The n-Category Café

Christmas is a time for giving, but it is also a time for topos theory. (At least, it is in my own private tradition; I don’t know about you.) Combining the two, I give you:

An informal introduction to topos theory

This came out of some impromptu talks I gave to a bunch of category theorists earlier in the year. In odd moments since then I’ve been typing up notes. Voilà!

I’ll be glad to hear your comments.

I’ll also be glad if anyone can suggest somewhere this might be publishable. While ideally I’d like to get it into a journal, I suspect it will probably just end up on the arXiv, because it’s purely expository and doesn’t really adopt a novel point of view. Its only possible claim to novelty is that it goes lightly over quite a lot of ground in not many pages. I had a look at the question about expository articles at MathOverflow, but no journal listed there seemed clearly appropriate. Any ideas?

guest post by Tim Silverman

Merry Xmas, and Happy Holidays, to all of you, but particularly those of you who, like me, are enduring the long, cold, dark evenings of a northern winter.

Well, here we are again, on our little outing into the world of modular curves. Last time, we looked at a lot of pictures of, or related to, the curves $X_1(N)$. These are the quotients of the complex upper half plane by the congruence subgroup of $\Gamma$ consisting of matrices of the form $\left(\array{1&b\\0&1}\right)$ mod $N$ for arbitary $b$. But only for $N\lt 6$.

So this time, we’ll look at the cases $N=6$ and above.

Earlier this year I joined the Boyd Orr Centre for Population and Ecosystem Health, at the kind invitation of my friend Richard Reeve in biology. ‘But Tom,’ I hear you say, ‘what do you know about population or ecosystem health?’ Fair question. But the Boyd Orr people are marvellously welcoming, and many of us share an interest in the quantification of biodiversity, so there I am.

I hope to tell you some time about my work on diversity with Christina Cobbold. Right now I want to talk about the incredibly interdisciplinary nature of the Boyd Orr Centre, and how that makes a mathematician feel.

Every year, the journal Environmental Microbiology publishes a choice selection of quotes from referees’ reports. This year’s is out now! The journal article is here, but since you can get the article for free if you register, I hope they won’t mind me putting another copy here.

What I especially appreciate are the passages that describe how effectively refereeing can crush your spirit:

The writing and data presentation are so bad that I had to leave work and go home early and then spend time to wonder what life is about.

I suppose that I should be happy that I don’t have to spend a lot of time reviewing this dreadful paper; however I am depressed that people are performing such bad science.

The biggest problem with this manuscript, which has nearly sucked the will to live out of me, is the terrible writing style.

The rest of my favourites follow…

Vi Hart has a great video called Doodling in Math Class: Snakes + Graphs. She has others, too. Some people are watching her stuff and enjoying math for the first time. Something about the hyperactive, slightly cynical delivery combined with pictures drawn in real-time reaches certain folks who are completely immune to the charms of Martin Gardner.

On somewhat short notice, for those in the vicinity of central Europe:

This Monday we have the first of a series of seminars:

• Quarterly Seminar on Topology and Geometry

  December 20, 2010,

  Univ. Utrecht, Netherlands

  (seminar website)

With the following talks:

• Hessel Posthuma (Univ. Amsterdam)

  Integrable hierarchies and Frobenius manifolds

  Abstract: I will explain the construction of an integrable hierarchy out of a Frobenius manifold, given in terms of a partition function of a cohomological field theory. In the homogeneous case, these hierarchies coincide with the ones constructed by Dubrovin and Zhang. Finally I will explain how one can use the action of the Givental group to deduce properties of these hierachies from those of the KdV equation when the underlying Frobenius manifold is semisimple. This is joint work with A. Buryak and S. Shadrin.

• Julie Bergner (UC Riverside)

  Generalized classifying space constructions

  Abstract: The process of constructing a classifying space for a group can be generalized so that we can find classifying spaces for categories. This construction is useful for many purposes, but also can be considered to lose much information if we use classical approaches in algebraic topology. In this talk, we give two approaches to strengthening the classifying space construction: one by changing how we think about spaces, and the other by changing the construction.

