The n-Category Café

There are two famous kinds of non-Euclidean geometry: hyperbolic geometry and elliptic geometry (which almost deserves to be called ‘spherical’ geometry, but not quite because we identify antipodal points on the sphere).

In fact, these two kinds of geometry, together with Euclidean geometry, fit into a unified framework with a parameter $s \in \mathbb{R}$ that tells you the curvature of space:

• when $s \gt 0$ you’re doing elliptic geometry

• when $s = 0$ you’re doing Euclidean geometry

• when $s \lt 0$ you’re doing hyperbolic geometry.

This is all well-known, but I’m trying to explain it in a course I’m teaching, and there’s something that’s bugging me.

It concerns the precise way in which elliptic and hyperbolic geometry reduce to Euclidean geometry as $s \to 0$. I know this is a problem of deformation theory involving a group contraction, indeed I know all sorts of fancy junk, but my problem is fairly basic and this junk isn’t helping.

Quick - what’s the $10^{100}$th digit of $\pi$?

If you’re anything like me, you have some uncertainty about the answer to this question. In fact, your uncertainty probably takes the following form: you assign a subjective probability of about $\frac{1}{10}$ to this digit being any one of the possible values $0, 1, 2, \dots 9$. This is despite the fact that

• the normality of $\pi$ in base $10$ is a wide open problem, and
• even if it weren’t, nothing random is happening; the $10^{100}$th digit of $\pi$ is a particular digit, not a randomly selected one, and it being a particular value is a mathematical fact which is either true or false.

If you’re bothered by this state of affairs, you could try to resolve it by computing the $10^{100}$th digit of $\pi$, but as far as I know nobody has the computational resources to do this in a reasonable amount of time.

Because of this lack of computational resources, among other things, you and I aren’t logically omniscient; we don’t have access to all of the logical consequences of our beliefs. The kind of uncertainty we have about mathematical questions that are too difficult for us to settle one way or another right this moment is logical uncertainty, and standard accounts of how to have uncertain beliefs (for example, assign probabilities and update them using Bayes’ theorem) don’t capture it.

Nevertheless, somehow mathematicians manage to have lots of beliefs about how likely mathematical conjectures such as the Riemann hypothesis are to be true, and even about simpler but still difficult mathematical questions such as how likely some very large complicated number $N$ is to be prime (a reasonable guess, before we’ve done any divisibility tests, is about $\frac{1}{\ln N}$ by the prime number theorem). In some contexts we have even more sophisticated guesses like the Cohen-Lenstra heuristics for assigning probabilities to mathematical statements such as “the class number of such-and-such complicated number field has $p$-part equal to so-and-so.”

In general, what criteria might we use to judge an assignment of probabilities to mathematical statements as reasonable or unreasonable? Given some criteria, how easy is it to find a way to assign probabilities to mathematical statements that actually satisfies them? These fundamental questions are the subject of the following paper:

Scott Garrabrant, Tsvi Benson-Tilsen, Andrew Critch, Nate Soares, and Jessica Taylor, Logical Induction. ArXiv:1609.03543.

Loosely speaking, in this paper the authors

• describe a criterion called logical induction that an assignment of probabilities to mathematical statements could satisfy,
• show that logical induction implies many other desirable criteria, some of which have previously appeared in the literature, and
• prove that a computable logical inductor (an algorithm producing probability assignments satisfying logical induction) exists.

Logical induction is a weak “no Dutch book” condition; the idea is that a logical inductor makes bets about which statements are true or false, and does so in a way that doesn’t lose it too much money over time.

You’ve probably met mathematicians at the University of Leicester, or read their work, or attended their talks, or been to events they’ve organized. Their pure group includes at least four people working in categorical areas: Frank Neumann, Simona Paoli, Teimuraz Pirashvili and Andy Tonks.

