The n-Category Café

Have fun in Split! I hope your high-school talk gets all those kids excited about $n$-categories!

Regarding this talk: when introducing people to the Eckmann–Hilton argument, I like to emphasize that this argument is really how we prove to kids that addition is commutative. Take 3 rocks and set them on the table next to 2 rocks:

$$\bullet \bullet \bullet \quad \bullet \bullet$$

This is $3+2$ rocks, by definition.

Now, slide the 3 rocks around the 2 rocks, so they trade places:

$$\bullet \bullet \quad \bullet \bullet \bullet$$

Now we have $2+3$ rocks, by definition. We have the same number of rocks — more precisely, the process of sliding the rocks around is an isomorphism of sets. Therefore, we have an equation

$$3 + 2 = 2 + 3$$

or more precisely, a braiding isomorphism

$$B_{3,2} : 3 + 2 \stackrel{\sim}{\to} 2 + 3$$

The fun thing is that with the Eckmann–Hilton argument, this proof becomes completely rigorous! The process of carrying out the proof becomes a braid. And, we see why we need at least 2 dimensions of space to carry out this argument!

If you’ve ever read Flatland, you may remember the scene in ‘Lineland’, where space is 1-dimensional and everyone is stuck talking to their two neighbors: their neighbor to the left and their neighbor to the right. There’s not enough room for them to trade places. For such people, the commutative law might seem counterintuitive — something only high-powered mathematicians could understand.

I have time to say more about the way to look at integration of Lie $n$-algebras which I mentioned. The following gives a couple of more details concerning the remark I made in the above entry and then ends with a question.

I am hoping that with the right way to think of the Lie $n$-group $G_{(n)}$ integrating a given Lie $n$-algebra $g_{(n)}$ the statement of local parallel transport which establishes an isomorphism between smooth functors

$$\mathrm{tra} : \Pi_n(X) \to \Sigma G_{(n)}$$

and certain differential form data on $X$ becomes a tautological statement.

So let $g_{(n)}$ be a given Lie $n$-algebra regarded as a one-object Lie $n$-algebroid.

Let $I^n$ be the $n$-dimensional standard cube.

As I argued above, I think we should think of the smooth space of smooth $n$-paths in the smooth space “$B G_{(n)}$” as the presheaf on manifolds which I denote

$$P_n(B G_{(n)})$$

and which is defined to send each smooth manifold $U$ to the set of smooth Lie $n$-algebroid maps from $T (U \times I^n)$ to $g_{(n)}$:

$$P_n(B G_{(n)}) : U \mapsto \mathrm{Hom}_{n\mathrm{Lie}}(T (U \times I^n), g_{(n)}) \,.$$

By the usual Yoga, this is nothing but the collection of differential form data $(A,B,C,\cdots)$ on $U \times I^n$ which models the differential graded commutative algebra Koszul dual to $g_{(n)}$.

For instance, for $g_{(n)} = g_{(1)} = g$ an ordinary Lie algebra, we simply have

$$P_1(B G_{(1)}) : U \mapsto \{ A \in \Omega^1(U\times I, g), F_A = 0 \} \,.$$

Now, a smooth local $n$-transport on some smooth space $X$ with values in $G_{(n)}$ is in particular a smooth map from the space of $n$-paths $P_n(X)$ of $X$ to $P_n(B G_{(n)})$.

Now, the smooth space of $n$-paths on $X$ is simply the sheaf

$$P_n(X) : U \mapsto \mathrm{Hom}_{\mathrm{smooth}}(U \times I^n, X) \,.$$

Hence a smooth parallel $n$-transport

$$\mathrm{tra} : P_n(X) \to P_n(B G_{(n)})$$

is in particular a morphism of sheaves:

$$U \mapsto ( \mathrm{tra}_U : \mathrm{Hom}_{\mathrm{smooth}}(U \times I^n, X) \to \mathrm{Hom}_{n\mathrm{Lie}}(T (U \times I^n), g_{(n)}) ) \,.$$

The question is hence how to characterize all such morphisms of sheaves.

