The n-Category Café

guest post by David Roberts

This post is about my forthcoming paper, extracted from chapter 1 of my thesis:

Internal categories, anafunctors and localisations

and is also a bit of a call for examples from $n$-category cafe visitors (see also these MO questions). I am most familiar with topological and Lie groupoids, but many interesting examples come from the world of schemes and algebraic stacks. I would like to know about these, but first I need to explain what I’m looking for. I would also appreciate to have any typos or inaccuracies pointed out. Please note that I haven’t written a final abstract yet, so what is there is just a placeholder!

Recall the notion of internal category. This comes with a natural definition of internal functor, and internal transformation, leading to a 2-category $Cat(S)$ of categories internal to $S$. However there are a number of settings where there are not enough 1-arrows between a pair of objects. One of these is when $S = Grp$, the category of groups. Then categories internal to $Grp$ are algebraic representations of pointed connected homotopy 2-types, but the natural hom-groupoid in $Cat(Grp)$ (yes, it’s a groupoid, as internal categories=internal groupoids here) does not represent the homotopy type of the mapping space.

This is the third and final episode of a little story about the foundations of quantum mechanics.

In the first episode, I reminded you of some basic facts about the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, and the quaternions $\mathbb{H}$.

In the second episode, I told you how Jordan, von Neumann and Wigner classified ‘formally real Jordan algebras’, which can serve as algebras of observables in quantum theory. Apart from the ‘spin factors’ $\mathbb{R}^n \oplus \mathbb{R}$ and the Jordan algebra of $3 \times 3$ self-adjoint octonionic matrices, $\mathrm{h}_3(\mathbb{O})$, these come in three kinds:

• The algebra $\mathrm{h}_n(\mathbb{R})$ of $n \times n$ self-adjoint real matrices with the product $a \circ b = \frac{1}{2}(a b + b a)$.
• The algebra $\mathrm{h}_n(\mathbb{C})$ of $n \times n$ self-adjoint complex matrices with the product $a \circ b = \frac{1}{2}( a b + b a)$.
• The algebra $\mathrm{h}_n(\mathbb{H})$ of $n \times n$ self-adjoint quaternionic matrices with the product $a \circ b = \frac{1}{2}(a b + b a)$.

In every case, even the curious exceptional cases, there is a concept of what it means for an element to be ‘positive’, and the positive elements form a cone. In this episode we’ll explore that further: we’ll meet the Koecher–Vinberg classification of convex homogeneous self-dual cones, and see how it’s really all about state-observable duality.

Spurred by an interest in metric spaces, I also got interested in integral geometry. But until quite recently I had no idea what it was. I’m guessing some of you have no idea either. The aim of this post is to give you a fast, very concrete explanation.

guest post by Tim Silverman

Welcome back!

Last time, I introduced the subgroup of $PSL(2, \mathbb{Z}_N)$ consisting of what I called the “affine transformations” of $X(N)$, of the form $\left(\array{a&b\\0&d}\right)$, which preserve the relation “having the same denominator”; and I also introduced two of its subgroups, the “translations”, of the form $\left(\array{1&b\\0&1}\right)$, which preserve denominators, and the “rescalings” of the form $\left(\array{a&0\\0&d}\right)$—between them, these two groups generate the affine transformations.

And I said that I intended to talk more about the latter group first. So that is what I am going to do.

This is the second part of a little story about the foundations of quantum mechanics.

In the first part, I introduced the heroes of our drama: the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, or the quaternions $\mathbb{H}$. I also mentioned their crazy uncle, who mainly stays locked up in the attic making strange noises: the octonions, $\mathbb{O}$.

When our three heroes were sent down from platonic heaven to tell the world about the algebraic structure of quantum mechanics, they took on human avatars and wrote this paper:

• Pascual Jordan, John von Neumann and Eugene Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1934), 29–64.

That’s what I’ll tell you about this time.

Then, in the final episode, we’ll meet the Koecher–Vinberg classification of convex homogeneous self-dual cones, and see how it’s really all about state-observable duality.

guest post by Zoran Škoda

The first entry of the n-Category Lab wiki appeared on November 28, 2008 02:18:19 marking the creation of the nLab; hence this Sunday morning it is 2 years old! Three days before the birthday its nominal count of pages reached 4500. This may be a proper moment to celebrate its wide usability already at its tender age and even more to invite people to use it more, and if possible to contribute. Like in our earlier update from May, we would like to point to some highlights in $n$Lab. But I somewhat run out of steam to dwell this time on the content and will rather outline some improvements in the content organization of the $n$Lab which may make it more attractive to you.

