Disintegrations Integrated and Operations on Categories of Sheaves

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:

measurability: the function $x \mapsto ||\phi_x||$ is measurable for all $\phi \in \mathcal{M}$
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 $\psi \in \mathcal{M}$
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.

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