The n-Category Café

To tell you what a core is, I need to take a run-up.

Two distinct points $x$ and $y$ of a metric space are adjacent if there’s no other point $p$ between them (that is, satisfying $d(x, y) = d(x, p) + d(p, y)$). For instance, if you view a graph as a metric space in which the distance between vertices is the number of edges in a shortest path between them, then two vertices are adjacent when they’re joined by an edge.

The inner boundary $\rho X$ of a metric space $X$ is the set of all points that are adjacent to something in $X$. For instance, the inner boundary of a closed annulus is its inner bounding circle. Here’s the inner boundary of another closed subset of the plane, shown in thick blue:

This picture should help you believe the following theorem about convex hulls, which I’ll write as $conv$:

Theorem   For all closed $X \subseteq \mathbb{R}^N$,

$$conv(X) = X \cup conv(\rho X).$$

(part of our Proposition 4.5). In other words, any point in the convex hull of $X$ that isn’t in $X$ itself must be expressible as a convex combination of points in its inner boundary. I mention this result just as motivation for the concept of inner boundary: it tells us something that a convex geometer who’s never heard of magnitude homology might care about.

The core of $X \subseteq \mathbb{R}^N$ is defined as

$$core(X) = \overline{conv(\rho X)} \cap X$$

— the intersection of $X$ with the closed convex hull of its inner boundary. For instance, here’s the core of the set I just showed you, shaded:

It turns out that the core construction is idempotent: once you’ve shrunk a set down to its core, taking the core again doesn’t shrink it any further.

Here’s an important result on cores (a consequence of our Proposition 5.7):

Theorem   Let $X$ be a nonconvex closed subset of $\mathbb{R}^N$. Then every point of $X$ has a unique closest point in $core(X)$.

It’s important because it means that the inclusion $core(X) \hookrightarrow X$ has a distance-decreasing retraction $\pi$, defined by taking $\pi(x)$ to be the closest point of $core(X)$ to $X$. And having these maps $core(X) \leftrightarrows X$ is key to proving the main theorem on magnitude homology equivalence.

(The weird hypothesis “nonconvex” has to be there! A convex set has empty inner boundary, so its core is empty too, which means the theorem will fail for trivial reasons.)

So now we know what the core is.

But what do I mean by saying that two spaces $X$ and $Y$ have the “same” magnitude homology? As for any homology theory of any kind of object, there are at least three possible interpretations:

• We could just ask that the groups $H_n(X)$ and $H_n(Y)$ are isomorphic for each $n$. As with other homology theories, this seems to be too loose a relationship to be really interesting.

• Or we could consider quasi-isomorphism, the equivalence relation on spaces generated by declaring $X$ and $Y$ to be equivalent if there exists a map $X \to Y$ inducing an isomorphism in homology.

• More demandingly still, we could ask that there exist back-and-forth maps $X \leftrightarrows Y$ that are mutually inverse in homology.

We take the third option, defining metric spaces $X$ and $Y$ to be magnitude homology equivalent if there exist maps $X \leftrightarrows Y$ whose induced maps $H_n(X) \leftrightarrows H_n(Y)$ are mutually inverse for all $n \geq 1$.

Here “map” means a map that is 1-Lipschitz, or short, or a contraction or distance-decreasing in the non-strict sense. When you view metric spaces as enriched categories, these are the enriched functors. And, incidentally, we exclude the case $n = 0$ because if not, that would make $X$ and $Y$ isometric and the whole thing would become trivial.

Main theorem   Let $X$ and $Y$ be nonempty closed subsets of Euclidean space. Then $X$ and $Y$ are magnitude homology equivalent if and only if their cores are isometric.

This is part — the most important part — of our Theorem 9.1, which also gives several other equivalent conditions.

I won’t say anything about the proof except that it makes crucial use of a theorem of Kaneta and Yoshinaga, which gives an explicit formula for the magnitude homology of subsets of Euclidean space.

So what does this theorem do for us?

For a start, it gives lots of examples of pairs of spaces that are magnitude homology equivalent. For instance, every closed set is magnitude homology equivalent to its core, so in the example

that we saw earlier, the whole grey set is magnitude homology equivalent to the shaded blue core. Or, all three of these closed subsets of the plane are magnitude homology equivalent:

(the last being the complement in $\mathbb{R}^2$ of the union of the disc and the square).

The main theorem also allows us to tell when two spaces are not magnitude homology equivalent. For instance, the set

is not magnitude homology equivalent to any of the three in the previous figure, simply because their cores are obviously not isometric. It’s an easy test to apply.

Postscript

I just want to mention one more thing. In order to get to our main theorem, we needed to prove a strengthening for closed sets of the classical Carathéodory theorem on convex sets, due to the extraordinary Greek mathematician Constantin Carathéodory. His famous theorem (well, one of them) is this:

Carathéodory’s theorem   Let $X$ be a subset of $\mathbb{R}^N$ and $a \in \conv(X)$. Then there exist $n \geq 0$ and affinely independent points $x_0, \ldots, x_n \in X$ such that

$$a \in conv\{x_0, \ldots, x_n\}.$$

Since an affinely independent set in $\mathbb{R}^N$ can have at most $N + 1$ elements, it follows that any point in the convex hull of $X$ is in the convex hull of some $(N + 1)$-element subset of $X$. That’s actually the version of Carathéodory’s theorem that people most often use. But it’s the full statement that’s most relevant to us here.

Now here’s a stronger theorem for closed sets, which appears as Theorem 3.1 of our paper:

Closed Carathéodory theorem   Let $X$ be a closed subset of $\mathbb{R}^N$ and $a \in \conv(X)$. Then there exist $n \geq 0$ and affinely independent points $x_0, \ldots, x_n \in X$ such that

$$a \in conv\{x_0, \ldots, x_n\}$$

and

$$conv\{x_0, \ldots, x_n\} \cap X = \{x_0, \ldots, x_n\}.$$

In other words, when $X$ is closed, you can find affinely independent points of $X$ whose convex hull not only contains the point $a$ (as in the classical Carathéodory theorem), but also, only intersects $X$ where it absolutely has to.

This theorem is absolutely classical in flavour and could easily have been proved by Carathéodory himself. I imagine it’s in the literature somewhere. But despite reading survey papers on Carathéodory-like theorems and asking around (including on Mathoverflow), we haven’t been able to find it. So we included a proof in our paper.