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.

So this paper of mine was recently accepted by the Journal of Homotopy and Related Structures (the version that was accepted should be reflected on the arXiv by tomorrow – i.e. July 10 – I’m not sure about the journal ). It’s been a while since I sent out the earliest version, and most of the changes have involved figuring out who the audience is, and consequently what could be left out. I guess that’s a side-effect of taking an excerpt from my thesis, which was much longer. In any case, it now seems to have reached a final point. Some of what was in it – the section about cobordisms – is now in a paper (in progress) about TQFT. I don’t see anywhere else to include the other missing bit, however, which has to do with Lawvere theories, and since I just wrote a bunch about MakkaiFest, I thought I might include some of that here.

The paper came about because I was trying to write my thesis, which describes an extended TQFT as a 2-functor (and considers how it could produce a version of 3D quantum gravity). The 2-functor

Z_G : nCob_2 \rightarrow 2Vect

(or into 2Hilb) is an ETQFT. The construction of the 2-functor uses the fact that you can get spans of groupoids out of cospans of manifolds – and in particular, out of cobordisms. One problem is how to describe nCob_2 so that this works. It’s actually most naturally a cubical 2-category of some kind. The strict version of this concept is a double category – which has (in principle separate) categories of horizontal and vertical of morphisms, as well as square 2-cells. Ideally, one would like a “weak” version, where composition of squares and morphisms can be only weakly associative (and have weak unit laws). A “pseudocategory” implements this where the only higher-dimensional morphisms are the squares, but it turns out to be strict in one direction, and weak in the other. As it happens, it’s a big pain to use only squares for the 2-morphisms.

Initially it seemed I would have to define a whole new structure to get weak composition in both directions, because in both directions, composition represents gluing bits of manifolds together along boundaries – using a diffeomorphism (or a smooth homeomorphism, depending on which kind of manifolds we’re dealing with). I called it a “double bicategory” and started trying to define it along the same lines as a double category. It then turned out that Dominic Verity had already defined a “double bicategory” – you can read the paper where I talk about how the notions are related. Here I want to talk about a few aspects which I cut out of the paper along the way.

The idea is that there are two ways of “categorifying”: internalization, and enrichment. A bicategory is a category enriched in Cat, the category of categories – for any two elements, there’s a whole hom-category of morphisms (and 2-morphisms). A double category is a category internal to Cat. This means you can think of it as a category of objects and a category of morphisms, equipped with functors satisfying all the usual properties for the maps in the definition of a category: composition functors, unit functors, and so forth. This definition turns out to be equivalent to the usual one. So I thought: why not do the same with bicategories?

Thus, the way I defined double bicategory was: “A bicategory internal to Bicat“. In the paper as it stands, that’s all I say. What I cut out was a sort of dangling loose end pointing toward Lawvere theories – or rather, a variant thereof – finite limit theories (for something more detailed, see this recent paper by Lack and Rosicky). As I mentioned in the previous post, a Lawvere theory is an approach to universal algebra – it formally defines a kind of object (e.g. group, ring, abelian group, etc.) as a functor from a category T which is the “theory” of such objects, while the functor is a “model” of the theory.

What makes it “universal” algebra is that it can involve definitions with many sorts of objects, many operations, given as arrows, of different arities (number of inputs and outputs). This last makes sense in the monoidal context, and in particular Cartesian. Making decisions like this – what class of categories and functors we’re dealing with – specifies which doctrine the theory lives in. In the case of bicategories, this is the doctrine of categories with finite limits. In a Lawvere theory in the original sense, the doctrine is categories with finite products – so if there’s an object G, there are also objects G^n for all n. Then there are things like multiplication maps m : G^2 \rightarrow G and so on. For a category or bicategory, multiplication might be partial – so we need finite limits. A model of a theory in this doctrine is a limit-preserving functor.

So what does the theory of bicategories look like? It’s easy enough to see if you think that a (small) bicategory is a “bicategory in Sets“, and reproduce the usual definition, omitting reference to sets. It has objects Ob, Mor, and 2Mor. (This fact already means this is a “multi-sorted” theory, which goes beyond what can be done with another approach to universal algebra based on monads). Funthermore, there are maps between these objects, interpreted as source, target, and identity maps of various sorts. These form diagrams, and since we’re in a finite limit theory, there must be various objects like Pairs = Mor \times_{Ob} Mor which for sets would have the interpretation “pairs of composable morphisms”. Then there’s a composition map \circ : Pairs \rightarrow Mor… and so on. In short, in describing the axioms for a bicategory in a “nice” way (i.e. in terms of arrows, commuting diagrams, etc.), we’re giving a presentation of a certain category, Th(Bicat), in generators and relations. Then a model of the theory is a functor Th(Bicat) \rightarrow \mathcal{C} – picking out a “bicategory in \mathcal{C}“.

