The n-Category Café

To explain this I’ll have to back up a bit.

First, Simon and I decided that ‘the cardinality of a metric space’ was too confusing a phrase: people hearing it for the first time think it means the cardinality of the set of points. So, we decided to switch to magnitude.

Second, magnitude was only defined for finite metric spaces. But there’s a rather hands-on strategy for extending the definition to compact metric spaces $A$: take a sequence $A_0 \subseteq A_1 \subseteq \cdots$ of finite subspaces, with $\bigcup A_n$ dense in $A$, and try to define

$$|A| = \lim_{n \to \infty} |A_n|.$$

There are all sorts of things that could go wrong with this strategy, but sometimes it works, or at least works well enough to enable us to get somewhere. For example, we use it to calculate the magnitudes of straight line segments, circles, and middle-third Cantor sets.

Third, a lot is known about ‘measures’ satisfying the inclusion-exclusion principle on subsets of $\mathbb{R}^n$. These are known as valuations. If you restrict to polyconvex sets (finite unions of compact convex sets) then the valuations are completely classified. This is Hadwiger’s Theorem. (I’m glossing over some details here: really we want the valuations to have a couple of further properties. See the link.) For example, Hadwiger tells us that any valuation on $\mathbb{R}^2$ must be of the form

$$\alpha_0 \cdot Euler characteristic + \alpha_1 \cdot perimeter + \alpha_2 \cdot area$$

where $\alpha_0$, $\alpha_1$ and $\alpha_2$ are constants. We’ll want to go outside the world of polyconvex sets, since there are many common non-polyconvex spaces for which Euler characteristic, perimeter etc. have obvious meanings (e.g. circles). But we’ll continue to use Hadwiger’s Theorem as a guide.

Finally, there’s a pesky technical point that I feel obliged to put in for anyone taking the trouble to follow in detail. Skip it if you’re not. The original definition of magnitude/cardinality involved some terms of the form $e^{-2d(a,b)}$. We switched to $e^{-d(a,b)}$. So if, in what follows, you’re ever surprised by a factor of $2$, you know why.

Now I can tell the story! We’ve known for a while that the magnitude of a line segment (closed interval) of length $L$ is

$$1 + L/2 = 1 \cdot Euler characteristic + \frac{1}{2} \cdot perimeter.$$

Let’s consider the magnitude of, say, ‘nice’ subsets of $\mathbb{R}^2$. If magnitude satisfies the inclusion-exclusion principle, then Hadwiger’s Theorem suggests that it is probably a linear combination of Euler characteristic, perimeter and area. The example of the line segment tells us what the first two coefficients must be. So, if circles are ‘nice’ then the magnitude of a circle $C_L$ of circumference $L$ should be

$$1 \cdot Euler characteristic + \frac{1}{2} \cdot perimeter = 1 \cdot 0 + \frac{1}{2} \cdot L = L/2.$$

Is that true? No! But it’s asymptotically true; that is,

$$\lim_{L \to \infty} (|C_L| - L/2) = 0.$$

Crudely put: for large $L$,

$$|C_L| \approx 1 \cdot Euler characteristic + \frac{1}{2} \cdot perimeter.$$

There’s an infinite family of sensible metrics you can put on the circle, and this result holds for all of them. The proof — even for the plain old subspace metric — uses some techniques of asymptotic analysis. Thanks to Bruce Bartlett for tracking down the key result.

Now let’s try another example: the Cantor set, defined as usual by successively removing middle thirds from a line segment. Let $T_L$ be the Cantor set whose endpoints are distance $L$ apart. This is a different kind of animal from the line or the circle: it’s not obvious what Euler characteristic and perimeter would even mean. (On the other hand, we do expect a nonzero measure of dimension $\log_3 2$, since that’s the Hausdorff dimension of $T_L$.) Let’s use the self-similarity of the Cantor set instead. Since $T_L$ consists of two disjoint copies of $T_{L/3}$, the inclusion-exclusion principle suggests

$$|T_L| = 2|T_{L/3}|.$$

Is that true? No! But it’s asymptotically true; that is,

$$\lim_{L \to \infty}(|T_L| - p(L)) = 0$$

for a certain function $p$ satisfying

$$p(L) = 2p(L/3).$$

(It follows that $|T_L|$ grows like $L^{\log_3 2}$.) Crudely put: for large $L$,

$$|T_L| \approx 2|T_{L/3}|.$$

Here’s an intuitive explanation. Magnitude respects coproducts, meaning that if $A$ and $B$ are metric spaces and we write $A + B$ for their ‘distant union’ (the disjoint union with $d(a, b) = \infty$ for all $a \in A$ and $b \in B$) then $|A + B| = |A| + |B|$. The union $T_L = T_{L/3} \cup T_{L/3}$ is disjoint. It’s not distant, since the two copies of $T_{L/3}$ are distance $L/3$, not $\infty$, apart — but when $L$ is large, it’s nearly distant, so the magnitude formula nearly holds.

(Incidentally, there’s a little surprise: it turns out that inside the Cantor set, a periodic function is hiding! But you’ll have to read the paper to find out about that.)

All this evidence led us to make a conjecture. Roughly, it says that the inclusion-exclusion principle is satisfied asymptotically for compact subsets of $\mathbb{R}^n$, and exactly for convex sets.

Strong Asymptotic Conjecture  There is a unique function $P: \{ compact subsets of \mathbb{R}^n \} \to \mathbb{R}$ such that

• $\lim_{t \to \infty} (|t A| - P(t A)) = 0$ for all $A$, where $t A$ denotes $A$ scaled up by a factor of $t$
• $P$ satisfies the inclusion-exclusion principle: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ and $P(\emptyset) = 0$
• when $A$ is polyconvex, $P(A)$ is a certain specific linear combination of the intrinsic volumes of $A$
• when $A$ is convex, $|A| = P(A)$ (i.e. $|A|$ is exactly that linear combination of intrinsic volumes).

Atiyah is supposed to have said something like ‘if you don’t understand a conjecture, generalize it’. Our conjecture might be too strong — and we have a Weak Asymptotic Conjecture too — but we believe that something like it is true.