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September 25, 2007

An Invitation to Higher Dimensional Mathematics and Physics

Posted by Urs Schreiber

I mentioned that I was aked to give a public lecture in the context of Categories in Geometry and Physics next Friday, to high school kids and journalists. I am imagining the lecture might proceed along the following lines.

Since this is still in progress, I would enjoy receiving comments and suggestions. The following already incorporates advice by John Baez and Tom Leinster.


An Invitation to Higher Dimensional Mathematics and Physics

In which sense is summing two numbers a 2-dimensional process? Everybody who knows that 2+32+3 is the same as 3+23+2 will be lead in this talk to a simple but profound result in a branch of mathematics known as nn-category theory. This simple insight in higher dimensional mathematics alone will be sufficient to make understandable some fancy ideas in modern theoretical high energy physics.

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 nn-category theory, if one wants to emphasize its nn-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 nn-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. 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

process \stackrel{process}{\rightarrow} .

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

totalprocess=process1process2. \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

getupsocksonshoesonopenwindow \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

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

different from

shoesonsockson \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

(windowclosed)openwindow(windowopen)stickouthead(freshbreeze). (window closed) \stackrel{open window}{\rightarrow} (window open) \stackrel{stick out head}{\rightarrow} (fresh breeze) \,.

Notice the difference: (windowclosed)stickouthead(headache). (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 (windowclosed)stickouthead(headache) (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

(windowclosed)openwindow(windowopen) (window closed) \stackrel{open window}{\rightarrow} (window open) and (windowopen)stickouthead(freshbreeze) (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:

makecoffee getnewspaper getnewspaper makecoffee \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+22 +3 = 3 + 2: let us write

\mathbb{N}

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

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

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

+2 \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+22 + 3 = 3 + 2 by drawing a square

+3 +2 +2 +3 . \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+22 + 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+22 + 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+23+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+32+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 +3 \mathbb{N} \stackrel{+3}{\to} \mathbb{N} we should now draw as standby +3 standby. \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 standby +3 standby. \array{ & \;\;\;\nearrow \searrow^{standby} \\ \bullet &\Downarrow^{+3}& \bullet \\ & \;\;\; \searrow \nearrow_{standby} } \,. standby +2 standby. \array{ & \;\;\;\nearrow \searrow^{standby} \\ \bullet &\Downarrow^{+2}& \bullet \\ & \;\;\; \searrow \nearrow_{standby} } \,.

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

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

and then we insert the remaining two coins:

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

+3 +2 . \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

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

which equals

+2 \array{ & \nearrow \searrow \\ \bullet &\Downarrow^{+2}& \bullet \\ & \searrow \nearrow } +3 . \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+22 + 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=52 + 3 = 5 and 3+2=53 + 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 AA and the other one BB and proves that AB=BA 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.

Posted at September 25, 2007 3:05 PM UTC

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Re: An Invitation to Higher Dimensional Mathematics and Physics

Great talk! There are some typos, which I may sneak in and fix…

I found I had to start thinking a bit starting around where you said “some state which the standby process of the coffee machine acts on. Like, say the existence of a power plant at the other end of the city.”

If I had to think a bit, it probably means something around here is too difficult for high school students. I see a number of difficulties:

