The n-Category Café

• Allyn Jackson, Comme apellé du néante - as if summoned from the void: the life of Alexandre Grothendieck. Part 1, Part 2.

This newly available document will be interesting to his fans, and also students of n-category theory:

• Alexander Grothendieck, extract of a letter to Ronald Brown dated 09/06/83.

We have seen ($\to$) that 11-dimensional supergravity is a gauge theory of a 3-connection, taking values in a certain Lie 3-algebra, $\mathrm{sugra}_{11}$, which is an extension of the super-Poincaré-1-algebra by a 4-cocycle.

I claimed ($\to$) that non-fake flat $n$-connections with values in an $n$-algebra $g$ are to be interpreted in terms of their curvatures, which are flat $(n+1)$-connections with values in

the $(n+1)$-algebra of inner derivations of $g$, where flatness encodes the $n$-Bianchi identity.

The task is hence to compute $\mathrm{inn}(\mathrm{sugra}_11)$.

That’s straightforward, but pretty hard. I am still hoping to figure out a shortcut, computing $\mathrm{inn}(g)$ directly at the level of FDAs. But I don’t see the pattern yet.

Meanwhile, it might be a good idea to study related but simpler examples. As John has already mentioned ($\to$) it might be easier to look at 10-dimensional supergravity first.

Here I present a discussion of what should be the bosonic part of the 2-connection governing 10-dimensional supergravity. The main point is to understand, from a categorical point of view, the relation

between the curvature 3-form $H$, the Kalb-Ramond 2-form $B$ and the $\mathrm{SO}(32) \oplus \mathrm{SO}(9,1)$-connection $A$, which governs the Green-Schwarz anomaly cancellation ($\to$) (Phys.Lett.B149:117-122,1984, ($\to$)).

This entry is an excuse to start talking about generalizations of cartesian closed categories (CCCs) suitable for quantum computation. In this discussion, let’s focus on symmetric monoidal categories with duals. Unlike CCCs, these don’t let us duplicate or delete information - as Wootters and Zurek put it, you can’t clone a quantum. But, they permit quantum entanglement, since a state of a two-part system needn’t be just a state of each of its two parts.

All this comes from the fact that the tensor product needn’t be cartesian. On the other hand, the presence of duals for objects lets us draw morphisms as diagrams with lots of input wires and output wires, where we can take any input and bend it around to become an output, or vice versa. These diagrams are a generalization of the Feynman diagrams that physicists know and love:

In quantum field theory, such diagrams describe how particles come in and go out… but in quantum computation, they describe how data flows in and flows out! They’re like “quantum flow charts”.

I’ll start by listing some references….

This entry is an excuse for discussing ways to categorify the notion of a cartesian closed category (CCC), so we can see computations in the λ-calculus as 2-morphisms.

To get what I’m talking about, start with the most fun introduction to the λ-calculus you can find:

• Raymond Smullyan, To Mock a Mockingbird, Alfred A. Knopf, New York, 1985.

It begins like this:

A certain enchanted forest is inhabited by talking birds. Given any birds A and B, if you call out the name of B to A, then A will respond by calling out the name of some bird to you; this bird we designate AB. Thus AB is the bird named by A upon hearing the name of B.

Next, take Smullyan’s stories and make movies of the process of computation in the λ-calculus, all in terms of birds.

Luckily, you don’ t need to do this yourself - someone already has:

• David Keenan, To dissect a mockingbird: a graphical notation for the lambda calculus with animated reduction.

If you stare carefully at these movies, and think hard, you’ll see they depict 2-morphisms in a cartesian closed 2-category. So, Keenan is secretly categorifying the concept of CCC, to see computation as a kind of process! That’s something I want to understand better.

I’ll start the discussion with a kind of confession.

This blog entry is an excuse for continuing our discussion of cartesian closed categories (CCCs) and the λ-calculus in a slightly more organized way. Let’s focus on more or less “traditional” aspects here. If you prefer to talk about categorified or quantized versions of CCCs and the λ-calculus, go to the relevant entry and talk there.

