The n-Category Café

Aha. Thanks for pointing that out. That is a vermiform-appendix of a bibliography entry, in the sense that it was introduced when I was expecting to use the category of smooth spaces, rather than manifolds, to work in. Then I decided to solve my problem in a different way. I’ll have to remove the reference, among other changes, before the paper is ready for a referee.

However, you may be correct.

Do I get the feeling we ought to be learning about iterated integrals?

As far as my expericence goes, Chen’s iterated integrals are the kind of structures that are encoded by smooth functors on strict $n$-categories of paths.

The “classical” iterated integral, involving iterations of just a single 1-form

is the “path ordered exponential” which describes smooth functors from 1-paths to a Lie group $G$.

As one goes to higher functors, more kinds of $p$-forms appear in the integrand.

2-functors from 2-paths into a 2-group come from a path ordered integral of a 1-form on loop space, which is itself an iterated integral, namely

Here $A$ takes values in a Lie algebra that acts on the Lie algebra that $B$ takes values in, and the expression is supposed to indicate the operation where we integrate the pullback of the 2-form $B$ over the loop $\gamma$, while continually parallel transporting it, using the path ordered exponential of the 1-form, to the origin of the loop.

This is discussed here.

There is a nice simple diagrammatic description behind this rather involved-looking iterated integral. It really expresses the limit of having lots of little squares inside the strict 2-group $G_2 = (H \to G)$ like this:

I don’t have a proof for this general statement, but I expect that what is going on is this:

A smooth $n$-functor is entirely defined by its derivatives at all identity $n$-morphisms. These derivatives can typically be expressed in terms of various $p$-forms. The functor’s value on a finite $n$-morphisms then is the result of composing lots of small $n$-morphisms, as indicated in the above diagram, each of which being describeable as the avaluation of some $p$-form.

Therefore, the value of the $n$-functor on a finite $n$-morphism is a an “iterated” sum of products of values of $p$-forms.

It’s like a generalization of the fundamental theorem of calculus from $n=1$ to arbitrary $n$.

Do I get the feeling we ought to be learning about iterated integrals?

Yes, they’re good things. But let me sketch why Chen is famous for two things: iterated integrals and smooth spaces.

Jeff cites Chen glancingly for his work on smooth spaces. Unlike the category of smooth manifolds, the category of smooth spaces has pushouts. This means you can always compose cospans without trouble. Jeff wants to compose cobordisms, which are specially nice cospans of smooth spaces. This would be a snap in the world of smooth spaces.

Chen developed the category of smooth spaces not primarily so it would have all pushouts, but so it would be cartesian closed. This implies that the space of paths in a smooth space is automatically a smooth space. This helped Chen develop the theory of iterated integrals with a minimum of fuss and muss.

I first learned iterated integrals back in the 90’s when I started work on loop quantum gravity. Such integrals play a big role there, and indeed throughout gauge theory, whenever you’re doing calculations with “path-ordered exponentials” - also known as “holonomies”. You can see this clearly in Gambini and Pullin’s book:

• Rodolfo Gambini and Jorge Pullin, Loops, Knots, Gauge Theories, and Quantum Gravity, Cambridge U. Press, Cambridge, 1996.

When Urs and I started working on higher gauge theory, we needed both aspects of Chen’s work! We needed the iterated integrals to compute the holonomy of a 2-connection along a path of paths… and we needed smooth spaces, to make sure the space of paths in a smooth space was again a smooth space!

OTOH funds for the Terrestrial Planet Finder and the Europa mission have been reinstated in June 2006.

Thanks for the info, Nathan and Torbjörn!

When researching this article I had some trouble finding the current state of all the different NASA programs. It was easy to find news articles, but with a constantly shifting situation, it’s hard to tell the latest articles from the outdated articles - and a lot of cancelled or indefinitely postponed NASA programs have webpages that make them look perfectly healthy… until you look at the dates.

Is there an easy place to overview all the programs and their funding history?

Yes, Blake. But if you look further back in time, and to less academic sources, you’ll see the cover story:

Jonathan V. Post, “Star Power for Supersocieties”, Omni, April 1980 (1st popular article to predict giant black hole in the center of Milky Way galaxy; 1st popular discussion of J. Post invention “gravity wave telegraph”).

A Gravity Wave Telegraph has a sender throwing a coded sequence of big asteroids and little asteroids into a black hole. Once we have LISA, if the sender is nearby, we become a receiver.

Is there any earlier citation to graviton SETI?

