## October 15, 2007

### This Week’s Finds in Mathematical Physics (Week 257)

#### Posted by John Baez

In week257 of This Week’s Finds, watch a sphere turn inside out, find out where dust came from in the early universe, and explore the Red Rectangle:

Then, learn how the integers secretly form a three-dimensional space, with prime numbers resembling knots. Read about some new work applying topos theory to quantum mechanics. Hear what Eugenia Cheng told me about monads on a train to Sheffield. And finally — watch the Tale of Groupoidification on video!

Posted at October 15, 2007 1:51 AM UTC

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### Re: This Week’s Finds in Mathematical Physics (Week 257)

Streater’s causes (lost and otherwise) can be found online here.

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Jeremy Bentham wasn’t mummified, he was stuffed! It’s an easier process, less likely to go wrong, and leaves the corpse looking a lot more lifelike.

(By one of those bizarre coincidences, I was just talking about this exact same thing in an entirely different context on usenet … )

Posted by: Tim Silverman on October 15, 2007 10:36 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Hmm. Everyone else told me he was mummified. Wikipedia seems agnostic on this issue, but entertaining nonetheless:

As requested in his will, his body was preserved and stored in a wooden cabinet, termed his “Auto-Icon”. Originally kept by his disciple Dr. Southwood Smith, it was acquired by University College London in 1850. The Auto-Icon is kept on public display at the end of the South Cloisters in the main building of the College. For the 100th and 150th anniversaries of the college, the Auto-Icon was brought to the meeting of the College Council, where he was listed as “present but not voting”. Tradition holds that if the council’s vote on any motion is tied, the auto-icon always breaks the tie by voting in favor of the motion.

The Auto-Icon has always had a wax head, as Bentham’s head was badly damaged in the preservation process. The real head was displayed in the same case for many years, but became the target of repeated student pranks including being stolen on more than one occasion. It is now locked away securely.

Where is Jeremy Bentham’s head kept? Inquiring minds want to know!

Posted by: John Baez on October 16, 2007 1:35 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

I just want to point out that “Jeremy Bentham’s Head” would make a great band name.

Posted by: John Armstrong on October 16, 2007 2:01 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Yeah, it would be great: then you could say things like “Last Friday I saw Jeremy Bentham’s Head in Amersham Arms, over on Newcross Road…”

And, they could do a song that goes like this:

Now, now now now it’s outside, looking in.

(Readers may need to be of a certain age to get that… I’m not sure.)

Posted by: John Baez on October 16, 2007 5:06 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

“… He’ll fly his astral hyperplane,
Takes you trips around the bay,
Brings you back the previous day….
Along the coast you’ll hear them boast
About a light they say that shines so clear.
So raise your glass, we’ll drink a toast
To the Auto-Icon who sells you thrills
along the pier.”

But I’m leery of such musical references to the younger generation.

Posted by: Jonathan Vos Post on October 16, 2007 9:05 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

John A, you’re not the first to think of that.

John B and Jonathan: I am of a certain age, but failed to recognize the reference, and googled to no avail. Please help!

Posted by: Todd Trimble on October 16, 2007 1:07 PM | Permalink | Reply to this

### Song (Spoiler warning); Re: This Week’s Finds in Mathematical Physics (Week 257)

Moody Blues, In Search Of The Lost Chord, 1968
Deram Records
Producer: Tony Clarke
Cover: Philip Travers

Legend Of A Mind
(Ray Thomas)

No, no, no, no, He’s outside looking in.
No, no, no, no, He’s outside looking in.
He’ll fly his astral plane,
Takes you trips around the bay,
Brings you back the same day,
Timothy Leary. Timothy Leary…

Hence my pun on “leery” was a hint.

I knew Tim Leary, had many nice conversations with him over the years, including at hackers 2.0 (or was it Hackers 4.0). Fascinating man, Wonderful song. But 1968, man. If you remeber it well, in context, you weren’t really in the sixties…

Posted by: Jonathan Vos Post on October 16, 2007 5:35 PM | Permalink | Reply to this

### Re: Song (Spoiler warning); Re: This Week’s Finds in Mathematical Physics (Week 257)

Oh, damn it! Of course I’ve heard that song many times (without paying attention to the lyrics, however).

Thanks for the straight dope, man.

Posted by: Todd Trimble on October 16, 2007 5:50 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

I did my postgrad study at UCL — his head is kept in a vacuum-sealed glass jar which, bizarrely enough, used to sit in-between his feet. It’s now kept locked away in the college vaults.

They had a picture of this old arrangement pinned up on the box a few years ago when Jeremy had been taken away to have his straw matting replaced (it had become infested). The collected ensemble looked slightly ghoulish to say the least!

Later I heard (and this may well be apocryphal) that the reason the head was taken off display in the case was that it had been kidnapped by a group of gadabouts from King’s College London as a merry rag-week stunt. One version of the story has them playing football with it (surely not true!)

Posted by: Allan E on October 16, 2007 3:07 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Yargh! Found an image of JB’s (Jeremy Bentham I mean!) auto-icon complete with head!