To kick off the discussion, let’s start with a sketchy and highly oversimplified history of CCCs and the λ-calculus.

I knew this would happen one day - sooner or later: I would be in need of differentiating a smooth 3-morphisms of a smooth 3-functor.

Now, with realizing non-fake flat 2-transport in terms of 3-functors ($\to$), that day has finally come.

(And I always thought I wouldn’t need that until seriously tackling Chern-Simons theory 3-functorially (cf. $\to$)).

Certainly, somewhere out there is a text which has everything about 1-, 2- and 3-morphisms of 3-functors in it that I would ever need. But there is nothing like doing it yourself. Plus, I need LaTeX templates for these 3-dimensional diagrams for my own use. Last not least, I might want to refer to this in some of my future entries on $n$-transport ($\to$) - which I am sure you are all looking forward to.

So I spent the better part of this evening with drawing 3D-diagrams. Or rather, programming LaTeX such as to draw these diagrams.

This takes a while, and of course I ran out of time before drawing the last diagram, that for 3-morphisms of 3-functors.

But anyway, I thought before going into the weekend I’d share some of the pictures.

Find seven pages of higher-order gauge transformations here:

$\;\;\;\;$ Morphisms of 3-Functors

Physicists are invited to think of everything 2-dimensional in there as an image of internal degrees of freedom of a charged string, and of everything 3-dimensional in there as an image of internal degrees of freedom of a charged membrane.

Comments are welcome. (I hope I don’t have any too obvious blunders in there, since at the end I was a little bit under time pressure.)

My student Mike Stay and I are working on category theory and quantum computation. But, I still have some catching up to do when it comes to category theory and ordinary computation!

• Mark Chu-Carroll, Lambda calculus
• Mark Chu-Carroll, Category theory
• Phil Scott, Some aspects of categories in computer science

The first two appear on Mark’s blog, Good Math, Bad Math.

I’m not going to explain this stuff here. I’m just going to dive in.

The Dijkgraaf-Witten model is a simple sort of topological quantum field theory where the only field is a gauge field, and the gauge group G is finite:

• Robbert Dijkgraaf and Edward Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990), 393.

Martins and Porter have a new paper on how to categorify this model, replacing the group G by a categorical group, or “2-group”.

I wrote about this stuff eleven years ago in week54 of This Week’s Finds - so if you want an elementary intro to these ideas, start there.

Since this is quite a serious blog as blogs go, I thought I’d leaven it with a little bit of fun. Eventually, if it becomes sufficiently fun, I won’t have to do anything else except read and post to this blog - except work now and then.

Some of you have seen my puzzles and already know the answers - but others of you probably haven’t. So, I’ll start with the old ones, and eventually move on to some new ones.

If you ever get completely frustrated, you can find answers for these puzzles on my website. But, it might be more fun to guess the answers here. Indeed, I’m hoping these puzzles can start some interesting conversations, because they touch upon all sorts of mysterious and funny aspects of our world… and my answers may not be right.

Okay:

What was Uncle Sam’s last name?

Lectures on n-Categories and Cohomology
John Baez and Michael Shulman

The goal of these talks was to explain how cohomology and other tools of algebraic topology are seen through the lens of n-category theory. Special topics include nonabelian cohomology, Postnikov towers, the theory of “n-stuff”, and n-categories for n = -1 and -2.

These talks were extremely informal, glossing over the difficulties involved in making certain things precise, just trying to sketch the big picture in an elementary way. It seemed useful to keep this informal tone in the notes. I cover a lot of material that seems hard to find spelled out anywhere, but nothing new here is due to me: anything not already known by experts was invented by James Dolan, Toby Bartels or Mike Shulman (who took notes, fixed lots of mistakes, and wrote the Appendix).

