The n-Category Café

guest post by Tim Silverman

Welcome to the second part of this series of posts on picturing modular curves.

Catching up

Last time, I zipped through some of the maths relating (among other things) Farey sequences, modular arithmetic, the hyperbolic plane, Platonic solids, the rational projective line, and the complex upper half plane, accompanied by pretty pictures of these often rather photogenic objects. But, during the attempts to get some of the svg for the pictures to work with the blog format, some of the pictures near the end got missed off! So I’m going to present them here.

But, also, I packed rather a lot of stuff into one post, and since I’m creating an extra post to put the pictures in, and since some people grumbled at the frenetic speed of the last post, I thought I’d take the opportunity to be more explicit about some of that maths I zipped through.

I think I’m gradually coming to understand why some people think $(\infty,1)$-toposes are the best thing since sliced bread. For me, I think a brief way to describe what’s so amazing about them is that they still have all the wonderful things I love about “good old” 1-toposes, but they also bring in homotopy theory and higher category theory in the “right” way, which fixes a number of problems (or inelegancies) that happen with good old 1-toposes.

In this post I want to describe one aspect of that, at an intuitive level that should hopefully make sense even if you have no idea what an $(\infty,1)$-topos is (yet). It’s closely related to stuff that Urs has been talking a lot about recently, but I prefer to think about it in terms of “petit toposes” instead of “gros” ones. From this perspective, the claim is that $(\infty,1)$-toposes elucidate the relationship between topological spaces and $\infty$-groupoids, by providing a context in which both can be embedded (almost) disjointly.

It’s not very often that one comes across blatant plagiarism, but I just have. I’m not going to give the details, partly because I’m not sure that public naming and shaming is the right thing to do, and partly because I don’t want to be sued. But I do want to say a little, because raising awareness may help to prevent this kind of thing from happening too often.

Here’s a gentle introduction to the work my students have been doing on categorification and physics:

• Sophie Hebden, The art of math: a pictorial branch of mathematics could help physicists draw new conclusions about quantum gravity and the nature of time.

It was put out by the Foundational Questions Institute, or FQXi. This is an organization that funds innovative research on hard questions like

what is the nature of time?

what is ultimately possible in physics?

and

how come there’s an ‘X’ in the acronym for ‘Foundational Questions Institute’?

A while back they gave me a grant to help out three of my grad students: John Huerta, Chris Rogers and Christopher Walker. It made a huge difference! Instead of working as teaching assistants all the time, they could write lots of papers, go to lots of conferences, and make progress much faster. They’re all finishing up this spring, and they’ll need jobs. You should hire them.

Unfortunately it’s a bit hard to describe their work in simple terms.

Fortunately, Sophie Hebden’s article does a great job! How do you explain categorification to people who haven’t studied math since high school? It may sound impossible, but this article does it.

But if you know some math, you’ll probably want to see more technical details: without the details, our work might sound like fluff with no substance. So: let me describe the papers we wrote with the help of this FQXi grant. For most I’ll include links, not only to the papers themselves, but to conversations about them here on the n-Category Café.

Benoît Mandelbrot died last week, aged 85. To mark the occasion, I will say something about the set that bears his name.

Tim Gowers, in the introduction to his wonderful little book Mathematics: A Very Short Introduction, writes:

I do presuppose some interest on the part of the reader rather than trying to drum it up myself. For this reason I have done without anecdotes, cartoons, exclamation marks, jokey chapter titles, or pictures of the Mandelbrot set. I have also avoided topics such as chaos theory and Gödel’s theorem, which have a hold on the public imagination out of proportion to their impact on current mathematical research

Mandelbrot was enormously successful in popularizing aspects of his work, as this quotation indicates. Unfortunately, the popular appeal of the Mandelbrot set, and fractals more generally, has sometimes made life hard for mathematicians who do serious work on such things. I don’t read Gowers’s statement as a criticism of those people or their work, but I think he’s expressing, at least, the feeling that the Mandelbrot set has become a kind of cliché.

Here I want to do a little bit to alter perceptions of the Mandelbrot set, by explaining how it fits into a wider mathematical context.

The concept of a

cohesive topos

axiomatizes properties of a general abstract context of geometric spaces. This has been proposed by Bill Lawvere.

Below is

1. a brief introduction to the definition of cohesive topos

2. an appreciation of Bill Lawvere’s work in this direction.

More is behind the above link.

How many people reading this blog also read Less Wrong — “a community blog devoted to refining the art of rationality”? Though I don’t know, I’d guess it’s just a small fraction. So here are some fun essays you might not have seen, written by the mathematician Jonah Sinick:

• Fields Medalists on School Mathematics.
• Great Mathematicians on Math Competitions and “Genius”.
• Vanity and Ambition in Mathematics.
• Beauty in Mathematics.

We’re pleased to announce the third meeting of the Scottish Category Theory Seminar, from 2.00 to 5.30 on Thursday 2 December, at the University of Strathclyde (in Glasgow city centre). All are welcome to attend. We have three invited speakers:

• Marcelo Fiore (Cambridge): On higher order algebra
• Peter Johnstone (Cambridge): Hochas and minimal toposes
• Bas Spitters (Nijmegen): Bohrification: topos theory and quantum theory

Abstracts are below. The speakers are chosen to represent category theory in, respectively, computer science, pure mathematics, and physics.

The meeting is generously supported by the Edinburgh Mathematical Society.

Information is on the Scottish Category Theory Seminar home page. Abstracts are there and, for your convenience, below.