• Christian Blohmann (MPI Bonn)

  Homotopy equivalence of correspondences and anafunctors of higher groupoids

  Abstract: An anafunctor (also called span or zigzag) in the category of higher groupoids can be viewed as local trivialization of a principal bundle on a hypercover of a higher stack. Correspondences, on the other hand, are the higher generalization of groupoid bibundles. In the setting of ordinary categories the notions of anafunctors and bibundles are equivalent, both providing a localization of the category of groupoids at Morita equivalences. I will report on the generalization of this result to the setting of quasi-categories.

More details are on the seminar website. If you’d like to attend but need more info, please contact me.

In a mocking, ironic nod to the concept of “open access”, Springer Verlag has made all papers in their journal Logica Universalis freely accessible from today until December 31st, 2010.

So, snatch as many papers you can before the great iron gate crashes closed again!

For example, if you’re curious about the interactions between logic and category theory, now is your chance to read this:

• Peter Arndt, Rodrigo de Alvarenga Freire, Odilon Otavio Luciano and Hugo Luiz Mariano, A global glance on categories in logic.

Many readers here will nod knowingly at the last sentence of this paper, which justifies some generalizations the authors have engaged in:

Second, on a more abstract level, it is a highly successful mathematical practice to admit pathological objects in a category in order to make (the global properties of) the category itself less pathological — the passage from manifolds to $C^\infty$-schemes in Differential Geometry illustrates well this point, as does the functorial approach to algebraic geometry, where one passes by the Yoneda embedding from schemes into a category of functors where most objects have no geometric appeal at all.

Last week I participated in the inaugural conference of the Association for the Philosophy of Mathematical Practice in Brussels. I decided to sketch some of my thinking on coalgebra, that Café people helped me formulate back here and here. This has turned into an article which will appear next year in Studies in History and Philosophy of Science.

Given twenty-five minutes to speak, there was only time to gesture at the connections with computer science, algebraic set theory, and analysis. It’s very tricky choosing a rich and interesting case study which is philosophically salient. To encourage the reader or listener to follow up the mathematics to understand what you’re saying, there must be a decent pay-off. An intricate twentieth century case study had better pack plenty of meta-mathematical punch. The trouble is that mathematics has become so enormously interconnected that when you pull on one strand, it’s easy to find yourself dragging along the whole edifice.

• Workshop and School on Higher Gauge Theory, TQFT and Quantum Gravity, Lisbon, February 10th-13th, 2011 (Workshop), February 7th-13th, 2011 (School). Organized by Roger Picken, João Faria Martins, Aleksandr Mikovic and Jeffrey Morton.

guest post by Tim Silverman

Welcome once more to our exploration of the structure of modular curves. So far, we’ve looked at the curves $X(N)$—that is, quotients of the complex upper half plane by the groups $\Gamma(N)$ for various $N$. We’ve seen that we can tile them with regular $N$-gons, three $N$-gons meeting at each vertex, and each labelled with a fraction (reduced mod $N$) at its centre; and that we can break this tiling up into several identical pieces, corresponding with the units in $\mathbb{Z}_N$ mod $\{1, -1\}$, which I’ve called the “projective” group of units. A unit $u$ corresponds to a matrix $\left(\array{v&0\\0&u}\right)$ (mod $N$) where $u v=1$, which acts on the fractions mod $N$ that lie at the centres of the $N$-gons, thus permuting the $N$-gons while preserving the tiling, and, in particular, carrying the piece centred on $\frac{1}{0}$ to the piece centred on $\frac{v}{0}$. These units, acting by multiplication, may be considered as the rescalings of the “projective line” over $\mathbb{Z}_N$.

Last time we classified irreducible unitary group representations into three kinds: real, complex and quaternionic. But what does this mean for physics?

Well, since elementary particles are often described using representations like this, particles must come in three kinds: real, complex and quaternionic!

Of course the details depend not just on the particle itself, but on the group of symmetries we consider. But still, it sounds pretty far-out. What sort of particle is quaternionic?

This time we’ll look at the simplest example: an electron, regarded as a representation of $SU(2)$. People usually describe its state with a pair of complex numbers. But in fact, it makes a lot of sense to use a single quaternion!