The talks were very informal, and so are these notes. A lengthy appendix clarifies certain puzzles and ventures into deeper waters such as higher topos theory. For readers who want more details, we include an annotated bibliography.

The concept of $n$-curvature of $n$-transport - and the nature of “fake” curvature.

As promised - or threatened - I want to go through all kinds of examples of $n$-transport with trivialization and transition ($\to$).

We already needed some of these examples in the comment sections (e.g. here or here) and they are relevant for most of the stuff that I plan to discuss here. I’d like to have a repository of worked examples.

One of my aims is to, eventually, give a coherent description of the $3$-transport that describes 11D supergravity ($\to$).

Transport - the way it is defined ($\to$) - by itself is an integral - or finite - notion. But most of the familiar examples of $n$-transport are known in their differential formulation only, involving differential forms. Therefore an important first step is to develop a notion of smooth transport, which may be differentiated and re-expressed in terms of differential form data.

The techniques for doing so can nicely be discussed in the context of the archetypical example that shall interest us, namely the $(n=2)$-analog of a principal bundle with connection.

Hence the content of this entry shall be smooth nonabelian fake-flat differential 3-cocycles, classifying principal 2-bundles with fake-flat connection. I shall follow our original discussion ($\to$) but use a recent streamlined formulation of the proofs ($\to$).

In fact, I’d dare to say that you can find here the quickest and most transparent derivation of the full cocycle data of a (fake flat) nonabelian gerbe with connection:

$\;\;\;$ Nonabelian Differential Cocycles.

(The non-fake flat case will be discussed elsewhere.)

Ever since John planted this idea into my brain, I have been thinking about (“parallel”) transport along $n$-paths.

As John already mentioned, part of the purpose of this blog is to have a place for the REALLY-DRY-DISCUSSIONS™ that we enjoy so much.

I guess it’s like with instant coffee. Dry by itself, but with some hot water added one can get pretty excited by it.

So before adding any genuinely new content here, we should allow for a place to coherently continue some discussions we are already having spread out over the blogosphere. Apart from the Klein 2-geometry meta-exercise, which David will take up us soon as he returns from his vacation next month, this involves in particular an exchange of observations concerning the relation between $n$-connections and (super)gravity theories that John and myself are involved in.

All is based on the age-old observation that certain free graded-commutative differential algebras - FDAs for short - constitute a surprisingly efficient tool for reasoning about supergravity.

Dry topic, isn’t it?

Let’s add some hot water.

It turns out that a couple of well-known sophisticated concepts in algebra that keep appearing all over the place in mathematical physics can actually be understood in a unified way as different incarnations of categorifications of elementary algebraic concepts.

For instance $L_\infty$ algebras. They are nothing but (semistrict) categorified Lie algebras.

Or free graded-commutative differential algebras. They are just the Koszul dual of the $L_\infty$ guys, something that follows from general abstract nonsense on operads.

You might argue that you don’t care how much abstract nonsense is equivalent to known structures. If it’s equivalent, why not stick to the familiar concepts?

The point is: only the $n$-category theoretic bird’s eye point of view reveals the message that god has written in large letters all over structures like supergravity. I claim.

For instance, in the Sugra-FDA community people have tried to heuristically understand the apparently unreasonable effectiveness of FDAs in terms of what they call “soft group manifolds”. That’s because the crucial algebraic structures in this business look almost - but not quite - like Maurer-Cartan equations of left-invariant differential forms on Lie groups.

So from this point of view one tries to regard the field content of some supergravity theory as a collection of something like differential forms on something like group manifolds. Except that everything is in a funny way “softer” than for honest groups.

I claim that this is not a useful point of view. The main reason is this:

These collections of fields that physicists usually write down, like the graviton field, the gravitino maybe, some 3-form field, etc., are usually really just local representatives of the true - globally defined - fields they represent. It’s like writing down a 1-form for representing the electromagnetic field. In general this only tells you what is going on in contractible patches of spacetime.

