The n-Category Café

Background, part I: finite spaces

The magnitude of a finite metric space $(A, d)$ is defined as follows. First, define the matrix $\zeta_A = \bigl[e^{-d(a,b)} \bigr]_{a,b \in A} \in \mathbb{R}^{A \times A}$. A weighting for $A$ is a vector $w \in \mathbb{R}^A$ such that for each $a \in A$,

If $w$ is a weighting for $A$, then the magnitude of $A$ is defined to be

In general, there are well-definedness issues with this definition as written. For the purposes of this post I’ll brush them aside by noting that if $\zeta_A$ happens to be a positive definite matrix, and hence invertible, then $A$ possesses a unique weighting. When this is the case, we call $A$ a positive definite metric space, or PDMS. For example, it turns out that all finite subsets of $\mathbb{R}^n$ (with the usual Euclidean metric) are positive definite metric spaces. PDMSs have the additional nice properties that their magnitude is always at least 1 (if they’re nonempty), and that magnitude is increasing with respect to inclusion.

Background, part II: compact spaces

Given an invariant of finite metric spaces, it’s natural to try to extend it to some class of infinite spaces, say compact spaces. There are two obvious general approaches: approximate a given compact space by finite spaces, or adapt the definition to apply to infinite spaces directly.

Let’s first consider approximation. We’d like to say that if $A$ is a given compact metric space, we can find a sequence of finite spaces $A_k$ which approximate $A$ in some sense and whose magnitudes $| A_k |$ converge to some limit, which we could then take as the definition of $| A \vert$. There are well-definedness issues here, in particular whether different approximating sequences would give different values of $| A |$. The usual way to handle such an issue, if you can, is by appealing to continuity: if magnitude were a continuous function of a finite metric space (with respect to an appropriate topology), then we could define magnitude on the family of compact spaces as its unique continuous extension.

Fortunately, there is a natural topology on the family of isometry classes of compact metric spaces. Unfortunately, magnitude is not a continuous function of a finite metric space. However, if we restrict to finite PDMSs, it turns out that magnitude is lower semicontinuous, or lsc for short. Like a continuous function, an lsc function has a canonical extension from a domain to its closure—not a unique lsc extension, but a minimal maximal one. So we can naturally define the magnitude of compact PDMSs (a compact space is a PDMS if each of its finite subspaces is) as the minimal lsc extension of the magnitude of finite PDMSs.

Concretely, this means that $| A |$ is well-defined as the limit of $| A_k |$, whenever $A_k$ is a sequence of finite PDMSs which approximate $A$ and for which $| A_k |$ is increasing. Since magnitude is increasing with respect to inclusion for finite PDMSs, the last condition can be guaranteed by taking an increasing sequence $A_k$. (Notice that this minimal lsc extension of magnitude a priori takes values in $[1, \infty]$. I don’t know whether every compact PDMS has finite magnitude. Tom proved that every compact subset of $\mathbb{R}^n$ has finite magnitude.)

Using roughly this approach (although before the semicontinuity result was proved), Tom and Simon showed, among other things, that the magnitude of an interval of length $\ell$ is $1 + \ell/2$. So it became clear that magnitude knows something about classical geometric invariants.

The approximation approach leads to some nice concrete results, and is justified by semicontinuity. But it’s rather unsatisfactory in at least two ways. On the one hand, it would simply be nicer from an aesthetic point of view to give a direct definition of the magnitude of a compact space, which extends the original definition; it might also shed new light on the original definition. On the other hand, such a direct definition would involve some notion of weighting for a compact space, which would provide a new tool for analyzing magnitude.

Later, Simon gave a definition of magnitude for certain compact spaces which extends the original definition. Given a compact metric space $(A, d)$, a weight measure for $A$ is a measure $\mu$ such that for each $a \in A$,

If $\mu$ is a weight measure for $A$, then we define

It turns out that this approach agrees with the approximation approach. That is, if $A$ is a compact PDMS and has a weight measure, then this definition of $| A |$ agrees with the earlier one. On the other hand, the domains of definition are different: not every compact PDMS has a weight measure, and many spaces which are not positive definite do possess weight measures.

Using this definition, Simon studied magnitudes of homogeneous Riemannian manifolds. Again, he found that magnitude knows about classical geometric invariants. However, the fact that many (probably most, in some sense) spaces don’t have weight measures makes the measure-based definition of magnitude unsatisfactory as well.

Weightings for compact PDMSs

More recently, I found a definition of magnitude for a compact PDMS which directly extends the original definition for a finite space. To begin with, we take inspiration from Simon’s definition and think of the weighting $w$ of a finite space as the measure (more precisely, as a signed measure, since its components need not be positive)

where $\delta_a$ is a point mass at $a$. We next take inspiration from the finite approximation definition and say that the weighting of a compact space $A$ should be a limit of weightings of finite spaces. That is, it should be a limit of some sequence of finitely supported signed measures on $A$. To make sense of this, we need to put some topology on the space $F M(A)$ of finitely supported signed measures. Since it’s a vector space, we’d probably like to make it a topological vector space. The nicest topological vector spaces are inner product spaces, so it makes sense to try to find an inner product on $F M(A)$ which will work nicely with magnitude.