It is clear that a large class of such morphisms comes from postcomposition with a fixed morphism of Lie $n$-algebroids from the tangent Lie algebroid of $X$

$$f : T X \to g_{(n)} \,.$$

That’s good. Such a morphism is nothing but the corresponding differential form data globally on $X$. For instance for our $n=1$ example such morphisms are in bijection with flat $g$-valued 1-forms on $X$.

I want to obtain a statement that this already exhausts all possible morphisms $$P_n(X) \to P_n(B G_{(n)}) \,.$$

Currently I am notr sure if all morphisms of the sheaves on manifolds which I have here need to compe from morphisms $f$ as above.

On the other hand: those coming from the maps $f$ as above are precisely those which fit into the philosophy of Chen-smooth structures:

morphisms of Chen-smooth spaces (which are presheaves with special properties) are required by definition to come from postcomposition with a map.

In fact, I could replace equivalently the set of smooth maps of smooth manifolds

$$\mathrm{Hom}_{\mathrm{smooth}}(U \times I^n, X)$$

by the canonically isomorphic set of morphisms of the corresponding tangent Lie algebroids

$$\mathrm{Hom}_{n\mathrm{Lie}}(T(U \times I^n), T X) \,.$$

Maybe then I want to pass to the obvious modification of Chen-smooth spaces where all occurences of ordinary manifolds and their smooth maps are replaced by the notion of their tangent Lie algebroids and their morphisms.

Doing so would tend to make the statement I am after a complete tautology. Which would be good. That’s what I am hoping it will be in the end. But there seems to be room for further tuning the way to say this best.

Q1. Your field of interest in mathematics is the relation between categories and quantum physics. During the Split categories conference you will be giving a popular lecture on this topic to the high school students, so can you briefly summarize that lecture for our readers?

We are all familiar with processes that take place in time. A bus takes us from here to there. The earth travels around the sun over the year. An electron travels through the vacuum tube of our TV set.

Such processes take place one after each other, forming a long chain of events. For a long time, physics was the study of such chains of processes occurring in nature.

Almost 300 years ago it was discovered that the right language to describe such processes are mathematical formulas. By cleverly manipulating strings of symbols, people were able to understand the causality of fundamental processes to an unprecedented accuracy.

We can compute to many decimal places the precise position of the earth in its orbit around our sun many billions of years ahead. And all of this just by manipulating strings of symbols. This success of the description of physical laws by mathematical formulas has led to a deep interaction and interrelation between mathematics and physics.

But remarkably, in the middle of the 20th century, physicists and mathematicians independently began trying to explore the possibility that linear chains of entities may not be the end of the story.

Physicists began speculating that the elementary particles, like the electrons in our TV sets, which appear to be just points that trace out 1-dimensional curves as they zip along, might maybe turn out to look like little loops when we look very, very closely. Such loops would trace out not 1-dimensional paths in space, but 2-dimensional surfaces.

Simple as it sounds, this speculation has led theoretical physicists to discover a whole new universe of ideas. And some of these ideas turned out to have no good description in terms of the kind of mathematics that has worked so well in physics for centuries.

But by a lucky coincidence of history, mathematicians had – completely independently – began formulating a kind of mathematical language which is to the one-dimensional formulas that we are taught in school like a 2-dimensional surface is to a line. Or like a 3-dimensional space. Or like something even higher dimensional.

The name of this language is category theory. Or $n$-category theory, if one wants to emphasize its $n$-dimensional nature.

As with the algebra which turned out to be so very useful for describing the movement of the planets, this piece of higher dimensional mathematics is very rich and interesting in its own right. But on top of that, it miraculously turned out to be precisely the right language to naturally describe the new ideas in theoretical physics.

And turns out. The study of interrelation between categories and $n$-categories with physics of point particles and their speculated loop-like generalizations is in its infancy.

As with all great and deep ideas, that of $n$-categories and the physics it describes has underlying it a couple of very beautiful and very simple ideas, understandable by every layman.

In my public talk I want to highlight some of these simple beautiful ideas in a way that requires no mathematical or physical education. Using simple but careful everyday reasoning, we will try to get a good conceptual understanding of what characterizes processes in nature and in mathematics, and what happens when we start passing from 1-dimensional chains of symbols for describing these to 2-dimensional pictures.