If you’re planning to come, it would help us if you sent a quick email saying so. Our email address is scotcats#cis,strath,ac,uk (making the obvious substitutions). This reaches the organizers: Neil Ghani, Alex Simpson, and me.

It’s my birthday, so I’d like to invite you all to a café. Guess which one?

This invitation is especially for those standing outside the Café with their nose pressed against the window. Recently I wrote this:

There are lots of great mathematicians who read this blog (some of whom I know personally) but never contribute. I think in some cases this is because they set themselves very high standards for what a comment should be, so they hardly ever make one. That’s a shame, because then the rest of us don’t get a chance to hear their thoughts.

John replied:

For a long time I’ve tried to encourage these people to post to the $n$-Café by posting hasty, ill-thought-out, joke-filled and error-ridden comments. […]

So, while it’s great that most people post really interesting and well-thought-out stuff here, it’s even better when we relax enough to make a few mistakes, as we would in normal conversation.

Some people don’t like the idea of making mistakes in public. If you’re one of them, you can use a pseudonym. Personally, I prefer real names to pseudonyms, but I’d much prefer to see interesting comments under a pseudonym than not to see them at all. (The comment form also asks for your email address. You can use a fake one if you want, but it’s only visible to the Café hosts and administrator anyway.)

Some people might wonder about typesetting. It’s easy: just type in plain text, as you would in an email. If you want to get fancy and incorporate Latex, you can: e.g. see this worked example. But plain text is fine.

Come in, come in!

guest post by Tim Silverman

Introduction—and a Kind of Apology

It is with considerable relief that I finally find myself in a position to make a (hopefully) more substantive contribution here: namely, the second in the series I started here, an alarmingly long time ago. I took the long way around to generate the pretty pictures I wanted, and did a lot of reading and calculation—mostly for articles that I intend to appear after this one! Somehow, it all took a lot longer than I expected—even after taking into account that, as I know all too well, these things tend to take a lot longer than expected.

But we’re here at last! My aim in this series is to give the most elementary discussion of modular curves and modular forms that I can manage at each stage, partly for my own benefit, and partly for the benefit of, well, people like me, I guess. One of the strange (but nice) things about the theory of modular curves and modular forms is that, although it lies at the confluence of many areas of mathematics, and leads directly on to some subjects that are very hard indeed, nevertheless, there is a surprising amount that one can do in it using only quite basic geometry and arithmetic. This aspect of the subject is perhaps sometimes somewhat obscured by all the other aspects, but one of my aims here is to bring it out more clearly.

I’m a little embarrassed that this article, and the next few ones at least, are not really categorical, let alone n-categorical, but if things go according to plan, I will at least end up talking a lot about the sorts of things that people often talk about here, albeit at a more elementary level than usual. In any case, I hope that it will provide some interest and amusement to the n-Café patrons, and ideally also to the crowds of urchins in the street outside with their noses pressed against the café window, not to mention the more elegant ladies and gentlemen who pass by and look in but do not enter.

With that warning out of the way, let me start by summarising where we got to in the first article.

Regular readers of this blog will be familiar with the notions of property, structure, and stuff. Less well-known is an intermediate notion between property and structure called “property-like structure.” This is structure which is essentially unique when it exists, such as the structure of finite products on a category, or the structure of an identity element in a semigroup (making it into a monoid). It is distinguished from a mere property (which is also unique, when it exists/holds) because it need not be preserved by all morphisms: not every functor between categories with products preserves products, and not every semigroup homomorphism between monoids is a monoid homomorphism.

We can also define, by analogy, a similar intermediate notion between structure and stuff, which it is natural to call “structure-like stuff.” But are there any examples?

My student Christopher Walker is groupoidifying Hall algebras. What’s a Hall algebra? We get such an algebra starting from any category with a sufficiently well-behaved concept of ‘short exact sequence’. In this algebra, the product of an object $A$ and an object $B$ is a cleverly weighted sum over all objects $X$ that fit into a short exact sequence

$$0 \to A \to X \to B \to 0$$

The ‘clever weighting’ is neatly explained by groupoidification, as sketched here. And if we pick our category in a nice way, the algebra we get is part of a quantum group!

But the Hall algebra is more than a mere algebra. It’s also a coalgebra! The algebra and coalgebra want to fit together to form a Hopf algebra, and they do, but only after a peculiar sort of struggle. Lately Christopher has been thinking about this, and he’s written a paper:

• Christopher Walker, A Hopf algebra structure on Hall algebras.
  
  Abstract: One problematic feature of Hall algebras is the fact that the standard multiplication and comultiplication maps do not satisfy the bialgebra compatibility condition in the underlying symmetric monoidal category $Vect$. In the past this problem has been resolved by working with a weaker structure called a ‘twisted’ bialgebra. In this paper we solve the problem differently by first switching to a different underlying category $Vect^K$ of vector spaces graded by a group $K$ called the Grothendieck group. We equip this category with a nontrivial braiding which depends on the $K$- grading. With this braiding, we find that the Hall algebra does satisfy the bialgebra condition exactly for the standard multiplication and comultiplication, and will also become a Hopf algebra object.

Comments, anyone? Corrections?

guest post by André Joyal

I have interrupted working in the CatLab last Spring because of some urgent things to do. I hope to return in some near future. There was recently a discussion in the category list about terminology used in the nLab (the “evil” terminology). The discussion drifted and John mentioned his involvement in fighting the ecological disaster. I applaud John’s involvement, but I believe his position is overly pessimistic. But my reply to John was blocked by the moderator of the category list for technical reasons that I do not understand. I am reproducing this reply here.