The n-Category Café

Hugo Hadwiger was a Swiss mathematician active in the middle part of the 20th century. An old-fashioned encyclopaedia might follow his name with something like “(fl. 1930–1970)”. Here “fl.” means “flourished”. I love that usage. (Are you flourishing right now?) Here’s a photo of Hadwiger, flourishing:

Hadwiger proved—among other things—a really fundamental theorem on the geometry of Euclidean space. Simon and I have both mentioned it on this blog before, but it’s so beautiful that it deserves its own space.

It’s all about ways of measuring the size of subsets of $\mathbb{R}^n$. For example, perimeter and area are both ways of measuring the size of (sufficiently nice) subsets of $\mathbb{R}^2$. Hadwiger’s theorem neatly classifies all such measures of size.

This is the first of a pair of posts. In this one, I’ll take my time explaining Hadwiger’s original theorem. In the next one, I’ll tell you something new.

Currently I am in Paris, visiting Frédéric Paugam. He is in the process of finishing a book with the title

Towards the mathematics of quantum field theory.

This book makes an admirable effort of collecting together for each aspect of QFT the most modern and most elegant mathematical formalism available.

For instance

• all smooth geometry is treated with diffeological spaces and generally smooth sheaves

  • for example a nice reformulation of the theory of distributions and functional calculus without use of topological vector spaces is given: “functorial analysis”;

• smooth infinitesimal and super calculus is thus accordingly treated with the synthetic differential geometry of smooth algebras;

• variational calculus of local action functionals is treated with $\mathcal{D}$-jet space technology following Beilinson-Drinfeld’s book;

• gauge systems are treated in the derived geometry presented by the BV-BRST complex.

And so on.

There’s something about the notion of flat functor that confused me vaguely in the background for a long time. Eventually I tracked it to its source: the same term is used with two slightly different meanings! As a preview of part of my CT2011 talk, today I’ll explain what those meanings are, and how a mutual generalization of them gives us a better notion of “morphism of sites”.

There’s been a lot of interest in entropy of late around here. I thought I’d record what I’d found since it’s spread over a few posts.

Entropy, we have seen, can provide a measure of information loss under coarse-graining. From a distribution over the restaurants in a town, if for each restaurant I specify a distribution over the dishes served there, then I can generate a distribution over all instances of restaurant and dish. On the other hand, from such a distribution over all dishes in the restaurants of the town, I can coarse-grain to give a distribution over restaurants. What Tom, John and Tobias show is that a sensible positive real-valued measure of what is lost is equal to the difference between the entropies of each distribution.

Domenico, Jim and myself are in the process of producing a new version of our article on differential characteristic maps. Since this is part of the bigger story of differential cohomology in a cohesive topos we had originally restricted attention to a particular construction. But reactions showed that this made readers tend to miss the impact. So now we have added a brief section with more indications of the applications, that are being described in more detail elsewhere.

And we have rewritten the extended abstract. That I want to hereby bounce off the $n$Café readership. For the latest pdf version and a hyperlinked version of the abstract see behind the link

Cech cocycles for differential characteristic classes.

Here is the

extended Abstract

What is called secondary characteristic classes in Chern-Weil theory is a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology: to bundles and higher gerbes with smooth connection. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and from Lie groups to higher connected covers of Lie groups by smooth $\infty$-groups: by smooth groupal $A_\infty$-spaces. This allows us to study the homotopy fibers of the differential characteristic maps thus obtained and to show how these describe differential obstruction problems. This applies in particular to higher twisted differential spin structures called twisted differential string structures and twisted differential fivebrane structures .

To that end we define for every $L_\infty$-algebra $\mathfrak{g}$ a smooth $\infty$-group $G$ integrating it, and define smooth $G$-principal $\infty$-bundles with connection. For every $L_\infty$-algebra cocycle 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.

When applied to the canonical 3-cocycle of the Lie algebra of a simple and simply connected Lie group $G$ this construction gives a refinement of the secondary first fractional Pontryagin class of $G$-principal bundles to cocycle space. Its homotopy fiber is the 2-groupoid of smooth $\mathrm{String}(G)$-principal 2-bundles with 2-connection, where $\mathrm{String}(G)$ is a smooth 2-group refinement of the topological string group. Its homotopy fibers over non-trivial classes we identify with the 2-groupoid of twisted differential string structures that appears in the Green-Schwarz anomaly cancellation mechanism of heterotic string theory.