  • Most people don’t think of a coffee machine sitting there as “acting some state, namely the existence of a power plant”. First of all, since when is “the existence” of something a “state” — and how does a distant coffee machine act on it? If I were a kid, I’d be thinking WHAT’S HE TALKING ABOUT NOW??? There should be some way to stick to more common-sense notions and terminology.
  • Here’s another mind-shattering sentence: “In fact, a little reflection reveals that what matters about the arrow which we are talking about is not so much the way in which we interpret them as processes, but just the way they compose”. I think here you are talking about the way category theory is an abstraction of the concepts of state and process… but that’s a subtle point — the sort of point you can only appreciate after some thought (as you note). So, SKIP IT. Stick to the concrete example at hand and get them to see how diagrams can be used to reason about it. The move towards ‘abstraction’ can come later — after they’ve fallen in love with mathematics and become mathematicians, thanks to your talk.
  • You write “the technical term is…” Again, now is not the time for you to clutter their minds with technical terms. The technical terms only become helpful later. Right now you’re in the middle of explaining a simple idea: the Eckmann–Hilton argument. Remember Rota’s advice: an audience is like a herd of sheep. You have to pick out a clear goal and keep them moving in that direction, or they will scatter. Your goal is to give a vivid proof that 2+3 = 3+2, which shows why 2d space is important for commutativity.
  • “… throw all intuition into the wind and just blindly follow the formalism”. It’s possible that you can say at each point what the diagram means, making it unnecessary to “blindly follow the formalism”. No? Surely it should be possible to give a clear proof that 3+2 = 2+3 without retreating into a ‘formalist’ approach.
  • “…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.” Personally I find this much easier than thinking about “processes between standby processes between coffee machines”, since the latter approach seems to involve two dimensions of time instead of two dimensions of space!
  • “… whatever identity process on whatever state you have, all processes of processes on that identity process…” If I were a kid I’d think THIS GUY IS LIKE A BROKEN RECORD! Hmm… I’m showing my age: there’s no such thing as a ‘record’ anymore. Maybe I mean a broken CD. Anyway, my point is that this sentence is too complicated for most people. Figure out a simpler way to say what you mean, unless your main goal here is to overwhelm the audience — which is occasionally fun, but only if they know they’re supposed to be overwhelmed.

After this, I think your audience will be completely exhausted and not able to really think anymore. The analytical part of their brains will be tired. But, they can probably still enjoy some good stories! So, you can switch to story-telling mode and tell them in a much less detailed and stressful way how this weird ‘higher-dimensional mathematics’ was invented, and how it’s now taking over string theory — basically because strings are higher-dimensional, more like those surfaces you drew than those line segments, which are like tracks of particles.

Posted by: John Baez on September 25, 2007 8:44 PM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

There is one place where you use the undefined (for this talk) term “morphism” instead of “process” which had been used all up to that point. Later you use “identity morphism”. These are the only places that use the term. So either you need to change both of these or define “morphism” earlier, probably right after you first describe an arrow as representing a process.

Posted by: Mark Biggar on September 25, 2007 9:43 PM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

There is one place where you use the undefined (for this talk) term “morphism” instead of “process”

Thanks! I’ll fix this tomorrow.

Probably the best strategy I should use would be to use the kindergarten terminology throughout, and then just append a small appendix to the file, only for those really interested, which provides a dictionary for translation to the standard terms.

Posted by: Urs Schreiber on September 25, 2007 11:24 PM | Permalink | Reply to this

A newbie’s POV

All of this is cristal clear to me until you start using ↗↘ and ↘↗ without introducing their meaning, and then I’m lost…

I assume that I’m supposed to guess what it means from the beginning of the text, but I don’t…

Some clarification would be welcome :-)

Posted by: Py on September 25, 2007 11:27 PM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

“2 coins” and “3 coins” are ambiguous.
Use e.g. “2 dollars” and “3 dollars” instead. Or another metaphor.

Posted by: ryanw on September 25, 2007 11:58 PM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

“2 kuna” and “3 kuna”, here.

Posted by: Tom Leinster on September 26, 2007 2:42 PM | Permalink | Reply to this

Processes vs violin strings

I think our disagreements in the past boils down to different ways of thinking about 1D curves. To you, a curve is always a process: you move a point along the curve. To me, a curve is a string which just sits there. You can move a point along a curve, or put down a violin string so that it fits the same curve. Same curve, but very different things to do, and described by different mathematics.