If $A$ is a PDMS, then such an inner product is staring us in the face (although it took many months before I noticed that): if $B \subseteq A$ is finite, then

Define $\mathcal{W}_A$ to be the Hilbert space completion of $F M(A)$ with this inner product. A weighting for $A$ will be an element of $\mathcal{W}_A$ which satisfies an appropriate analogue of the original condition $\zeta_A w = 1$.

When $v = \sum_{b \in B} v_b \delta_b \in F M(A)$, for some finite $B \subseteq A$, we can straightforwardly define the function

for $a \in A$. It’s not hard to show that $Z v$ is a bounded continuous function, and that $\| Z v \|_\infty \le \| v \|_{\mathcal{W}}$, so the linear map $Z:F M(A) \to C(A)$ has a unique continuous linear extension $Z:\mathcal{W}_A \to C(A)$. We can now define a weighting of $A$ to be a $w \in \mathcal{W}_A$ such that $Z w(a) = 1$ for each $a \in A$.

The map $Z$ is injective, so $A$ has at most one weighting. Moreover, $A$ has a weighting if and only if there is a uniform bound on the magnitudes of its finite subsets. In other words, $A$ possesses a weighting iff the magnitude $| A |$, as defined by finite approximations, is finite. (This also means that I don’t know whether every compact PDMS has a weighting, but I do know that every compact subset of $\mathbb{R}^n$ does.)

Now that we know what a weighting for $A$ is, we’re ready to define the magnitude of $A$. This is slightly less straightforward, since if $w \in \mathcal{W}_A$ is not in $F M(A)$, then there is no meaning to $w_a$. To get around this, we notice that the original definition for a finite space can be rewritten as

and the latter sum fits right into our Hilbert space framework. So, if $A$ is a compact PDMS with weighting $w \in \mathcal{W}_A$, then we define the magnitude of $A$ to be

This fits with all the previous definitions: if $A$ is a finite PDMS, this simply repeats the original definition in a more complicated way; if $A$ is a compact PDMS, this gives the same value of magnitude as finite approximation; and if $A$ is a compact PDMS with a weight measure $\mu$, then $\mu$ is a weighting for $A$ and $\mu(A) = \| \mu \|_{\mathcal{W}}^2$.

So what does this give us that we didn’t have before? As hoped for, it’s a definition which works for a broad class of spaces, and is (to me, at least) more elegant than approximation by finite spaces. It may shed a little bit of light on the original definition of magnitude for finite spaces—it suggests that one should think of the magnitude not, as originally written, as the sum of the entries of the weighting $w$, but as the double sum

In other words, it suggests that the matrix $\zeta_A$ should be thought of, not as defining a linear map, but as defining a bilinear form. (The same vague, speculative remark applies more generally to the definition of the magnitude of an enriched category.)

What about the hope that the weighting $w$ of $A$ would turn out to be a useful tool? Before addressing that, let’s first specialize to subsets of $\mathbb{R}^n$.

Weightings and magnitude in Euclidean space

If $A \subseteq \mathbb{R}^n$, then the linear operator $Z$ is a convolution operator, which suggests using a bunch of Fourier mumbo-jumbo. If you do that, it turns out that $\mathcal{W}_A$ is a subspace of the Sobolev space (or more properly, Bessel potential space) $H^{-(n+1)/2}$, and the norm is, up to a constant depending on $n$, the standard Sobolev norm on $H^{-(n+1)/2}$. This isn’t the most familiar, classical sort of thing—it’s a Sobolev space, but with a negative, fractional (half the time) exponent—but it is a known and well-understood thing. Its elements are distributions, not functions, but a lot of machinery exists to work with such things.

Even better, $Z$ maps $\mathcal{W}_A$ into the Sobolev space $H^{(n+1)/2}$. This is a Sobolev space whose elements are functions. (Real, honest, continuous functions—not even almost-every-equal equivalence classes!) If $w$ is the weighting of $A$, then $h = Z w \in H^{(n+1)/2}$ is, by definition, the unique function in $Z(\mathcal{W}_A)$ such that $h(a) = 1$ for all $a \in A$. It’s not hard to show that $| A | = c_n \| h \|^2_{H^{(n+1)/2}}$, for an explicit dimensional constant $c_n$. With a bit more work, one can show that

and the infimum is uniquely achieved when $f = h$.

This reformulation of magnitude in terms of a variational problem for functions turns out to be extremely useful. It opens the door to using tools from Hilbert space geometry, Fourier analysis, potential theory, and PDEs to study magnitude. But this post has gotten to be rather long, so I’ll leave it to you to read the paper (and hopefully future papers and posts) to see how!