The n-Category Café

Introductory remarks

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.

Processes

Everybody knows that the order in which one adds two numbers is irrelevant:

$$3 + 2 = 2 + 3 \,.$$

This sounds like the most obvious thing in the world. But instead it is a rather peculiar fact.

Most processes which we encounter in our lives are not at all like this: the order does matter.

To see this, just imagine that in the morning, instead of first putting on your socks and then your shoes, you’d first put on your shoes, and then your socks. The order of these two operations does matter.

Or just imagine what would happen if instead of first opening the window and then sticking your head out, you’d do that the other way around.

On the other hand, while most processes do not allow us to change their order, they all share at least the property that one can be performed after the other.

This sounds trivial. And it is. But that shall not prevent us from thinking about it.

In 1945 Eilenberg and MacLane revolutionized mathematics by inventing a very sneaky notation for a process: an arrow

$$\stackrel{process}{\rightarrow}$$.

Then they introduced an even more clever notation for the result of one process occurring after the other: two arrows:

$$\stackrel{total process}{\rightarrow} = \stackrel{process 1}{\rightarrow} \stackrel{process 2}{\rightarrow} \,.$$

Suppose you get up in the morning, then put on your socks, then put on your shoes, then open the window, then stick out your head, then decide it is a horrible day to go to work, then go back to bed – then notice that you still have your shoes on.

This little story may be read in arrow language like this

$$\stackrel{get up}{\rightarrow} \stackrel{socks on}{\rightarrow} \stackrel{shoes on}{\rightarrow} \stackrel{open window}{\rightarrow} \cdots$$

Okay. But so what is it that makes

$$\stackrel{socks on}{\rightarrow} \stackrel{shoes on}{\rightarrow}$$

different from

$$\stackrel{shoes on}{\rightarrow} \stackrel{socks on}{\rightarrow}$$

?

It’s clear: the problem is that the processes do not just occur, they also change the state. Like the state that you are bare-footed.

Luckily, Eilenberg and MacLane also took care of that: in between their arrows, they drew symbols indicating the state a process acts on, and the state the process results in.

So

$$(window closed) \stackrel{open window}{\rightarrow} (window open) \stackrel{stick out head}{\rightarrow} (fresh breeze) \,.$$

Notice the difference: $$(window closed) \stackrel{stick out head}{\rightarrow} (headache) \,.$$

Of course the last statement is a joke. It is traditional not to incorporate too many jokes in mathematical notation, Mathematicians are sober people. They’d rather want to forbid undesirable things like $(window closed) \stackrel{stick out head}{\rightarrow} (headache)$.

And so they do: the convention is that every process must carry the information which two states it relates. And to compose two processes the resulting state of the first must be an admissable starting point for the second.

So given the two processes

$$(window closed) \stackrel{open window}{\rightarrow} (window open)$$ and $$(window open) \stackrel{stick out head}{\rightarrow} (fresh breeze)$$

we may compose them one way. But not the other.

But of course there are also processes which do commute. For instance, suppose you do decide to get up and get to work after all. Reluctantly. So you come to the breakfast table. You can either first get a coffee and then pick up the newspaper, or the other way around. In both cases you end up reading the news and drinking coffee.

While drawing arrows next to each other was already a stroke of genius, Eilenberg and MacLane’s biggest achievement was arguably the realization that a situation like this is best depicted by drawing a square of arrows:

$$\array{ &\stackrel{make coffee}{\to}& \\ \;\;\;\;\;\downarrow^{get newspaper} && \;\;\;\;\;\downarrow^{get newspaper} \\ &\stackrel{make coffee}{\to}& }$$

This says that the process obtained by composing the upper and rightmost process is the same as that obtained by composing the leftmost and lower morphism.

This finally allows us to come back to the original observation that $2 +3 = 3 + 2$: let us write

$$\mathbb{N}$$

for the state “a bunch of coins in the coffee machine”. Then write

$$\mathbb{N} \stackrel{+3}{\to} \mathbb{N}$$

for the process of inserting 3 coins into the coffee machine, and

$$\mathbb{N} \stackrel{+2}{\to} \mathbb{N}$$

for the process of inserting two coins. This way we can now show off and state the simple fact that $2 + 3 = 3 + 2$ by drawing a square

$$\array{ \mathbb{N} &\stackrel{+3}{\to}& \mathbb{N} \\ \downarrow^{+2} && \downarrow^{+2} \\ \mathbb{N} &\stackrel{+3}{\to}& \mathbb{N} } \,.$$

Mathematicians say that $2 + 3 = 3 + 2$ comes from the fact that addition is a commutative operation. This is to distinguish it from operation whose order does matter. These are called non-commutative.

