### Magnitude of Metric Spaces: A Roundup

#### Posted by Tom Leinster

**Update (November 2018):** *I’m maintaining a bibliography of publications on magnitude at* www.maths.ed.ac.uk/~tl/magbib.

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Mark Meckes and I have just arXived one paper each on the magnitude of metric spaces:

- Mark Meckes, Positive definite metric spaces
- Tom Leinster, The magnitude of metric spaces.

This blog has played an important part in the development of the theory. I tried to send Jacques an email thanking him for his role in that, but his spam-blocking software blocked it. So Jacques, if you’re reading: thanks!

It’s a tangled story: you can find online at least 8 papers, 10 blog posts and 12 talks related to this subject, and they really didn’t appear in logical order. The purpose of this post is simply to round up the available material.

I hope that Mark, or I, or both of us, will feel like writing something expository about our new papers—but that’s for another day.

First I should clarify the…

### Terminology

From the beginning (p. 38), the idea was that an enriched category should have associated to it a certain fundamental quantity, in a way that I won’t describe now.

Enriched categories include ordinary categories, and in that case it makes sense to call that quantity “Euler characteristic”. This was the first case to be investigated in detail. In turn, categories include sets, and in that case the quantity is cardinality; they also include groupoids, where the quantity is cardinality in the sense of Baez and Dolan.

Enriched categories also include metric spaces. At some point (January 2008)
it became clear to me that it was really worth thinking about this quantity for
metric spaces. Since the quantity is analogous to cardinality of sets, I made
the mistake of calling it “cardinality”, not realizing that “the cardinality of
a metric space” would be understood by most people to mean the cardinality of
its set of points. (Oddly though, I realized that “Euler characteristic” would
be similarly ambiguous.) About a year later Simon Willerton and I discussed
alternative names. After rejecting such contenders as “fundamental invariant”,
“characteristic” and “bigness”, we settled on—or rather,
he rightly persuaded me that we should settle on—*magnitude*.

So now the convention is to use “magnitude” to refer to the quantity associated
to an enriched category. In particular, this applies in the special case of
metric spaces
(categories
enriched in $[0, \infty]$). For ordinary categories (enriched in **Set**)
and algebroids or linear categories (enriched in **Vect**) it’s appropriate
to use “Euler characteristic” as a synonym.

**Upshot**: *in the various writings on this, the terms
“cardinality”,
“Euler characteristic” and “magnitude” are more or less interchangeable.*

(To digress slightly, this terminological slipperiness arises from an interesting mathematical phenomenon: forgetful functors don’t preserve magnitude, although free functors do. That’s a rule of thumb rather than a theorem, but it means you need to be super-careful. For example, if $X$ is a metric space and $U(X)$ its set of points then you need to distinguish between the magnitude of $X$ and the magnitude/cardinality of $U(X)$.)

Here now is the list of online materials on magnitude of enriched categories (including categories and metric spaces). Of course there are many precursors in the literature: this didn’t come out of thin air. But I’ll begin where my own involvement began, which is also where (to my knowledge) the definition of the magnitude of an enriched category first appeared.

### Papers

I’ll give the arXiv date of each paper, but the chronological order is nothing like the logical order. (E.g. much of my recent paper is really a logical precursor to the other ones about metric spaces, but I took a long time writing it up.)

- The Euler characteristic of a category (me, October 2006)
- The Euler characteristic of a category as the sum of a divergent series (Clemens Berger and me, July 2007)
- On the asymptotic magnitude of subsets of Euclidean space (Simon and me, August 2009)
- Heuristic and computer calculations for the magnitude of metric spaces (Simon, October 2009)
- A maximum entropy theorem with applications to the measurement of biodiversity (me, October 2009)
- On the magnitude of spheres, surfaces and other homogeneous spaces (Simon, May 2010)
- Positive definite metric spaces (Mark, December 2010)
- The magnitude of metric spaces (me, December 2010).

### Blog posts

- Euler characteristic of a category (David, October 2006)
- This week’s finds in mathematical physics (week 244) (John, February 2007)
- Return of the Euler characteristic of a category (David, July 2007)
- Metric spaces (me, February 2008)
- Entropy, diversity and cardinality (part 1) (me, October 2008)
- Entropy, diversity and cardinality (part 2) (me, November 2008)
- Asymptotics of the magnitude of metric spaces (me, about the joint paper with Simon, August 2009)
- More magnitude of metric spaces and problems with penguins (Simon, October 2009)
- An adventure in analysis (me, November 2009)
- Intrinsic volumes and Weyl’s tube formula (Simon, March 2010)
- On the magnitude of spheres, surfaces and other homogeneous spaces (Simon, April 2010)

### Talks

- A whole bunch of talks by me (March 2006—September 2010), including the prequel to…
- Magnitude of metric spaces II (Simon, CRM, Barcelona, September 2010)
- The magnitude of a metric space (Mark, Fields Institute, November 2010)

I might have missed some things. Let me know!

*Update (7 April 2013)*

I’ll add a few more resources that have appeared since I wrote this post.

### Papers

- Notions of Möbius inversion (me, January 2012)
- Spread: a measure of the size of metric spaces (Simon, September 2012)

### Blog posts

- Möbius inversion for categories (me, May 2011)
- The magnitude of an enriched category (me, June 2011)
- The spread of a metric space (Simon, September 2012)
- Mark on magnitude (me, March 2013)
- The convex magnitude conjecture (me, March 2013)

### Talks

- Magnitude and other measures of metric spaces (Simon, July 2012)
- The magnitude of metric spaces (Mark, March 2013)

If I’m *really* organized, I’ll keep this updated.

## Re: Magnitude of Metric Spaces: A Roundup

If “the chronological order is nothing like the logical order”, is there a good order in which to approach these papers and blog posts, if we’re coming to this for the first time? It looks like the October 2006 paper and post would be a good starting point (or is there something else we should read before those?), but after that where does the story most naturally continue?