Everybody who knows that $3 + 2$ equals $2 + 3$ will thereby be led to discover a simple but profound result in $n$-category theory – the so-called Eckmann-Hilton argument.

If time permits, I might be tempted to end the talk with entertaining the audience by making some remarks on how this is relevant for some of the modern ideas which pervaded theoretical high energy physics in the last couple of decades.

Q2. You are also one of the admins at $n$-category cafe blog. As it has become an interesting and influential internet based communication and discussion tool for researchers in category theory, can you tell me a bit more about it - how did the idea start, how it developed, and what is its influence on the category maths community professional work?

Before the $n$-category Café came into existence, the three hosts, John Baez, David Corfield and myself, had shared the interest in $n$-categories and their role in math, physics and philosophy, and the interest in sharing and discussing these issues on the internet with people from all around the world – but what we didn’t share was a single site to suit those needs.

John Baez, who is not only the urblogger of both math and physics (or was he the überblogger?), but also the one who initiated thinking about applications of higher category theory to physics, had his own web-based column This week’s finds in mathematical physics since 1993. David Corfield and myself a while ago started using two blogs to communicate on the web (now dormant: Philosophy of Real Mathematics and The String Coffee Table, respectively) when we thought that it might make all of us happier if we had a single blog unifying these activities.

Jacques Distler kindly offered to provide the required technology and since then the $n$-category café is running on his equipment.

That’s how it started. More details on this can be found in Two Café Owners Interviewed.

I am rather pleased with how the $n$-category café developed. For me, it has become a primary source for intellectual and scientific exchange.

I believe there is evidence that the kind of discussion we and our numerous and esteemed visitors at the $n$-category Café had and have helped demonstrate what a couple of years ago was not widely considered probable or even desireable: that useful and high-quality scientific exchange in public web-based sites is possible, useful and enjoyable and not un-comparable in relevance for the participants with other scientific exchange as on conferences and in seminars.

Lately the web has seen a considerable increase in the number of mathematical blogs concerned with serious scientific discussion and exchange. Prominent examples (without the aim of completeness) are Fields medalist Terence Tao’s blog, Fields medalist Tim Gower’s blog, Fields medalist Richerd Borcherds’ blog, The secret blogging seminar, John Armstrong’s blog and a couple more of them.

I think it is clear that a new medium for serious high-quality scientific communiation is being explored here.

On the other hand, I don’t feel in the position yet to speculate about what

its influence on the category maths community professional work

is in general.

But I can make a couple of observations:

I believe we have seen some young students become aware of the topics which we are discussing here first and maybe only due to this blog. From my own time as a student I remember how very difficult up to almost impossible it was, without the presence of senior researchers on the web, to get access to insider information about truly actual research, that which has not yet made it into textbooks. It seems that bridging a certain information barrier between students (and we all remain students in most of the topics that are out there for all our lives) and experienced researchers is one of the things serious scientific blogs can be very good for.

Then I think we have seen exchange of information among experts here, which is unlikely to have occured elsewhere than on the blog, if only because the persons involved would not have come into contact at all otherwise. I consider myself as having benefited immensely from information that was provided here, as a reaction to things I said on the blog or questions that I asked, by commenters. I have found collaborators on the blog which I would hardly have come into contact with otherwise, some of whom I have still never met in person (which however needs to change).

On conferences, I see a certain benefit in the fact that people who I have never met before tend to know what I am up to from this blog. This is helpful for getting into contact in person even if there was no online contact before.

I do imagine that this kind of effect the $n$-category café has on me, personally, as a member of some community, might in the long run become an effect that scientific blogging activity has on the entire community. But I don’t feel to be in the position to speculate on that far beyond what applies to me personally.

With a little luck John Baez and David Corfield may find and take the time to say more on this issue.

Last night at about 3 am I learned that I am supposed to give a public talk to high school kids and journalists next Friday.

Good luck Urs! Looks like the Croatians don’t mess about when it comes to higher categories :-)

Parts of the discussion of Chern-Simons-like obstructions to lifts of $G$-bundles through String-like extension can now be found in the section

String- and Chern-Simons $n$-Transport

in the set of slides of (almost) the same title: On String- and Chern-Simons $n$-Transport.