Now this department is under severe threat. A colleague of mine writes:

24 members of the Department of Mathematics at the University of Leicester — the great majority of the members of the department — have been informed that their post is at risk of redundancy, and will have to reapply for their positions by the end of September. Only 18 of those applying will be re-appointed (and some of those have been changed to purely teaching positions).

It’s not only mathematics at stake. The university is apparently on a process of “institutional transformation”, involving:

the closure of departments, subject areas and courses, including the Vaughan Centre for Lifelong Learning and the university bookshop. Hundreds of academic, academic-related and support staff are to be made redundant, many of them compulsorily.

If you don’t like this, sign the petition objecting! You’ll see lots of familiar names already on the list (Tim Gowers, John Baez, Ross Street, …). As signatory David Pritchard wrote, “successful departments and universities are hard to build and easy to destroy.”

I’m down in Bristol at a conference – HoTT and Philosophy. Slides for my talk – The modality of physical law in modal homotopy type theory – are here.

Perhaps ‘The modality of differential equations’ would have been more accurate as I’m looking to work through an analogy in modal type theory between necessity and the jet comonad, partial differential equations being the latter’s coalgebras.

The talk should provide some intuition for a pair of talks the following day:

• Urs Schreiber & Felix Wellen: ‘Formalizing higher Cartan geometry in modal HoTT’
• Felix Wellen: ‘Synthetic differential geometry in homotopy type theory via a modal operator’

I met up with Urs and Felix yesterday evening. Felix is coding up in Agda geometric constructions, such as frame bundles, using the modalities of differential cohesion.

I’m now trying to announce all my new writings in one place: on Twitter.

Why? Well…

In Tom’s recent post he mentioned that Juan Antonio Barceló and Tony Carbery had been able to calculate the magnitude of any odd-dimensional Euclidean ball. In this post I would like to give some idea of the methods they use for calculating the magnitude.

Tony and Juan Antonio calculate the magnitude of an odd dimensional ball of a given radius using a potential function rather than a weighting, I think that if you know much about magnitude then you will have some idea what a weighting is but not much idea about what a potential function is, so I will explain that below, the theory having been developed by Mark Meckes.

I intend to brush over technical details about distributions, I hope that I do not do so in too egregious a fashion.

In Tony and Juan Antonio’s paper, various aspects of mine and Tom’s Convex Magnitude Conjecture are confirmed [see the comments below]; however, the calculations for the five-ball provide a counterexample to the conjecture in general. This raises lots of new and interesting questions, but I won’t go into them in this post.

David Madore has a lot of great stuff on his website - videos and discussion of rotating black holes, a math blog whose only defect is that half is in French, and more.

He has has an interesting story that claims to tell you the Ultimate Question, and its Answer:

• David Madore, Toward enlightenment.

No, it’s not 42. I like it, but I can’t tell how much sense it makes. So, I’ll ask you.

I’m excited that over on this thread, Mike Shulman has proposed a very plausible theory of magnitude homology. I think his creation could be really important! It’s general enough that it can be applied in lots of different contexts, meaning that lots of different kinds of mathematician will end up wanting to use it.

However, the story of magnitude homology has so far only been told in that comments thread, which is very long, intricately nested, and probably only being followed by a tiny handful of people. And because I think this story deserves a really wide readership, I’m going to start afresh here and explain it from the beginning.

Magnitude is a numerical invariant of enriched categories. Magnitude homology is an algebraic invariant of enriched categories. The Euler characteristic of magnitude homology is magnitude, and in that sense, magnitude homology is a categorification of magnitude. Let me explain!

John Regehr writes: “holy cow this Cousot+Cousot paper achieves a density I’ve never before seen.” Me neither!

Much of the paper looks like the snippet shown, except for the part where they take the time to explain that “e.g.” means “for example”. Read this Twitter thread for speculation on how this state of affairs came to be.

Okay, let’s look at some examples of topological crystals. These are what got me excited in the first place. We’ll get some highly symmetrical crystals, often in higher-dimensional Euclidean spaces. The ‘triamond’, above, is a 3d example.

