,\mathcal{A},\mu)$ to $(Y,\mathcal{B},\nu)$ consists of:

• a measurable function $f : X \rightarrow Y$
• for each $y \in Y$, the preimage $f^{-1}(y) = X_y$ becomes a measure space (with the obvious subspace sigma-algebra $\mathcal{A}_y$), with measure $\mu_y$

such that $\mu$ can be recovered by integrating against $\nu$: that is, for any measurable $A \subset X$, (that is, $A \in \mathcal{A}$), we have

$\int_Y \int_{A_y} d\mu_y(x) d\nu(y) = \int_A d\mu(x) = \mu (A)$

where $A_y = A \cap X_y$.

So the point is that such a morphism gives, not only a measurable function $f : X \rightarrow Y$, but a way of “disintegrating” $X$ relative to $Y$. In particular, there is a forgetful functor $U : \mathbf{Disint} \rightarrow \mathbf{Msble}$, where $\mathbf{Msble}$ is the category of measurable spaces, taking the disintegration $(f, \{ (X_y,\mathcal{A}_y,\mu_y) \}_{y \in Y} )$ to $f$.

Now, $\mathbf{Msble}$ is Cartesian; in particular, the product of measurable spaces, $X \times Y$, is a categorical product. Not true for the product measure space in $\mathbf{Disint}$, which is just a monoidal category1. Now, in principle, I would like to describe what to do with groupoids in (i.e. internal to), $\mathbf{Disint}$, but that would involve side treks into things like volumes of measured groupoids, and for now I’ll just look at plain spaces.

The point is that we want to reproduce the operations of “direct image” and “inverse image” for fields of Hilbert spaces. The first thing is to understand what’s mean by a “measurable field of Hilbert spaces” (MFHS’s) on a measurable space $X$. The basic idea was already introduced by von Neumann not long after formalizing Hilbert spaces. A MFHS’s on $(X,\mathcal{A})$ consists of:

• a family $\mathcal{H}_x$ of (separable) Hilbert spaces, for $x \in X$
• a space $\mathcal{M} \subset \bigoplus_{x \in X}\mathcal{H}_x$ (of “measurable sections” $\phi$) (i.e. pointwise inverses to projection maps $\pi_x : \mathcal{M} \rightarrow \mathcal{H}_x$) with three properties:
1. measurability: the function $x \mapsto ||\phi_x||$ is measurable for all $\phi \in \mathcal{M}$
2. completeness: if $\phi \in \mathcal{M}$ and $\psi \in \bigoplus_{x \in X} \mathcal{H}_x$ makes the function $x \mapsto \langle \phi_x , \psi_x \rangle$ then $\psi \in \mathcal{M}$
3. separability: there is a countable set of sections $\{ \phi^{(n)} \}_{n \in \mathbb{N}} \subset \mathcal{M}$ such that for all $x$, the $\phi^{(n)}_x$ are dense in $\mathcal{H}_x$

This is a categorified analog of a measurable function: a measurable way to assign Hilbert spaces to points. Yetter describes a 2-category $\mathbf{Meas(X)}$ of MFHS’s on $X$, which is an (infinite dimensional) 2-vector space – i.e. an abelian category, enriched in vector spaces. $\mathbf{Meas(X)}$ is analogous to the space of measurable complex-valued functions on $X$. It is also similar to a measurable-space-indexed version of $\mathbf{Vect^k}$, the prototypical 2-vector space – except that here we have $\mathbf{Hilb^X}$. Yetter describes how to get 2-linear maps (linear functors) between such 2-vector spaces $\mathbf{Meas(X)}$ and $\mathbf{Meas(Y)}$.