Accordingly, squares as the above one are known as commutative squares.

While you can try to impress your friends at next saturday’s party by telling them that you know and understand what a commutative square in category theory is, let’s try to instead impress ourselves by noticing that what is really remarkable here is not the funny “commutative”, which is just terminology, but rather the fact that we are suddenly talking about squares.

Category theory is often said to be very abstract. But all we have done so far is that we looked at a bunch of arrows. If that is too abstract for you, you might want to consider trying to find a hobby other than math or physics.

Instead, what is really hard about category theory is – typesetting your papers.

Even though one starts out talking about arrows,which are inherently 1-dimensional, one ends up discussing squares, which are 2-dimensional. And few typesetting systems know how to deal with content that does not appear as long 1-dimensional chains of symbols.

How can we understand this shift in dimension which we have run into, simply by pondering the fact that $2 + 3 = 3 + 2$?

Well, while we have been asserting this fact, let’s try to prove it. Try proving it to your 5-year old sister, using language and concepts she understands. Maybe this works:

put 3 rocks on a table $$\bullet \bullet \bullet$$ then add another two rocks $$\bullet \bullet \bullet \;\;\;\; \bullet \bullet \,.$$ That’s $3+2$ rocks. Next slide these rocks around on the table to obtain first

$$\array{ & \bullet \bullet \\ \bullet \bullet \bullet }$$

then

$$\array{ \bullet \bullet \\ \bullet \bullet \bullet }$$

then

$$\array{ \bullet \bullet \\ & \bullet \bullet \bullet }$$

and finally

$$\bullet \bullet \;\;\;\; \bullet \bullet \bullet \,.$$

That’s clearly $2+3$ rocks. And it’s clearly the same number of rocks as before!

But in order to go through this proof, we crucially needed the two dimensions of the table the rocks are sitting on.

Imagine you were an ant living on a blade of grass. Imagine carrying around some dead beetle for next breakfast. Imagine you run into two fellow ants on that blade of grass, each carrying itself a beetle. But – unfortunately – carrying it in the other direction than you are.

Now, even though one beetle plus two beetles is the same as two beetles plus one beetle, abstractly speaking, in order to prove that you need to evade to a second dimension. But that’s not possible if you are an ant on a blade of grass.

Higher processes

We originally regarded our coffee machine $$\mathbb{N}$$ containing a bunch of coins as a state. Something that the process of inserting coins acts on.

But notice that a coffee machine which just sits there on standby as time passes is itself already a process! A boring process, true, wich just consumes some electricity to keep a red light glowing, but still a process.

To emphasize this we should write $$\stackrel{\mathbb{N}}{\to}$$ for “a coffee machine containing a bunch of coins which just sits there on standby as time passes”.

If we don’t care about how long exactly the coffee machine just sits there and does nothing useful, we write $$\stackrel{\mathbb{N}}{\to} \stackrel{\mathbb{N}}{\to} = \stackrel{\mathbb{N}}{\to} \,.$$

With the coffee machine itself thus being a (boring) process, this means that filling coins into it is a process that acts on a process!

Hence what we originally denoted by $$\mathbb{N} \stackrel{+3}{\to} \mathbb{N}$$ we should now draw as $$\array{ & \;\;\;\nearrow \searrow^{standby} \\ \bullet &\Downarrow^{+3}& \bullet \\ & \;\;\; \searrow \nearrow_{standby} } \,.$$

Now, if we want to insert first three and then two coins into the machine, we find we have two options:

Either we do it quickly, whithout letting time pass between inserting first three coins, then two coins. This means doing $$\array{ & \;\;\;\nearrow \searrow^{standby} \\ \bullet &\Downarrow^{+3}& \bullet \\ & \;\;\; \searrow \nearrow_{standby} } \,.$$ $$\array{ & \;\;\;\nearrow \searrow^{standby} \\ \bullet &\Downarrow^{+2}& \bullet \\ & \;\;\; \searrow \nearrow_{standby} } \,.$$

Or else, we first insert three coins, and then wait a while:

$$\array{ & \nearrow \searrow \\ \bullet &\Downarrow^{+3}& \bullet &\to& \bullet \\ & \searrow \nearrow }$$

and then we insert the remaining two coins:

$$\array{ & \nearrow \searrow \\ \bullet &\Downarrow^{+3}& \bullet &\to& \bullet \\ & \searrow \nearrow }$$ $$\array{ &&& \nearrow \searrow \\ \bullet &\to&\bullet &\Downarrow^{+2}& \bullet \\ && & \searrow \nearrow } \,.$$

