The n-Category Café

Where to start reading about magnitude

Incidentally, if you’re curious about magnitude but don’t know what to read first, one possible starting point is these colloquium slides. The basic shape of the theory — glossing over technicalities — is this:

• Magnitude is a numerical invariant of enriched categories. It is arguably the canonical measure of the size of an enriched category, whatever “size” means.

• Magnitude homology is an algebraic invariant of enriched categories. Indeed, it is arguably the canonical homology theory of enriched categories. It is also intended to be a categorification of magnitude.

• Enriched categories include, among other things, ordinary categories and metric spaces. The magnitude of a category is often called its Euler characteristic. For metric spaces, magnitude and magnitude homology are something really new, so that’s where a lot of the interest lies.

For more detail, here’s a 2016 survey of magnitude, excluding magnitude homology:

Tom Leinster and Mark Meckes. The magnitude of a metric space: from category theory to geometric measure theory. arXiv:1606.00095; in Nicola Gigli (ed.), Measure Theory in Non-Smooth Spaces, de Gruyter Open, 2017.

Magnitude homology was first introduced for graphs:

Richard Hepworth and Simon Willerton. Categorifying the magnitude of a graph. arXiv:1505.04125; Homology, Homotopy and Applications 19(2) (2017), 31–60.

Graphs can be seen as special metric spaces, where the distance between two vertices it the number of edges in a shortest path between them. In turn, metric spaces are special enriched categories. The extension of magnitude homology from graphs to enriched categories was done here:

Tom Leinster and Michael Shulman. Magnitude homology of enriched categories and metric spaces. arXiv:1711.00802; Algebraic & Geometric Topology 21 (2021), 2175–2221.

Several people have pointed out that it would be really useful to have a modern survey of magnitude and magnitude homology. A bunch of us are working on it! But for now, these are my best suggestions.

Hotspots

Right now, it’s undoubtedly magnitude homology that’s receiving the most attention. The magnitude homology of graphs is especially active, although there’s been a lot on general metric spaces too.

Here are a few of the topics in this sphere that people have been investigating:

• The relationship between magnitude homology and persistent homology of a metric space — which detect very different information about a space (Nina Otter; Simon Cho)

• Applications of magnitude homology to the analysis of networks (Chad Giusti and Giuliamaria Menara)

• A theory of magnitude cohomology (Richard Hepworth)

• A comprehensive spectral sequence approach that encompasses both magnitude homology and path homology (Yasuhiko Asao; Richard Hepworth and Emily Roff; Kiyonori Gomi; Sergei Ivanov, …)

• A concept of magnitude homotopy (Yu Tajima and Masahiko Yoshinaga)

• An analysis of the inherent complexity of magnitude homology of graphs (Radmila Sazdanovic and Victor Summers; Luigi Caputi and Carlo Collari)

• New results on magnitude homology equivalence of subsets of $\mathbb{R}^n$, involving convex geometry (Adrián Doña Mateo and myself).

In need of attention

Although some topics in magnitude have received intense attention, there are others that remain wide open. Doubtless they won’t all be equally fruitful, but I think there are real opportunities to do new and exciting things. Some might not even be that hard.

Here’s my list, in no particular order.

• The magnitude of graphs   The magnitude of a graph is an invariant taking values in the ring of rational functions over $\mathbb{Q}$ in one variable. Or, from another viewpoint, it takes values in power series over $\mathbb{Z}$. From some angles it looks a bit like the Tutte polynomial, but neither is a specialization of the other.

  The magnitude homology of graphs is being thoroughly investigated, but magnitude itself not so much. As far as I know, it has received zero attention from graph theorists, or combinatorialists more generally, presumably because it hasn’t been used to solve an open problem they care about.

  I developed the theory myself as far as I could, but I’m not remotely skilled in graph theory. I’d love to see what the magnitude of graphs would look like in the hands of an expert graph theorist. What would they want to prove? What questions would they ask? What use could they put it to?

• Effective sample size   In statistics, there’s a notion of effective sample size, which I learned of long ago from Paul Blackwell and blogged about here. And in a certain context, the effective sample size is the magnitude of the correlation matrix! That is, it’s the sum of all the entries of the inverse matrix.

  This is very likely not a coincidence, as the world of magnitude already involves concepts like “effective number of points” and “effective number of species”. But the connection between magnitude and effective sample size is almost completely uninvestigated.

• The magnitude of higher categories   The way magnitude works is that given a notion of the size of each object of some monoidal category $V$, one automatically obtains a notion of the size of any finite category enriched in $V$. For instance, starting with the notion of the cardinality of a finite set, one automatically obtains the notion of the magnitude of a finite category.

  And we can keep going! From the notion of the magnitude of a finite category, we automatically get a notion of the magnitude of a finite 2-category, since a 2-category is a category enriched in $\mathbf{Cat}$. And so on, to get a notion of the magnitude of a finite $n$-category.