Massey’s book “Singular Homology Theory” does cubical homology theory. One reason that cubical is good is that the product of two cubes is a cube, so, if I remember correctly, you don’t have to worry about Eilenberg-Zilber type of things.

In the realm of gerbes, Cheeger-Simons groups were defined using cubical complexes, and that makes doing transgression to loop spaces much less hassle. If you do that with a simplicial approach you end up having to decompose the prism of the interval cross an $n$-simplex into $n+1$ $(n+1)$-simplices which is quite nifty but messier.

David wrote:

What can be said about the relationship between the approaches to weak n-categories according to the different shapes one opts for: globular, cubical, simplicial, opetopic, etc.?

Ugh - that’s too big of a question!

Is it that one wants them all to be developed, and shown to be equivalent in some sense, since each finds some area of application in which they are most convenient?

Yes, exactly.

I’ll try to say just a little about this enormous issue.

Right now it seems that for each different category of ‘shapes’ which one might use to develop $n$-categories — globes, simplices, cubes, multisimplices, opetopes, etc. — there is a presheaf category: globular sets, simplicial sets, cubical sets, multisimplicial sets, opetopic sets, etc..

A good question is thus: what properties must a presheaf category have in order to be a suitable foundation for a theory of (weak) $n$-categories?

This was already tackled by Grothendieck back in 1983, in his Pursuing Stacks — see the discussion of ‘modelizers’. But, I don’t know anybody who understands what he did!

More recently, Clemens Berger and Denis-Charles Cisinski have made a lot of progress on this issue in their (not yet available) paper on ‘geometric Reedy categories’. I’ve seen Berger give a couple of excellent talks on this subject.

Warning: finding the class of presheaf categories that allow the development of a theory of weak $n$-categories is not so cut-and-dry as I’m making it sound, because different approaches to $n$-categories make very different use of the ‘shapes’ involved. Berger and Cisinski are investigating just one type of approach, and focusing on $n$-groupoids more than $n$-categories.

But, people believe that every definition of $n$-category should eventually come with a concept of the ‘nerve’ of an $n$-category. This will be a simplicial set, and it should be a simplicial weak $n$-category in the set of Street.

Since simplicial sets are so fundamental in mathematics, the nerve seems like a natural ‘hub’ to go through when translating between different definitions of weak $n$-category. It’s a bit like when you fly in the US using Delta Airlines: every flight seems to go through Atlanta.

But, getting this to work seems to be taking a lot of effort on the part of some very smart people — it may take a decade or two.

Simon wrote:

Massey’s book “Singular Homology Theory” does cubical homology theory.

Yes. Amusingly, this is the book they used in my first course on homology as an undergrad.

One reason that cubical is good is that the product of two cubes is a cube, so, if I remember correctly, you don’t have to worry about Eilenberg–Zilber type of things.

Right. For those who don’t know what ‘Eilenberg–Zilber types of things’ are, let me explain. Suppose you define homology using simplices. Then, to relate the homology groups of a product of spaces to the product of their homology groups, you need to break a product of simplices into a bunch of simplices. This is a combinatorial pain in the butt, which Eilenberg and Zilber figured out how to do.

The problem of breaking the product of two simplices into a bunch of simplices is actually quite fascinating if you let yourself get interested in it. But, kids are usually forced to confront this problem in the middle of trying to do something else: namely, compute the homology of a product of spaces, like $S^n \times S^m$. And then, the Eilenberg–Zilber combinatorics seems like a huge nightmarish digression.

Working with cubes instead of simplices allows us to avoid this, because it’s trivial to break a product of cubes into cubes: the product is a cube. So, Massey wrote a book taking the cubical approach to homology.

As far as I know, this is about the only way in which cubical homology is better than simplicial homology. It’s a great advantage. But, there are more significant advantages to using simplices.

A rather minor one is that you don’t need to mod out by degenerate simplices when doing singular homology — miraculously, you still get the right answer. With cubical homology, you really do need to mod out by degenerate cubes.

But this miracle is sort of misleading: you really should mod out by degenerate simplices, and you need to in fancier situations.

A more significant advantage of simplices is that the category of simplices (including the $-1$-simplex!) is the free monoidal category on a monoid. This is what gives rise to the bar construction, which we use to define the cohomology of groups, Lie algebras, rings, and all other gadgets defined by monads. (If you don’t know about the bar construction, see the stuff around page 18 in my universal algebra notes.)

Luckily, we don’t need to work with just one kind of shape: we can use cubes or simplices, depending on what we’re interested in.