This total process of first inserting three coins, then waiting for a while, and then inserting the remaining two coins is this:

$$\array{ & \nearrow \searrow && \nearrow \searrow \\ \bullet &\Downarrow^{+3}& \bullet &\Downarrow^{+2}& \bullet \\ & \searrow \nearrow && \searrow \nearrow } \,.$$

But we knwo that this is also the same as

$$\array{ &&& \nearrow \searrow \\ \bullet &\to&\bullet &\Downarrow^{+2}& \bullet \\ && & \searrow \nearrow }$$ $$\array{ & \nearrow \searrow \\ \bullet &\Downarrow^{+3}& \bullet &\to& \bullet \\ & \searrow \nearrow }$$

which equals

$$\array{ & \nearrow \searrow \\ \bullet &\Downarrow^{+2}& \bullet \\ & \searrow \nearrow }$$ $$\array{ & \nearrow \searrow \\ \bullet &\Downarrow^{+3}& \bullet \\ & \searrow \nearrow } \,.$$

These four diagrams, which are all equal, say that it doesn’t matter whether

- we first insert three coins, then two of them

- we first insert three coins, then do nothing for a while, and then add the remaining two coins

- we first insert two coins, then do nothing for a while, and then add the remaining two coins

- we first insert two coins, then three of them.

The result is always the same: five new coins in the coffee machine – and hopefully a freshly brewed coffee in our hands.

This means we have proven once again that $2 + 3 = 3 + 2$, now as a statement about processes of processes!

Notice two things:

first, if you think of the above disk-shaped diagrams as containing two rocks and three rocks respectively, you can literally see that we are doing, once again, nothing but sliding rocks over a table.

second, notice that all along above we had really used our knowledge that $2 + 3 = 5$ and $3 + 2 = 5$ to be able to impose the condition that addition is a commutative operation. But now we suddenly find much more:

the above manipulation of processes of processes did not use any propery of addition, except that it can consistently be interpreted as a process on some identity process.

This means: whatever identity process on whatever state you have, all processes of processes on that identity process have a composition for which the order is irrelevant.

This is known as the Eckmann-Hilton argument.

[At this point the speaker pulls out two party balloons from his pocket, inflates them, ties them together at their tips, labels one of them $A$ and the other one $B$ and proves that $$A B = B A$$ by sliding the two balloons alongside each other, keeping their tips attached.]

Some fancy implications for physics

Now let’s suppose we talk to one of those physicsists who are speculating that, once we look really, really closely, elementary particles

$$\bullet$$

– like electrons, quarks, neutrinos, photons, and the like – which trace out paths as time goes by

$$\bullet \to \bullet$$

start to look like little 1-dimensional extended entities

$$\array{ & \nearrow \searrow }$$

which trace out surfaces as time goes by

$$\array{ & \nearrow \searrow \\ & \Downarrow& \\ & \searrow \nearrow } \,.$$

If true, this situation is likely to follow the rules of higher dimensional logic, one of which we have discovered above. Hence we immediately know:

if in the life of such a “string”, nothing interesting would happen along the string itself

$$\array{ & \nearrow \searrow }$$

but that physically interesting things happen only as the string moves

$$\array{ & \nearrow \searrow \\ & \Downarrow& \\ & \searrow \nearrow }$$

then this would mean that the string behaves a little bit like a coffee machine on standby! Which in turn implies that what happens as the string moves cannot be all too interesting: it has to be a commutative process.

That could be interesting already, but might be a little dull in the long run. Maybe it might worry us that fundamental physics, which seems to be very rich, would seem to be forced to be very dull, really, by this argument.

So we would maybe conclude that it is likely that interesting physics is happening not just along the 2-dimensional path which a string traces out as it propagates, but that interesting processes are already associated to the string itself. If so, the processes taking place as it moves have a chance of being rich and interesting.

“Hm”, the physicist we are talking to says, “how did you come to that conclusion? Interestingly, this is essentially the conclusion that we have come to, too. But apparently from very different reasoning.”

Indeed, it turns out that assuming particles to be strings implies that there are interesting effects taking place on the boundaries of these strings. Physicist refer to these as “D-branes”. Maybe you have heard about them in the news. String theory is that part of higher dimensional algebra which people like to have discussions about in the mass media.