According to Eckmann-Hilton, if you assume that the vertical and horizontal products belong to the same group, then this group is necessarily abelian. One way around this is to to assume that the vertical and horizontal products belong to different groups. AFAIU, this uniquely leads to 2-groups.

However, there is another possibility: the horizontal product does not generate a group. The vertical product always forms a group, because it describes a process: moving the curve perpendicular to itself. There is an inverse because you can move the curve back. In contrast, gluing curves along their endpoints does not have an inverse, so is not a group.

Mathematically, this horizontal product is the tensor product or its continuum analog. It is not a group because it does not have an inverse, but a different property which I call unique factorization. If A in Vm and B in Vn, their products AB in Vm+n. Once you know AB and m (or n), you know the factors A and B uniquely, up to a scalar. In contrast, if A and B belongs to some group G, knowledge of AB does not determine A and B, because AB = A’B’ where A’ = AC and B’ = C-1B.

Posted by: Thomas Larsson on September 26, 2007 7:49 AM | Permalink | Reply to this

Re: Processes vs violin strings

To you, a curve is always a process

No. I am just emphasizing the process imagery here since it seems to lend itself to intuitive access.

Higher morphisms in nn-categories can have very different interpretations than in terms of processes.

A very common interpretation is in terms of nn-dimensional spaces with corners, their (n1)(n-1)-dimensional boundaries, the (n2)(n-2)-dimensional boundaries of these boundaries and so on.

I thought about using this connection of nn-categories to topology in the talk. But then didn’t. But I guess I could have.

Posted by: Urs Schreiber on September 26, 2007 12:46 PM | Permalink | Reply to this

Re: Processes vs violin strings

Ah, but the word “morphism” by itself carries the connotation of a process: an arrow with an inverse. The point, which I think I never managed to get across, is that you can have multiplication without an inverse (tensor product), and this is a natural choice for gluing violin strings.

Posted by: Thomas Larsson on September 26, 2007 1:11 PM | Permalink | Reply to this

Re: Processes vs violin strings

Sorry, I don’t follow. How does either the term “morphism” or the term “process” connote the existence of an inverse?

There are plenty of processes that are irreversible on their faces. We can let gas out of a box to fill up a room, but it’s pretty hard to reverse the flow. Is gas diffusion not a “process”?

Posted by: John Armstrong on September 26, 2007 1:44 PM | Permalink | Reply to this

Re: Processes vs violin strings

I was thinking about 2-gauge theories. On the one hand, a curve can be thought of as the path of a particle, and a surface as a path of paths. This naturally makes the curve into a process of parallel transport with an inverse: transport in the opposite direction. On the other hand, a curve can be a string in its own right which transports nothing. Then you have parallel transport for surfaces but not for curves, and instead of having an inverse the curve can be factorized uniquely.

But we have discussed this many times, and I don’t think I ever managed to get the point across that these are very different situations and should be described by different math, despite involving the same curves and surfaces.

Posted by: Thomas Larsson on September 26, 2007 2:00 PM | Permalink | Reply to this

Re: Processes vs violin strings

Sorry, I didn’t look at the name of the poster, just assumed it was Urs. The last paragraph would only have made sense then.

Posted by: Thomas Larsson on September 26, 2007 2:04 PM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

If I can just add my twopenneth (which would give the same result whether I added it after or before my threepenneth), you should probably avoid the term ‘trivial’ as it means something specific to mathematicians and not to the general public. I would opt for ‘obvious’ or ‘common sense’.

Posted by: Simon Willerton on September 26, 2007 10:33 AM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

you should probably avoid the term ‘trivial’

Okay, good. Thanks.

I am collecting all the suggestions. As soon as I find the time, I’ll try to create a new polished version of the above notes, incorporating all this feedback.

Right now I need to do something else.

Posted by: Urs Schreiber on September 26, 2007 12:22 PM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

I was really enjoying things until the diagonal arrows started appearing. i.e after:

“…which used to be a process between states itself – now becomes a process between processes.