  This appears to be a reasonable notion. For instance, there’s an $n$-category $\mathbf{S}^n$ that looks like the $n$-sphere (e.g. $\mathbf{S}^1 = (\bullet \rightrightarrows \bullet)$), and its magnitude is the Euler characteristic of the topological $n$-sphere $S^n$ (either $0$ or $2$, depending on the parity of $n$). But apart from this single example and one paper by Kohei Tanaka on bicategories, there has barely been any research at all on the magnitude of $n$-categories. And the question of the correct definition of the magnitude of a finite $\infty$-category is still wide open.

• The magnitude of linear categories   By a “linear category”, I mean a category enriched in vector spaces. As in the previous bullet point, the notion of the dimension of a finite-dimensional vector space automatically gives rise to the notion of the magnitude of a finite linear category.

  Probably there are interesting theorems to be proved about the magnitude of linear categories. But as far as I know, there’s very little work on this so far: just the two papers listed in the magnitude bibliography. What we now know about magnitude homology may be especially useful here.

• Spectral geometry and semiclassical analysis   Some of the most analytically sophisticated work on the magnitude of subsets of $\mathbb{R}^n$ has used techniques from these subjects. For example, this is how it was shown that for nice enough sets $X, Y \subseteq \mathbb{R}^n$, an asymptotic inclusion-exclusion principle holds:

  $$mag(t(X \cup Y)) + mag(t(X \cap Y)) - mag(t X) - mag(t Y) \to 0$$

  as $t \to \infty$. Here $t X$ is the space $X$ scaled by a factor of $t$.

  Heiko Gimperlein, who did this work with Magnus Goffeng and Nikoletta Louca, has expressed the view that a lot more about magnitude could be proved with spectral geometry techniques, if only enough people with that expertise knew about the open problems and put the time in. He and his collaborators have already done a lot, but they’re only three people!

  As an example of what could be achieved by mathematicians with these skills, see the next bullet point…

• Maximum entropy   Off to the edge of magnitude-world are the almost-synonymous notions of entropy and diversity. In particular, there is a notion of the entropy of a probability distribution on a compact metric space.

  Given any compact metric space $X$, one can ask what the maximum entropy of $X$ is — that is, the maximum among the entropies of all possible probability distributions on $X$. (Emily Roff, Mark Meckes and I established that such a thing is well-defined.) Like magnitude, this is a real number associated with any compact metric space. One can also ask which distribution on $X$ achieves that maximum.

  It turns out that maximum entropy is closely related to magnitude. (For example, the maximum entropy of $X$ is equal to the magnitude of the support of the maximizing distribution.) That’s how it fits into the world of magnitude.

  So: maximum entropy is a numerical invariant of metric spaces. But almost no examples are known! For instance, apart from the trivial one-dimensional case, no one knows the maximum entropy of a Euclidean ball — not in any dimension, not of any radius. No one knows what the maximizing distribution on it is either. No one even knows the support of the maximizing distribution. And if that’s the sorry state of knowledge about balls, you can imagine how ignorant we are about more complex spaces.

  I understand from Heiko that it should be possible to apply the analytic techniques mentioned in the previous bullet point to solve this kind of problem. But so far, no one’s tried.

• The categorification problem   The final item on my list is different. It’s not so much an overlooked or understudied area as a really important problem that hasn’t been solved.

  Magnitude homology is intended to be a categorification of magnitude. There is a theorem making this precise for finite metric spaces, as follows.

  Magnitude homology of metric spaces $X$ is a real-graded homology theory: there is one group $H_{n, \ell}(X)$ for each integer $n \geq 0$ and real $\ell \geq 0$. This means that for each real $\ell \geq 0$, we have an Euler characteristic

  $$\chi_\ell(X) = \sum_n (-1)^n rank(H_{n, \ell}(X)).$$

  Now assemble these Euler characteristics into a single generating function in a formal variable $q$:

  $$\sum_{\ell \in \mathbb{R}^+} \chi_\ell(X) q^\ell$$

  The categorification result is this: when $X$ is finite, this generating function evaluated at $q = e^{-t}$ gives exactly the magnitude of $t X$, for all $t \gt 0$.

  But that only works for finite spaces. For infinite spaces, nothing like this can possibly true, at least with the current definitions. For example, the magnitude homology of a convex subset of $\mathbb{R}^n$ is trivial in homological degrees $n \gt 0$, which means we can’t possibly recover anything from it. On the other hand, convex sets have very different magnitudes: among other things, the function $t \mapsto mag(t X)$ determines the volume and dimension of $X$. So it’s certainly impossible to recover the magnitude of a convex set from its magnitude homology.

  The moral: something is wrong in the foundations of magnitude homology!

  The challenge is to fix it, in such a way that the categorification theorem extends from finite to infinite metric spaces. No one knows how.

In this post I’ve just listed my subjective impressions of which areas are particularly active in the world of magnitude, which are most in need of attention, and where I think there might be exciting new results to be found. I’m sure others would have different lists, and I’d probably write different ones myself if I did this tomorrow rather than today. I hope no one will be offended by me mentioning A rather than B, and I also hope that others with other ideas will put them in the comments.

Happy magnituding! Here’s to the next 100.