We’ll see why in a while. But first, in case you fell asleep last time, let me remind you what we proved — we need it now. There are three choices for an irreducible unitary representation of a group $G$ on a complex Hilbert space $H$:

1. Our representation may not be isomorphic to its dual, in which case we call it truly complex.
   
   
2. It may be isomorphic to its dual thanks to an invariant antiunitary operator $J: H \to H$ with $$J^2 = 1.$$ In this case we call it real, because it’s the complexification of a representation on a real Hilbert space. And in this case there’s an invariant nondegenerate bilinear form $g : H \times H \to \mathbb{C}$ with $$g(v,w) = g(w,v) ,$$ also known as an orthogonal structure on $H$.
   
   
3. It may be isomorphic to its dual thanks to an invariant antiunitary operator $J: H \to H$ with $$J^2 = -1 .$$ In this case we call it quaternionic, because it comes from a representation on a quaternionic Hilbert space. In this case there’s an invariant nondegenerate bilinear form $g : H \times H \to \mathbb{C}$ with $$g(v,w) = -g(w,v) ,$$ also known as a symplectic structure on $H$.

This is the three-fold way.

Jam de longe, Cauchy pruvis ke kontinua funkcio $f(x)$ kiu verigas la funkcian ekvacion $$f(x + y) = f(x) + f(y)$$ kiuj ajn estu la nombroj $x, y$, necese estas homogena, unuagrada funkcio $f(x) \equiv A x$.

• Robert Darnton, The library: three jeremiads, The New York Review of Books, December, 2010.

It seems that only in the last 7 years or so have people working on the foundations of quantum theory started seriously talking to computer scientists using the language of categories. A lot of this was due to the burst of excitement about quantum computers and other forms of quantum information processing… but the Oxford group led by Samson Abramsky and Bob Coecke also played a crucial role.

Now Bob Coecke and Jamie Vicary have started a mailing list on “quantum foundations”.

guest post by Tim Silverman

And here I am again, back with more of “A Child’s Garden of Modular Curves”.

Where We’ve Got to and Where We’re Going Next

We’ve been looking at the modular curves $X(N)$ by way of their tilings by regular $N$-gons, each $N$-gon being labelled with one of the “fractions reduced mod $N$”. So far, we’ve been trying to understand their structure by looking at the action of the subgroup of $PSL(2, \mathbb{Z}_N)$ consisting of matrices of the form $\left(\array{a&0\\0&d}\right)$ (mod multiplication by $\{1,-1\}$) where $a$ and $d$ are units with $a d=1$. This is obviously isomorphic to the group of units of $\mathbb{Z}_N$ mod change of sign: ${\mathbb{Z}_N}^*/\{1, -1\}$. (I think I’ll call this latter the projective group of units for short.)

In the post before last (which stood for a while as a lone promontory in an almost postless desert), we divided up some tilings using colour to distinguish denominators. In the last post (which, by contrast, hid like a small woodland flower in a luxuriant forest of other posts), we saw how the projective group of units acts on the coloured tilings, but only for prime $N$. You might want to glance back at that to refresh yourself with colour before continuing with the present comparatively dry post.

This time, we will continue looking at the action of the projective group of units, but extending our examination to composite $N$.

Last time I described some problems with real and quaternionic quantum theory — or at least, ways in which they’re peculiar compared to good old complex version of this theory.

This time I’ll tell you about the three-fold way, and you’ll begin to see how real and quaternionic Hilbert spaces are lurking in complex quantum theory.

The name ‘three-fold way’ goes back to Dyson:

• Freeman Dyson, The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, Jour. Math. Phys. 3 (1962), 1199–1215.

But the idea goes back much further, to a paper by Frobenius and Schur:

• F. G. Frobenius and I. Schur, Über die reellen Darstellungen der endlichen Gruppen, Sitzungsber. Akad. Preuss. Wiss. (1906), 186–208.

I’ll admit I haven’t read this paper, so I’m not quite sure what they did, but everyone cites this and mentions the ‘Frobenius–Schur indicator’ when discussing the fact that irreducible group representations come in three kinds.

And that’s what I’ll explain now. As you’ll see, the trinity of ‘real’, ‘complex’ and ‘quaternionic’ goes hand-in-hand with another famous trinity: ‘orthogonal’, ‘unitary’ and ‘symplectic’!