Like”

With previous diagrams it looked like, starting from top left, I could go right or down. With the diagonal arrows I wasn’t sure if I could follow them or not. The dot symbol was also newly introduced then, and I wasn’t sure about it. Does the “standby” text also go with an arrow, like “coffeemachineonstandbyforawhile” - if so, does it go along with both of the diagonal arrows? I can’t think why an arrow would have no process, but I just don’t know.
Cheers,
Paul

Posted by: Paul on September 26, 2007 11:54 AM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

With the diagonal arrows I wasn’t sure if I could follow them or not.

Okay, thanks for letting me know.

First: when drawn on a blackboard, these diagonal arrows look like curved arrows. It’s only the typesetting restrictions here that make them look diagonal.

So, yes, you can follow them. The diagonal arrows that appear so far are all such that following them denotes “carrying out the standby process of a coffee machine for a while”.

The dot symbol was also newly introduced then, and I wasn’t sure about it.

Yes, I need to replace that by something. I was talking with John about this: what kind of state is it that an electric device on standby acts on?

Posted by: Urs Schreiber on September 26, 2007 12:12 PM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

You could start with a one-slot coffee machine for sequential input and move to a two-slot coffee machine for simultaneous input … but going to n-slot machines…?
Charlie

Posted by: Charlie C on September 26, 2007 4:34 PM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

I like this idea; it seems like a nice compromise between Urs’s coffee machine and John’s basket. You have a coffee machine with two slots, and the coins fall into the same place either way. (Say the extra slot is there for people who are in a huge hurry to get their coffee!)

Posted by: anon. on September 26, 2007 5:51 PM | Permalink | Reply to this

Hands-on; Re: An Invitation to Higher Dimensional Mathematics and Physics

Based on my experience teaching multidimensional geometry to graduate students of Art and Design (at the Art Center College of Design in Pasadena) and to university students in business, fashion design, animation, and other subjects (Woodbury University): be sure, as soon as possible, to allow the students to touch and handle 3-D polytopes, made out of straws or toothpicks or cardbaord, or whatever. Tetrahedron, Cube, etcetera, labeled as desired. Then projections of 4-D polytopes to 3-D.

You and I are used to looking at diagrams on paper and on-screen. But public schools have delayed and dumbed down geometry, and the 19th century kids who worked through Euclid for Solid geometry might as well be from another country. The past is another country. They do things differently there.

Posted by: Jonathan Vos Post on September 26, 2007 5:14 PM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

How does mathematics describe the process of meeting a mugger on the bus? Math provides a model of some very limited aspects of the bus journey.

“We can compute to many decimal places the precise position of the earth…” is a contradiction. If you are ‘computing to many decimal places’ then you are making an approximation to however many decimal places you get to. That is not a precise position, it is an approximate position.

Electrons cannot be points otherwise they would trace out infinitely thin lines. You can *model* certain aspects of electrons using points, but those points are not the electrons.

“mathematical language which is to the one-dimensional formulas that we are taught in school like a 2-dimensional surface is to a line.”

What? You may lose your audience here.

“algebra … very rich and interesting

Is school algebra that interesting? x + y = 2, what’s y when x is 3. Yeah, a real blast.

“understandable by every layman…”

So the poor sod who doesn’t understand it is going to be left thinking he’s subhuman. (So far that’s me! Pass me a banana:-)

Eilenberg and MacLane’s works looks like a cumbersome version of Tony Hoare’s CSP notation. CSP stands for Communicating Sequential Processes and it was designed to handle parallel processes like the coffee situation. It would go something like (n->C|C->n). Which is much more concise than your diagram.

Arrows are not inherently one dimensional, otherwise you would not be able to see them.

The rock proof is much easier in one dimension - slide a rock from three row to two row.

[Student listens to boring coin stuff – thinks, yeah but what is category theory and what has all this got to do with finding the solution to QG. Looks at watch. Hungers for dinner.]