That’s fine. But one needs more. There needs to be a way to glue all these local fields together to well-defined global thing.

In phenomenological physics, one usually gets away with completely ignoring this aspect. There is as yet no robust observable evidence of our surrounding spacetime having nontrivial topology. So who cares?

One should care for two reasons:

1) For practical matters, it might well be - who knows? - that there really are small extra dimensions. If so, one naturally expects these to be compact. Hence most likely they will have nontrivial topology. And the effects that small extra dimensions have on observations at practical energy scales are all entirely due to the global topological effects. The effects of the local physics of small extra dimensions would be obserble only at ever higher energies.

In a word - if you are at all interested in a theory of supergravity that lives in, well, eleven dimensions, you should better not ignore the implications of nontrivial topology of spacetime.

2) The other reason is much better. If it doesn’t work globally even in theory - even if you will never be able to check it experimentally - even if the theory has nothing to do with the real world - it’s bound to be nonsense.

So the question is this: does thinking of the local field content of supergravity as a collection of “differential forms on a soft group manifold” tell you how to lift your theory from local patches to the full thing?

If it does, I don’t see it. I’d say it does not.

Instead, I claim that what is called a “soft group manifold” in supergravity is precisely - in disguise - the data of a local $n$-connection with values in some Lie $n$-algebra.

So, in particular, I should maybe add that the problem I am referring to here goes beyond understanding spinors as sections of spin bundles. We need to understand not sections of ordinary 1-bundles - but of 3-bundles (or 2-gerbes, if you like). And indications are that we need 2-gerbes coming from twisted nonabelian 1-gerbes. So we better get this formalism under control.

As an example, a $\mathrm{Lie}(G)$-valued 1-form on a contractible patch of spacetime would be a local 1-connection with values in a 1-algebra. If you want to see instanton effects in your Yang-Mills theory, you will have to be able to glue two of such guys consistently on overlapping domains of definition.

And this generalizes. From 1-forms to 2-forms to … $n$-forms, taking values in 2-algebras, 3-algebras… $n$-algebras.

And this immediately tells us what’s really going on. The theory for how to turn this into something globally defined has been worked out.

You may or may not believe in what I am saying here. If not, you are lacking the hot water to turn our dry discussion into something thrilling.

But in any case, this is part of the reason why John and I think it is interesting to identify, for instance, the gauge 3-group of supergravity as such, and to understand what it all means. And that’s what we are doing in this discussion.

Below the fold I simply compile some of the already existing parts of the disucssion, taken from the String Coffee Table (I, II) and John’s This Week’s Finds (I, II).

The idea is to reserve the comment section of this entry for further discussion along these lines. Sort of as a sub-forum. So it may happen that we post comments to this thread not today, not tomorrow, but maybe in five weeks, or in half a year. Depending on how things develop. If you are intersted in following the discussion, you might want to subscribe to the $n$-Category Café comment RSS feed. This will alert you automatically when new comments come in.

(I am just saying this because with all these non-dry blogs around I got the impression that people tend to abandon a comment section of an entry just because it is no longer on top of the index page. )

David Corfield, Urs Schreiber and I have decided to join forces and take over the universe - with Jacques Distler providing invaluable technical support.

But we’re just getting started here…so for now, check out this picture of the "Bullet Cluster" - actually two galaxy clusters colliding at 5000 kilometers per second:

You can read more about this in week238 of This Week’s Finds.

The basic idea: on Wednesday August 15th, a press release came out entitled NASA Announces Dark Matter Discovery. It didn’t say what they found. They’re trying to build up suspense for a mysterious press conference on Monday the 21st. But since one of the folks at the teleconference is Maxim Markevitch, who has found dark matter in the Bullet Cluster, I can guess what this press conference will be about - and I’ll tell you! It’s pretty cool.

Welcome to John Baez, David Corfield and Urs Schreiber. Their brand-new blog, The n-Category Café, will focus on that heady interface between Physics, Mathematics and Philosophy.