This describes a 2-vector space – that is, a $\mathbf{Vect}$-enriched abelian category – whose objects are MFHS’s, and whose morphisms are the obvious (that is, fields of bounded operators, whose norms give a measurable function). One thing Wendt does is to show that a MFHS $\mathcal{H}$ on $X$ gives rise to measurable Hilbert sheaf – that is, a sheaf of Hilbert spaces on the site whose “open sets” are the measurable sets in $\mathcal{A}$, and where inclusions and “open covers” are oblivious to any sets of measure zero. (This induces a sheaf of Hilbert spaces $H$ on the open sets, if $X$ is a topological space and $\mathcal{A}$ is the usual Borel $\sigma$-algebra). If this terminology doesn’t spell it out for you, the point is that for any measurable set $A$, there is a Hilbert space:

$H(A) = \int^{\oplus}_A \mathcal{H}_x d\mu(x)$

The descent (gluing) condition that makes this assignment a sheaf follows easily from the way the direct integral works, so that $H(A)$ is the space of sections of $\coprod_{x \in A} \mathcal{H}_x$ with finite norm, where the inner product of two sections $\phi$ and $\psi$ is the integral of $\langle \phi_x, \psi_x \rangle$ over $A$.

The category of all such sheaves on $X$ is called $\mathbf{Hilb^X}$, and it is equivalent to the category of MFHS up to equivalence a.e. Then the point is that a disintegration $(f, \mu_y) : (X,\mathcal{A},\mu) \rightarrow (Y,\mathcal{B},\nu)$ gives rise to two operations between the categories of sheaves (though it’s convenient here to describe them in terms of MFHS: the sheaves are recovered by integrating as above):

$f^* : \mathbf{Hilb^Y} \rightarrow \mathbf{Hilb^X}$

which comes from pulling back along $f$ – easiest to see for the MFHS, so that $f^*\mathcal{H}_x = \mathcal{H}_{f(x)}$, and

$\int_f^{\oplus} : \mathbf{Hilb^X} \rightarrow \mathbf{Hilb^Y}$

the “direct image” operation, where in terms of MFHS, we have $(\int_f^{\oplus}\mathcal{H})_y = \int_{f^{-1}(y)}^{\oplus}\mathcal{H}_x d\mu_y(x)$. That is, one direct-integrates over the preimage.

Now, these are measure-theoretic equivalents of two of the Grothendieck operations on sheaves (here is the text of Lipman’s Springer Lecture Notes book which includes an intro to them in Ch3 – a bit long for a first look, but the best I could find online). These are often discussed in the context of derived categories. The operation $\int_f^{\oplus}$ is the analog of what is usually called $f_*$.

Part of what makes this different from the usual setting is that $\mathbf{Disint}$ is not as nice as $\mathbf{Top}$, the more usual underlying category. What’s more, typically one talks about sheaves of sets, or abelian groups, or rings (which give the case of operations on schemes – i.e. topological spaces equipped with well-behaved sheaves of rings) – all of which are nicer categories than the category of Hilbert spaces. In particular, while in the usual picture $f_*$ is left adjoint to $f^*$, this condition fails here because of the requirement that morphisms in $\mathbf{Hilb}$ are bounded linear maps – instead, there’s a unique extension property.

Similarly, while $f*$ is always defined by pulling back along a function $f$, in the usual setting, the direct image functor $f_*$ is left-adjoint to $f^*$, found by taking a left Kan extension along $f$. This involves taking a colimit (specifically, imagine replacing the direct integral with a coproduct indexed over the same set). However, in this setting, the direct integral is not a coproduct (as the direct sum would be for vector spaces, or even finite-dimensional Hilbert spaces).

So in other words, something like the Grothendieck operations can be done with 2-Hilbert spaces, but the categorical properties (adjunction, Kan extension) are not as nice.

Finally, I’ll again remark that my motivation is to apply this to groupoids (or stacks), rather than just spaces $X$, and thus build Extended TQFT’s from (compact) Lie groups – but that’s another story, as we said when I was young.

1 Products: The fact that we want to look at spans in categories that aren’t Cartesian is the reason it’s more general to think about spans, rather than (as you can in some settings such as algebraic geometry) in terms of “bundles over the product”, which is otherwise equivalent. For sets or set-groupoids, this isn’t an issue.