Read the first three Feynman lectures to see how to motivate students to suffer through the boring stuff.

—————Comments on Baez 1—

1) I was happy with the introduction of a coffee machine as state and process.

2) Definitely right about skipping the composing arrows. I don’t get it, and still don’t know what category is… gotta check wikipedia!

3) Baaa. Baaaa. You can do it more easily in 1D. [kick sheep] Baaaaaaaaaaaaaaa.

4) Gowers likes just following the formalism.

5) Yeah the rocks are better, stick with them.

6) CDs are so last decade.

—————–Comments on URS 2 —

‘1-morphism in an abelian (semi-)group’.

Is there an easy introduction to this stuff… and n-category theory? Maybe you can try telling ME what they are with simple examples. If I get it there’s a good chance those bright kinds will…

I thought you explained very well why a coffee machine is a process! The electricity keeping the red light on was a good touch… “Acts on the coffee?” – bad move – leave as is! Does the coffee machine have to be thought of as acting on anything? Do you act on anything when you are on standby? Does a process have to do anything? Think of the Main Event Loop in a computer operating system. It just sits there waiting for you to do something.

——Comments on Baez 2——

“Does putting coins in the coffee machine leave it unchanged?” Good point. I spotted that but abstracted out the change and was a happy bunny again…

Money basket? Wazat? Keep it real…

Do you remember those fairground games where you have multiple slots where several people drop in money at the same time, it piles up, and you never get the money back :-) Maybe that can be used?

Wick rotation? Boy am I rusty…
————–Willerton———-

Don’t call anything obvious, you’ll just upset the audience if they don’t find it obvious! I got this advice from Gowers’ “Mathematics a very short intro.” Good advice! It’s a great book to read to see how to explain things to bright high school kids. Also, you might learn something. I certainly did.

Posted by: paul grieg on September 26, 2007 5:40 PM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

On a related subject, what would be a list of problems outside category theory that category theory has helped to solve? (aiming this list at a professional physicist). For example, if the subject were complex analysis, I could mention that it gives an easy proof of the fundamental theorem of analysis, or that it gives a nice way to evaluate certain real integrals by deforming the integration contour. Simple results that I can state to anyone who knows real analysis but doesn’t know complex analysis. So, what would be analogous for category theory? The closest application of it near what I do would be to description of topological order, but that always sounds more like a description of an existing result than a new result.

Posted by: matt on September 27, 2007 3:17 PM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

I’d love to have a list like this too. Somehow people don’t find it as convincing when I point out how categories simplify some concepts and elegantly unify others.

Posted by: John Armstrong on September 27, 2007 6:19 PM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

That’s one of the problems with our mathematical culture: that the activities of conceptual simplification and theory-building aren’t generally considered as valuable or important as “problem-solving”. (Despite the fact that that’s what people spend a lot, perhaps most of the time doing: shaking down theorems and arguments to simple and conceptually comprehensible form.) And a lot of people somehow have very decided opinions about category theory and category theorists to begin with.

But anyway, there is no question about the fact that category theory has been instrumental to the development of incredibly powerful tools which have been used in some of the best mathematics in the past sixty years. A lot of mathematics is literally unthinkable without the language and tools provided by category theory.

Some of the main examples which come to my mind include Grothendieck topologies, descent theory and Grothendieck’s reformulation of Galois theory, along with the idea of a classifying topos. These ideas continue to be developed to this day, e.g., with the introduction of powerful new topologies, as in work of Voevodsky which in part won him a Fields Medal.

Then there is the wonderful notion of closed model category developed by Quillen and others, which opened up whole new worlds and was vital to the solution of problems in algebraic topology.

I think it is fair to say that the combined techniques of operads, monads, clubs, props, etc. have become extremely important in recent years.

Dana Scott made crucial use of categories in his groundbreaking work in domain theory (extensional models of lambda calculus).