It’s a wonderful fact that nature is described using complex Hilbert spaces. We can take a beam of electrons and split it. If we do it right, each electron goes both ways! Then we can insert a tightly wound coil of wire between the two beams. By running some current through this wire, we can make a magnetic field that’s mostly trapped inside the coil. By this method, we can multiply the part of the electron taking one route by $i$, as compared to the part that takes the other route. And we can check that this is true by studying the interference patterns that appear as the beams recombine! Indeed we can do this for any complex number on the unit circle, say $exp(i \theta)$.

But what’s so great about the complex numbers? You can set up a theory of Hilbert spaces based on any normed division algebra. And as you’re undoubtedly sick of hearing, there are three choices: the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, and the quaternions $\mathbb{H}$. So mathematically, at least, there are three possible kinds of quantum mechanics!

Only three? There could be more, but Solèr’s theorem picks out these three from among a vast set of alternatives, based on some simple axioms about how infinite-dimensional Hilbert spaces should work.

What about the finite-dimensional case? The Jordan–von Neumann–Wigner theorem classifies the possibilities in an approach based on algebras of observables. The Koecher–Vinberg theorem starts from seemingly different assumptions, but leads to the exact same conclusions. Both these theorems leave room for some exotic possibilities involving octonions and spin factors — but the overall message of all these results seems to be: real, complex and quaternionic quantum mechanics are equally good.

However, for some reason — or perhaps no good reason — nature is best described by complex quantum mechanics. We can take an electron and multiply it by $i$, so real quantum mechanics is out. But we can’t multiply it by $j$ or $k$ — or at least that’s what everyone says. So quaternionic quantum mechanics is out too, apparently.

This has led people to look for mathematical ways in which complex quantum mechanics is ‘better’ than the real or quaternionic theories. Lucien Hardy proved a nice result along these lines:

• Lucien Hardy, Quantum theory from five reasonable axioms.

But I want to tell you about two others.

I need to be preparing tomorrow’s session of our seminar on derived differential geometry, but I can’t concentrate on this with all the puzzle discussion here. So I thought I’d counteract this by forcing some Hochschild discussion on you all. ;-)

At $n$Lab: Hochschild cohomology I am preparing some notes. It starts with a very-general-abstract definition. But then in the Examples-section I have a very-specific-concrete discussion – supposed to be expositional – of how to understand the ordinary Hochschild complex of a commutative associative algebra as being the $\infty$-function algebra on the corresponding categorical loop space object.

While I keep working on this, I’d be grateful for any comments.

James Dolan gave me another puzzle today.

This one is a bit more sophisticated. Can you find a really nice solution? I’m also curious to hear how well-known this fact is. (Neither of us know a reference.)

One nice thing about category theory is that despite its soaringly ambitious nature, it still contains hundreds of satisfying little puzzles to entertain the problem-solver in us. James Dolan likes to give me these puzzles and see how long it takes me to solve them. While I find it a bit distressing to be put on the spot like that, they’re still fun.

Here’s the one he gave me yesterday. See how long it takes you.

David Corfield likes theorems that say what’s special about the real numbers — especially theorems where $\mathbb{R}$ emerges unexpectedly at the end, like a rabbit from a magician’s hat. He enjoys guessing how the rabbit got into the hat!

So, his ears perked up when I mentioned my favorite theorem of this type: Solèr’s Theorem. In 1995, Maria Pia Solèr proved a result with powerful implications for the foundations of quantum mechanics. She starts with what sounds like a vast generalization of the concept of infinite-dimensional Hilbert space: a generalization that replaces the complex numbers with an arbitrary ‘division $\ast$-ring’. But then, she shocks us by showing that this ring must be one of these three:

• the real numbers $\mathbb{R}$,
• the complex number $\mathbb{C}$,
• the quaternions $\mathbb{H}$!

These are our old friends, the famous trio: the associative normed division algebras!

Let me tell you what Solèr’s Theorem actually says. I’ll do little more than give the necessary definitions and state the result. I won’t say anything about the proof, and only a bit about the implications for quantum theory. For that, you can read this wonderful paper:

• Samuel S. Holland Jr., Orthomodularity in infinite dimensions; a theorem of M. Solèr, Bull. Amer. Math. Soc. 32 (1995), 205–234. Also available as arXiv:math/9504224.

and then:

• Maria Pia Solèr, Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra 23 (1995), 219–243.