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.