We at the mathematics department at the University of Edinburgh are doing more and more things in conjunction with our sisters and brothers at Heriot–Watt University, also in Edinburgh. For instance, our graduate students take classes together, and about a dozen of them are members of both departments simultaneously. We’re planning to strengthen those links in the years to come.

The news is that Heriot–Watt are hiring.

Monoidal categories are often introduced as an abstraction of categories with products. Instead of having the categorical product $\times$, we have some other product $\otimes$, and it’s required to behave in a somewhat product-like way.

But you could try to abstract more of the structure of a category with products than monoidal categories do. After all, when a category has products, it also comes with special maps $X \times Y \to X$ and $X \times Y \to Y$ for every $X$ and $Y$ (the projections). Abstracting this leads to the notion of “monoidal category with projections”.

I’m writing this because over at this thread on magnitude homology, we’re making heavy use of semicartesian monoidal categories. These are simply monoidal categories whose unit object is terminal. But the word “semicartesian” is repellently technical, and you’d be forgiven for believing that any mathematics using “semicartesian” anythings is bound to be going about things the wrong way. Name aside, you might simply think it’s rather ad hoc; the nLab article says it initially sounds like centipede mathematics.

I don’t know whether semicartesian monoidal categories are truly necessary to the development of magnitude homology. But I do know that they’re a more reasonable and less ad hoc concept than they might seem, because:

Theorem   A semicartesian monoidal category is the same thing as a monoidal category with projections.

So if you believe that “monoidal category with projections” is a reasonable or natural concept, you’re forced to believe the same about semicartesian monoidal categories.

I’m happy to announce that this paper has been published:

• Mike Stay, Compact closed bicategories, Theory and Applications of Categories 31 (2016), 755–798.

Abstract. A compact closed bicategory is a symmetric monoidal bicategory where every object is equipped with a weak dual. The unit and counit satisfy the usual ‘zig-zag’ identities of a compact closed category only up to natural isomorphism, and the isomorphism is subject to a coherence law. We give several examples of compact closed bicategories, then review previous work. In particular, Day and Street defined compact closed bicategories indirectly via Gray monoids and then appealed to a coherence theorem to extend the concept to bicategories; we restate the definition directly.

We prove that given a 2-category $C$ with finite products and weak pullbacks, the bicategory of objects of $C$, spans, and isomorphism classes of maps of spans is compact closed. As corollaries, the bicategory of spans of sets and certain bicategories of ‘resistor networks” are compact closed.

The notion of the magnitude of a metric space was born on this blog. It’s a real-valued invariant of metric spaces, and it came about as a special case of a general definition of the magnitude of an enriched category (using Lawvere’s amazing observation that metric spaces are usefully viewed as a certain kind of enriched category).

Anyone who’s been reading this blog for a while has witnessed the growing-up of magnitude, with all the attendant questions, confusions, misconceptions and mess. (There’s an incomplete list of past posts here.) Parents of grown-up children are apt to forget that their offspring are no longer helpless kids, when in fact they have a mortgage and children of their own. In the same way, it would be easy for long-time readers to have the impression that the theory of magnitude is still at the stage of resolving the basic questions.

Certainly there’s still a great deal we don’t know. But by now there’s also lots we do know, so Mark Meckes and I recently wrote a survey paper:

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

Here I’ll tell you some of the highlights: ten things we used not to know, but do now.

In real analysis you get just what you pay for. If you want a function to be seven times differentiable you have to say so, and there’s no reason to think it’ll be eight times differentiable.

But in complex analysis, a function that’s differentiable is infinitely differentiable, and its Taylor series converges, at least locally. Often this lets you extrapolate the value of a function at some faraway location from its value in a tiny region! For example, if you know its value on some circle, you can figure out its value inside. It’s like a fantasy world.

Algebraic geometry has similar miraculous properties. I recently learned about two.