Now, a bicategory in Sets is a bicategory. But a bicategory in Bicat is another matter. First of all, I should say there’s something kind of odd here, since Bicat is most naturally regarded as a tricategory. However, we can regard it as a category by disregarding higher morphisms and taking 2-functors only up to equivalence to make Bicat into an honest category with associative composition. Thus, if we have a functor F : Th(Bicat) \rightarrow Bicat, we have:

  • Bicategories F(Ob), latex $F(Mor)$, and F(2Mor)
  • 2-Functors F(s), F(\circ) and so on
  • satisfying conditions implied by the bicategory axioms

But each of those bicategories (in Sets!) has sets of objects, morphisms, and 2-morphisms, and one can break all the functors apart into three collections of maps acting on each of these three levels. They’ll satisfy all the conditions from the axioms – in fact, they make three new bicategories. So, for example, the object-sets of the bicategories F(Ob), F(Mor) and F(2Mor) form a bicategory using the object maps of the 2-functors F(s) and so on.

So if we say the original bicategories F(Ob) and so on are “horizontal”, and these new ones are “vertical”, we have something resembling a double category, but weak (since bicategories are weak) in both directions. The result is most naturally a four-dimensional structure (the 2-morphisms in 2Mor are most conveniently drawn as 4d, which is shown in Table 2 of the paper).

Now, the paper as it is describes all this structure without explicitly mentioning the theory Th(Bicat) except in passing – one can define “internal bicategory” without it. This is why this is a “loose end” of this paper: a major benefit of using Lawvere-style theories is the availability of morphisms of theories, which don’t come up here.

In any case, with this 4D structure in hand, what I do in the paper is (a) get some conditions that allow one to decategorify it down to Verity’s version of “double bicategory” (and even down to a bicategory); and (b) show that couble cospans are an example (double spans would do equally well, but the application is to cobordisms, which are cospans). My own reason for wanting to get down to a 2D structure is the application to extended TQFT, which means we want a 2-category of cobordisms, thought of in terms of (co)spans.

Maybe in a subsequent post I’ll talk about the example itself, but one point about internalization does occur to me. Double cospans give an example of a double bicategory in the sense above – a strict model of Th(Bicat) in Bicat. In fact, they consist of “(co)spans of (co)spans” in a way that Marco Grandis formalized in terms of powers \Lambda^n, where \Lambda is the diagram (i.e. category) \bullet \leftarrow \bullet \rightarrow \bullet. One can actually think of this in terms of internalization: these are spans in a category whose objects are spans in \mathcal{C}, and whose morphisms are triples of maps in C linking two spans (likewise for the span-map 2-morphisms). Yet it’s manifestly edge-symmetric: both the horizontal and vertical bicategories are the same.

As I mentioned in the previous post, there are lots of nice examples of double categories which are not edge-symmetric – sets, functions, and relations; or rings, homomorphisms, and bimodules, say. In fact, the second is only a pseudocategory – weak in one direction (composition of bimodules by tensor product is really only defined up to isomorphism). This is a significant thing about non-edge-symmetric examples. There’s much less motive for assuming both directions are equally strict. It’s also more natural in some ways: a pseudocategory is a weak model of Th(Cat) in Cat – equations in the theory are represented by (coherent) isomorphisms. This is the most general situation, and a strict model is a special case.

In the bicategory world, as I said, Bicat is a tricategory, so weaker models than the one I’ve given are possible – though they’re not symmetric, and so while one direction has composition and units as weak as a bicategory, the other direction will be weaker still. Robert Paré, in a conversation at MakkaiFest, suggested that a nice definition for a cubical n-category might have each direction being one step weaker than the previous one – a natural generalization of pseudocategories. Maybe there’s a way to make this seem natural in terms of internalization? One can iterate internalizing: having defined double bicategories, collect them together and find models of Th(Bicat) in DblBicat, and so forth. Maybe doing this as weakly as possible would give this tower of increasing weakness.

Now, I don’t have a great punchline to sum all this up, except that internalization seems to be an interesting lens with which to look at cubical n-categories.

So one of the things I’ve been doing recently is finishing up a version, and talking about, this paper which I’ve now put on the arXiv. While at it, I figured I should update a previous paper – the current version cuts out part of the original subject (cobordism categories) and expands on the category-theory side of things, giving more detailed proofs, etc. That part will then be out of the way when the topology side shows up in another paper, yet to appear, which will also use the stuff about 2-vector spaces and groupoids from the “new” paper.

Ironically, although I fixed the “issue” which arose when I was posting on the subject – and I’ll come back to that – I’ve already talked about most of what’s in the “new” paper, whereas I never got around to talking about what’s in the “old” one, updated version or not. That’s the one called “Double Bicategories and Double Cospans”, which is the most strictly category-theoretic thing I’ve produced: all the motivation from physics has been abstracted away.  So when I have some time, I’ll write something about that one.

For now, I just wanted to link to this new stuff.