Finally, when our construction is applied to the canonical 7-cocycle on the Lie 2-algebra of the String-2-group, it produces a secondary characteristic map for $\mathrm{String}$-principal 2-bundles which refines the second fractional Pontryagin class. Its homotopy fiber is the 6-groupoid of principal 6-bundles with 6-connection over the Fivebrane 6-group . Its homotopy fibers over nontrivial classes are accordingly twisted differential fivebrane structures that have beeen argued to control the anomaly cancellation mechanism in magnetic dual heterotic string theory.

Enriched category theory is a machine of great power. On the front of that machine is a dial:

Turning the dial selects the category in which you’re enriching — or more poetically, selects a branch of mathematics. Turning it to $\{ true,  false \}$ selects order theory. Turning it to $[0, \infty]$ selects metric geometry. Turning it to $Ab$ selects homological algebra. Turning it to $Set$ selects category theory itself. Turning it to $n Cat$ selects $(n + 1)$-category theory. Any definition framed in the full generality of enriched category theory is, therefore, extremely general.

Most of my work over the last few years has been directed towards understanding size. Many mathematical objects come with a canonical ‘size’ invariant: the cardinality of a set, the Euler characteristic of a topological space, the entropy of a probability space. There is a familial resemblance between these invariants, and I’ve been trying to pin it down.

In Cambridge on Tuesday I’ll give a talk about my best effort so far: a definition of the magnitude of an enriched category. This produces, automatically, many size-like invariants: some old and familiar, some half-familiar, and some that are completely unexplored. I thought I’d prepare for the seminar by telling the story here.

On Friday May 20th my student Christopher Walker successfully defended his thesis, which is now available here:

• Christopher Walker, A Categorification of Hall Algebras.

Now he’s looking for a job. If you know anyone who needs a good algebraist who is also a topnotch teacher (he has extensive teaching experience), please let them know about Christopher. Or let him know about them!

But let me tell you what he’s done.

A Hall algebra is a magically efficient way to read off a quantum group… or at least ‘half’ a quantum group… directly from a Dynkin diagram. This trick works for the simply-laced Dynkin diagrams: $A_n, D_n, E_6, E_7$, and the magnificent $E_8$:

We spoke earlier here on the n-Café about a paper in which Christopher Walker elegantly described Hall algebras as Hopf algebras in braided monoidal categories.

In his thesis, Walker categorifies this idea, describing Hall algebras as living inside certain braided monoidal bicategories. This is a very natural thing to do, since Hall algebras are obtained in the first place by a process of decategorification, or more precisely, ‘degroupoidification’. This process turns groupoids into vector spaces.

This means that while a Hall algebra is normally thought of as a vector space equipped with extra structure—multiplication, comultiplication and so on—at a more fundamental level it’s really a groupoid. The trappings of linear algebra removed, we see the bare bones here.

The previous entry on ∞-Dijkgraaf-Witten theory has turned into a more general discussion of $\sigma$-model quantum field theories. I kept posting in the comment section supposedly bite-sized bits of a general exposition as it incrementally appears in the $n$Lab entry sigma-model .

The next installment I will instead post here, as a separate entry. See below the fold. This is about understanding Chas-Sullivan’s “string topology operations” as an example for a $\sigma$-model, as previously discussed. The main conceptual new ingredient is that for this we can no longer restrict attention to spaces of states that are $n$-vector spaces over some field, but need fully fledged $(\infty,n)$-vector spaces over an $\infty$-ring (albeit $n = 1$ will do for the present purpose).

So while the main thrust here is the $\sigma$-model story, the main content of the following is rather more generally a tiny little bit of exposition of the beautiful theory by Ando-Blumberg-Gepner-Hopkins-Rezk of discrete $\infty$-bundles of module spectra.

Some of us recently tried an experiment here: writing a paper on the blog. It was a great success, and now we’re basically done:

• John Baez, Tobias Fritz and Tom Leinster, A characterization of entropy in terms of information loss.

While the discussion leading up to this paper (here, here and here) was intense and erudite, the final result is simple and sweet. If the paper itself doesn’t make that clear to you, maybe this summary will:

• John Baez, A characterization of entropy, Azimuth.

All this concerns entropy for classical systems. But now Tobias and I are trying to generalize our work to quantum systems. They work a bit differently. Let me explain.

In what follows, I won’t assume any knowledge of quantum theory, or physics at all. I’ll try to define all my terms and state the problem we’re facing in a purely mathematical way. It’s pretty simple.