Oh, and let’s not forget the uses made of categorification!

Obviously the list goes on, and on, and on.

But yeah, John, I do feel your pain. On a very, very modest level: when I arrived as a post-doc at Macquarie, as the new kid on the block, I was invited to give a colloquium talk on category theory. I was trying to think of something to say which might not make the analysts in the crowd roll their eyes; specifically, I was trying to think of an example where category theory could be used to effectively solve an easily stated problem, which would make sense to a general audience, and wasn’t too trivial.

There’s a problem stated by Paul Halmos in his book I Want to Be a Mathematician; he mentions it as a problem from a take-home exam for a course he gave on topological groups (“Geniuses are expected to solve all fifteen [problems].”) The problem asks: does there exist a connected (Hausdorff) topological group of exponent 2? That was the problem I settled on.

I started off by giving some possible lines of thinking which suggested the answer could go one way or another, and then led them sort of surreptitiously through some categorical ways of reformulating the problem, in terms of the idea of a group of exponent 2 in a general category with products. I then explained how you could create new groups from old by applying product-preserving functors. Then came the kicker: start with the simplest such group of exponent 2 in Set: 2\mathbb{Z}_2. Then apply a series of product-preserving functors which lead you from sets to topological spaces:

SetCatSimplicialsetsTop.Set \to Cat \to Simplicial sets \to Top.

I’ll leave it to the reader to figure out which functors I mean, but if you apply them starting with 2\mathbb{Z}_2, you end up in Top with an infinite-dimensional sphere which carries such a topological group structure. Aha! There you go!

Well, it’s sort of a silly problem, but it did seem to go over well, in that people said, “Hey! You really can use category theory to solve problems!” :-)

Posted by: Todd Trimble on September 27, 2007 7:50 PM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

Matt asked:

what would be a list of problems outside category theory that category theory has helped to solve?

I came to think that there are historical analogues to this question, which keeps being asked:

there was a time when the concept of differential forms was new and fancy and suspicious. (This time is being preserved by some physicists and their textbooks into the present day.)

Now: which problem can you solve using differential forms that you cannot solve without them?

Maxwell found his equations without using any differential forms. He filled an entire page. Today we know differential forms, and rewrite this page equivalently in half a line.

But, more importantly, in the remaining half of the line we find their nonabelian generalization.

Then in the next line we generalize them to higher dimensions both of spacetime and of the sources.

Then in 5 more lines we find their generalization from local data to global data. Thus first finding line bundles with connection.

Then it depends. If you don’t use categories, you need now to fill 20 pages for describing what Maxwell’s equations for higher dimensional sources mean globally. If you know your categories, that’s just half a page, where its mostly the picture of the 3-simplex that you draw and decorate in your category which takes up the space.

Gotta run now. Hope that illustrates the point.

Posted by: Urs Schreiber on September 27, 2007 7:38 PM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

what would be a list of problems outside category theory that category theory has helped to solve?

I’m not sure if it would class as category-theory per se, but generalized homology theories (namely K-theory) got used to solve problems in operator theory to do with “essentially normal” operators (i.e. T commutes with T* modulo compacts). This is, more or less, Brown-Douglas-Fillmore theory if memory serves me correctly. MathSciNet gives me Bull AMS 79 (1973), 973–978 as a good place to start.

The reason I think this might count as an application of the category-theoretic mindset, if not of its main results, is that as I recall it becomes crucial that various `Ext’ invariants behave functorially, both for calculations and in setting up the general machinery.

(Disclaimer: this is well outside my area of competence, so corrections from more knowledgeable readers most welcome)

Posted by: yemon choi on September 28, 2007 12:09 AM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

According to Ross Street’s website, category theory is being used at the aerospace company Boeing, but I don’t know if cats have been used there already to solve any actual problems:

http://www.math.mq.edu.au/~street/Boeing.html

Posted by: Charlie Stromeyer Jr on September 29, 2007 5:28 PM | Permalink | Reply to this

Re: An Invitation to Higher Dimensional Mathematics and Physics

The work described at the page Charlie Stromeyer mentions, www.math.mq.edu.au/~street/Boeing.html , could well be being used to solve actual problems, but related work has been going on for a long time, and not only at Boeing.