In my impression, in its first year, the $n$Lab was focused on our daily research needs and central areas of our interest: category theory, including higher, topoi, homotopy theory, topology, sheaves, stacks, simplicial objects, descent, cohomology (including differential), foundations and categorical aspects of physics. I have received signals from some users of $n$Lab that they do not contribute because they “do more concrete things”, say Lie algebras, representation theory, mathematical physics and so on and feel they do not wish to write about categories. But this is a misunderstanding: more stuff in related areas is very welcome and we need contributors telling us the story in nearby areas of algebra, mathematical physics, differential geometry and so on (of course, not that far an area that we can not understand, appreciate and connect to).

I’m writing a paper called “Division algebras and quantum theory”, which is mainly about how quantum theory can be formulated using either the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, or the quaternions $\mathbb{H}$ — and how these three versions are not really separate alternatives (as people often seem to think), but rather three parts of a unified structure.

This is supposed to resolve the old puzzle about why Nature picked $\mathbb{C}$ when it was time for quantum mechanics, while turning up her nose at $\mathbb{R}$ and $\mathbb{H}$. The answer is that she didn’t: she greedily chose all three!

But sitting inside this paper there’s a smaller story about Jordan algebras and the Koecher–Vinberg classification of convex homogeneous self-dual cones. A lot of this story is ‘well-known’, in the peculiar sense that mathematicians use this term, meaning at least ten people think it’s old hat. But it’s still worth telling — and there’s also something slightly less well-known, the concept of state-observable duality, which is sufficiently lofty and philosophical as to deserve consideration on this blog, I hope.

So I’ll tell this story here, in three parts. The first is just a little warmup about normed division algebras. If you’re a faithful reader of This Week’s Finds, you know this stuff. The second is also a warmup, of a slightly more esoteric sort: it’s about an old paper on the foundations of quantum theory written by Jordan, von Neumann and Wigner. And the third will be about the Koecher–Vinberg classification and state-observable duality.

Urs and I recently got into a discussion about the correct definition of “locally constant ∞-sheaf.” In trying to sort it out, I realized that I don’t even know the right definition of a locally constant 1-sheaf, in general! Specifically, there are two definitions of “locally constant sheaf” which are the same when the base space (or, more generally, topos) is locally connected—but otherwise they’re not the same. Can anyone tell me which is the right definition in general, and why?

When dealing with categories of more than one size—that is, belonging to more than one set-theoretic universe—we are sometimes faced with the need to replace some given category by a version of “the same” category defined at a different size level. Usually, it is fairly obvious how to do this, but for theoretical reasons one would like a general construction that always works. One very nice way to perform such an enlargement is described in sections 3.11–3.12 of Kelly’s book, using Day convolution. But he doesn’t mention that in many cases, this general construction actually gives the same result as the naive one.

guest post by Tim Silverman

Well, here we are again, trying to understand modular curves in the simplest possible way. The last couple of times, I displayed lots of pictures, and talked about tilings. This time, I’m going to display better pictures—with colour!—and I’m going to talk about denominators. Denominators? Yes, denominators: those things on the bottom of fractions.

Why? Well, we want to understand a little bit more about how the $N$-gons are patched together—not so much topologically as from as a number-theoretical point of view (though that’s a rather grandiose phrase for some mostly simple arithmetic). I also want to look at some larger groups of transformations than just $\Gamma(N)$. And I’m going to explore this by considering particularly the denominators of the mod-$N$-reduced fractions.

guest post by Tim Silverman

Welcome back to this series of posts: Pretty Pictures of Modular Curves

The Story So Far

The last two times, we started looking at the curves $X(N)$, that is, the quotients of the upper half of the complex plane by the groups $\Gamma(N)$, the latter being defined as those subgroups of $PSL(2, \mathbb{Z})$ which consist of the matrices congruent to $\left(\array{1&0\\0&1}\right)$ mod $N$. We discovered that for $N=3$, $N=4$ and $N=5$, the resulting quotients (to be strictly accurate: once we have compactified by cusps) can basically be thought of as (the surfaces of) Platonic solids, respectively the tetrahedron, cube and dodecahedron, perhaps best thought of as the spherical versions of those solids. These are curved surfaces tiled regularly with a finite number of regular $N$-gons, with $3$ of the tiles meeting at each vertex. This view in terms of tiling also works for the $N=2$ case, which involves a tiling by $3$ bigons. The residual action of $PSL(2, \mathbb{Z})$ on these quotient surfaces is then precisely the symmetry of the tiling.

In a series of posts I want to give a flavour of the idea, well known to experts, that integral transforms, given in terms of kernels, can be viewed from a pull-push perspective, and to tie this in to the the idea of enriched profunctors as transforms between presheaf categories. I would like to discuss Fourier transforms, the Legendre transform, the Radon transform, Fourier-Mukai transforms and many other things; I’m not sure how far I will get, however, as I want to try to do this quite gently.

These sorts of things have been discussed here at the Café on various occasions.

This time I will discuss the Fourier transform, or rather Fourier series, in terms of a pull-push operator. Next time I will talk about the composition of kernels and the relevance of the Beck-Chevalley Condition and the Projection Formula (also known as the Frobenius Identity).