I say this because the page talks about using colimits to put together software specifications written in some kind of logic. The idea has a long history, and dates back to research by Rod Burstall and Joseph Goguen written up in their 1980 paper “The semantics of Clear, a specification language”. (In “Abstract Software Specifications”, edited by D. Bjorner. Lecture Notes in Computer Science, Volume 86, Springer, 1980.)

According to that paper, the idea was inspired by work started by Goguen in the 1960s, and described in his paper “Sheaf Semantics for Concurrent Interacting Objects”. (In “Mathematical Structures in Computer Science”, Vol. 2, No. 2. (1992); online at Citeseer and CiteseerX.)

This used limits to represent the behaviour of systems of interacting parts. Each part was represented as, in essence, a set of event traces or histories. Arrows between parts represented the way attributes of one part mapped on to those of another. The limit of the resulting diagram represented the possible event traces of the entire system, once constraints between the objects had been taken into account. If there were no such constraints, the behaviour would just be the product of the behaviours of the parts.

(I remember this because I once coded an implementation of it in Prolog, using a limit-calculation algorithm similar to that in David Rydeheard and Rod Burstall’s book “Computational Category Theory”, www.cs.man.ac.uk/~david/categories/book/book.pdf.)

Actually, the attributes of the parts could range over domains other than time, and Goguen used sheaves to represent the structure of permitted observations. So what I said about parts being represented as event traces is a simplification. The work was pretty general.

Anyway, it inspired the use of colimits in software specification. The Kestrel Institute mentioned in the original URL was set up to exploit such research, but it’s certainly not the only place that has done so. The OBJ family of algebraic software specification languages - see www.cs.ucsd.edu/~goguen/sys/obj.html - were based on it. So, I believe, is a current language, the Common Algebraic Specification Language, described at www.informatik.uni-bremen.de/cofi/wiki/index.php/CASL .

The author of www.math.mq.edu.au/~street/Boeing.html , Michael Healy, has experimented with colimits for combining not only logical specifications, but also neural nets. He’s written this up at “Category Theory Applied to Neural Modeling and Graphical Representations”. (In “Proceedings of the International Joint Conference on Neural Networks”, 2000; available at Citeseer and CiteseerX.). He sets up a category of “concepts” (theories) and a category whose objects are nodes in a neural network, and has functors that map arrows representing the “is a subconcept of” relation onto paths in the network.

He also uses natural transformations to weld different mappings of concept hierarchy-to-network into one. I don’t know whether Boeing is using any of this work. I’d really like to see it more generally known, though, because it might inspire other neural network researchers to try the same kind of thing. Neural networks badly need such a principled approach, and someone ought to be trying to promote it to them.

Incidentally, Ronald Brown and Timothy Porter make some suggestions about colimits for understanding brain activity in terms of the component neurons: “Category Theory and Higher Dimensional Algebra: potential descriptive tools in neuroscience”. arxiv.org/PS_cache/math/pdf/0306/0306223v1.pdf .

Jocelyn Paine

Posted by: Jocelyn Paine on July 22, 2008 6:23 PM | Permalink | Reply to this

Pathological curve; Re: An Invitation to Higher Dimensional Mathematics and Physics

I’ve got to ask here. I just drove my son back to USC tonight, after a weekend visit home. I’ve mentioned him here before: eighteen years old, already has B.S. in Math and B.S. in Computer Sci, and is youngest American in a top-10 law school. Anyway, I just told him, and he immediately got it and drew corollaries.