The main point this time is the notion of the push-forward of a function. (Category theorists who don’t know the term ‘push-forward’ might find it useful to think of Kan extensions.) I don’t know enough analysis to be able to give the general setting of push-forwards in this situation here, but it is only necessary, for the Fourier transform, to understand push-forwards along projections.

The previous entry mentioned that Chern-Weil theory exists in every cohesive $\infty$-topos $\mathbf{H}$. For $\mathbf{H} = \infty LieGrpd$ the topos of smooth $\infty$-groupoids, this reproduces ordinary Chern-Weil theory – and generalizes it from smooth principal bundles over Lie groups to principal $\infty$-bundles over Lie $\infty$-groups.

Some basics of this $\infty$-Chern-Weil theory in the smooth context we have been trying to write up a bit more. Presently the result is this

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff,

  Cocycles for differential characteristic classes - An $\infty$-Lie theoretic construction

  (pdf)

Abstract We define for every $L_\infty$-algebra $\mathfrak{g}$ a smooth $\infty$-group $G$ integrating it, and define $G$-principal $\infty$-bundles with connection. For every $L_\infty$-algebra coycle of suitable degree we give a refined $\infty$-Chern-Weil homomorphism that sends these $\infty$-bundles to classes in differential cohomology that lift the corresponding curvature characteristic classes.

As a first example we show that applied to the canonical 3-cocycle on a semisimple Lie algebra $\mathfrak{g}$, this construction reproduces the Cech-Deligne cocycle representative for the first differential Pontryagin class that was found by Brylinski-MacLaughlin. If its class vanishes there is a lift to a $\mathrm{String}(G)$-connection on a smooth String-2-group principal bundle. As a second example we describe the higher Chern-Weil-homomorphism applied to this String-bundle which is induced by the canonical degree 7 cocycle on $\mathfrak{g}$. This yields a differential refinement of the fractional second Pontryagin class which is not seen by the ordinary Chern-Weil homomorphism. We end by indicating how this serves to define differential String-structures.

A cohesive $\infty$-topos is a big $\infty$-topos $\mathbf{H}$ that provides a context of generalized spaces in which higher/derived geometry makes sense.

It is an $\infty$-topos whose global section $\infty$-geometric morphism $(Disc \dashv \Gamma): \mathbf{H} \to \infty Grpd$ admits a further left adjoint $\Pi$ and a further right adjoint $CoDisc$:

$$(\Pi \dashv Disc \dashv \Gamma \dashv Codisc) : \mathbf{H} \to \infty Grpd$$

with $Disc$ and $Codisc$ both full and faithful and such that $\Pi$ moreover preserves finite products.

The existence of such a quadruple of adjoint $\infty$-functors alone implies a rich internal higher geometry in $\mathbf{H}$ that comes with its internal notion of Galois theory , Lie theory , differential cohomology and Chern-Weil theory .

In order of appearance:

• $\infty$-Groups

• Cohomology

• Concordance

• Homotopy and Galois theory

• van Kampen theorem

• Paths

• Universal coverings and Whitehead towers

• Flat differential cohomology and local systems

• de Rham cohomology

• Concrete objects: Cohesive $\infty$-groupoids

• Infinitesimal objects: $\infty$-Lie algebroids

• Lie theory

• Maurer-Cartan forms and curvature characteristic forms

• Differential cohomology

• Chern-Weil theory

• Chern-Simons theory

So cohesive $\infty$-toposes should be a fairly good axiomatization of higher/derived geometry in big $\infty$-toposes. There is also an axiomatization of higher/derived geometry on little $\infty$-toposes, called structured $\infty$-toposes.

There ought to be a good way to connect the big and the little perspective on higher/derived geometry. For instance given a cohesive $\infty$-topos $\mathbf{H}$, one would hope that there naturally is associated with it a geometry (for structured $\infty$-toposes) $\mathcal{G}$ such that for every concrete object $X \in \mathbf{H}$ the over-$\infty$-topos $\mathbf{H}/X$ is naturally a little $\mathcal{G}$-structured $\infty$-topos.

I have some ideas on this, but am not really sure yet.

I have a project afoot to gain some historical understanding on the rise of appreciation for mathematical duality. Something I need to know more about is the process whereby the duality involved in Fourier analysis came to be seen as arising through a pairing of a space with its dual, and so allowing comparison to other such dualities.

This has got me thinking once again about transforms, something we’ve discussed many times at the Café before. In as general a setting as possible, we could take some pairing

$$A \times B \to C,$$

and then if we have a map $g$ from $C$ to a rig $D$, we can transform certain functions from $A$ to $D$ to ones from $B$ to $D$. We would do this by forming the $D$-sum over $a$ of $D$-products of $g((a, b))$ and $f(a)$.