I said that one can have a curve in an infinite dimensional space (can’t in finite dimensional space) where the curve can be considered parametrically as a point moving in time, where at every instant it moves perpendicular to all motion in its past, i.e. to every direction that it had moved before.

Now I can’t find the citation. Weirdly, when I googled on key words from the above, up popped a paragraph from a 1987 interview with Jerry Garcia of the Grateful Dead…

When I was at Boeing 1979-1984 (in Kent Space center, and at JPL) I never heard anyone mention Category Theory, except myself… But may have been in the wrong places. They did shut down Boeing Research labs in Washington State when I was there in the aerospace bust that had the billboard saying: “will the last person leaving Seattle please turn off the light.”

Posted by: Jonathan Vos Post on October 15, 2007 9:50 AM | Permalink | Reply to this

Kenneth R. Davidson; Re: Pathological curve; Re: An Invitation to Higher Dimensional Mathematics and Physics

Found it! In “An Elementary and Superficial Introduction to Operator Algebra”, by Kenneth R. Davidson of the University of Waterloo, FieldsNotes, Sep 07, Volume 8:1, Fields Institute, reviewed by Juris Steprans, p.4:

“As an illustration of the unexpected behaviour encountered in infinite dimensional spaces, Davidson posed the following question:

Can a continuous curve be drawn in Hilbert space which is always moving perpendicular to earlier portions of the curve? Since the question refers to curves that are merely continuous, but not differentiable, a precise formulation asks for a curve such that the lines connecting distinct pairs of points along the curve are perpendicular so long as the two intervals along the curve between these two pair of points are disjoint. In finite dimensional Euclidean space this is impossible because there is not enough room in k dimensional space to have even k+1 mutually perpendicular lines. However, Hilbert space does allow for such exotic behaviour. There is a wrinkled curve which, loosely speaking, moves in a direction perpendicular to all previous directions at any point in time.”

The musical quote:

Jerry Garcia Interview
November 12, 1987
Part 2 of 4

http://www.yoyow.com/marye/garcia2.html

“Have you ever checked out any of the Grateful Dead clone bands?

Yeah, but I don’t think that’s where it’s at, exactly. Really, it’s people who have to invent their own version of what the Grateful Dead is, starting now. Not doing what we’ve done, but digging the way we’ve done it, and doing what they’re going to do – continuing this notion.
But somebody else has to see that – it isn’t going to work just by following our footsteps. It’s gonna work by taking off perpendicular to every direction we’ve gone off in.”


I feel that this begs for categorification.

Posted by: Jonathan Vos Post on October 16, 2007 8:57 AM | Permalink | Reply to this

Re: Former comments with an explosively huge subject line

I don’t know much about this stuff, but I’m wildly guessing this is related to “the resolution of the identity” as a technique in the Borel functional calculus. In this technique, two disjoint Borel subsets of the real line correspond to Hilbert space projections which are orthogonal. So if you consider the curve (defined wrt non-zero vv in an infinite-dimensional Hilbert space HH)

ϕ:tχ (,t)(1 H)(v)\phi: t \mapsto \chi_{(-\infty, t)}(1_H)(v)

where χ E\chi_E denotes the characteristic function of a Borel set EE, then I guess ϕ(w)ϕ(u)\phi(w) - \phi(u) is orthogonal to ϕ(t)ϕ(s)\phi(t) - \phi(s) for ss < tt < uu < ww.

Again, I don’t know much about this, but I suspect this resolution of the identity is also connected with this continuous geometry business (one can define continuous chains of subspaces by considering images of operators χ (,t)(1 H)\chi_{(-\infty, t)}(1_H), I guess).

So while Jonathan was just kidding (I think!) about categorifying Jerry Garcia’s comments, I wouldn’t be surprised if large portions of this kind of classical functional analysis really was begging for categorification!

Posted by: Todd Trimble on October 16, 2007 5:45 PM | Permalink | Reply to this

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