Posted by: Allan E on October 16, 2007 3:24 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

JB assured me

Everyone else told me he was mummified.

Noooo! Not another thing I know that ain’t so! At this rate, my quest for knowledge will reach completion when I know nothing at all except that I know nothing. What a dismal prospect!

Posted by: Tim Silverman on October 16, 2007 9:50 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Tim wrote:

Noooo! Not another thing I know that ain’t so!

Wait a minute there — don’t jump off that bridge yet! You could be right — it’s possible the people I knew just took one look at Bentham’s corpse and thought “mummy”, without bothering to investigate precisely how it reached its current condition.

He’s listed in the Wikipedia article on self-mummification:

In the 1830s, Jeremy Bentham, the founder of utilitarianism, left instructions to be followed upon his death which led to the creation of a sort of modern-day mummy. He asked that his body be displayed to illustrate how the “horror at dissection originates in ignorance”; once so displayed and lectured about, he asked that his body parts be preserved, including his skeleton (minus his skull, for which he had other plans), which were to be dressed in the clothes he usually wore and “seated in a Chair usually occupied by me when living in the attitude in which I am sitting when engaged in thought.” His body, outfitted with a wax head created because of problems preparing it as Bentham requested, is on open display in the University College London.

But, does this really rule out the possibility that his innards were removed and his body stuffed? I don’t know.

Posted by: John Baez on October 17, 2007 4:20 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

A few addenda that folks may have missed if they read this Week’s Finds shortly after I posted it:

1. The draft paper by Kapranov and Smirnov is available on the web:

It begins:

The analogies between number fields and function fields have been a long-time source of inspiration in arithmetic. However, one of the most intriguing problems in this approach, namely the problem of the absolute point, is still far from being satisfactorialy understood. The scheme $Spec(\mathbb{Z})$, the final object in the category of schemes, has dimension 1 with respect to the Zariski topology and at least 3 with respect to the etale topology. This has generated a long-standing desire to introduce a more mythical object $P$, the “absolute point”, with a natural morphism $X \to P$ given for any arithmetic scheme $X$ […]

2. For the proof of the Gelfand-Naimark theorem inside a topos, see:

They show that any commutative C*-algebra A in a Grothendieck topos is canonically isomorphic to the C*-algebra of continuous complex functions on the compact, completely regular locale that is its maximal spectrum (that is, the space of homomorphisms $f: A \to \mathbb{C}$). Conversely, they show any compact completely regular locale X gives a commutative C*-algebra consisting of continuous complex functions on X. Even better, they explain what all this stuff means.

3. I wrote “monad on a category” at various points where I should have written “monad in a 2-category” in discussing the work of Street and Cheng. It’s fixed now.

Posted by: John Baez on October 16, 2007 1:47 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

By the way, the Kapranov–Smirnov paper talks about the field with one element!

Posted by: John Baez on October 16, 2007 1:49 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

I remember being entranced by one of your earlyish TWFs where you point out how two interesting things that can happen to 1 dimensional things in a plane correspond to important category theoretic constructions: pulling a zigzag straight and adjunctions; two lines merging and monoidal product.

It made me wonder at the time if you could list all the interesting things which could happen to surfaces in a higher space and then search for all the corresponding interesting constructions in 2-categories.

It looks like this work of Eugenia you report is a step in this direction. Is there anything to look for beyond ways that instances of three planes merging can be moved about, and the removal or creation of various folds, caps and cups?

What class of surfaces is this that allows three half-planes to meet?

Posted by: David Corfield on October 16, 2007 11:18 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Branched surface, it looks like. But can anything else interesting happen?

Posted by: David Corfield on October 16, 2007 3:42 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

The work of Eugenia is certainly highly relevant. The class of diagrams to which I think you (David) may be referring were dubbed “surface diagrams” by Dominic Verity, and AFAIK were first developed in conversations between Verity and Street back in 1994 or 1995; somewhat later I began working on a general theory of them in (still unpublished) work with Margaret McIntyre.

Examples of three planes coming together can be related both to associators and to Yang-Baxterators. If for example you take an arrangement of 3 planes in 3-space (in generic position), and consider the movie whose frames are slices by a generic pencil of parallel planes, the slices are frames of a Reidemeister III move (naturally, these frames are planar projections of tangles – to get a truer picture one must consider an arrangement of three 2-planes in 4-space, as in the Baez-Langford paper). In the 3-space picture, this arrangement would be interpreted as a 3-cell in a Gray-category, which I am calling a Yang-Baxterator.

There is a surface diagram sense in which an associator is a truncated Yang-Baxterator: if instead of three planes, you take one full plane (say $z = 0$), and then half of a second plane on one side of the first plane (say $y = 0$ and $z \geq 0$), and then a quarter of a third plane (say $x = 0$ and $y, z \geq 0$), then the movie whose frames are slices by a generic pencil of parallel planes is the movie of an associator.

These and many other examples (such as the swallowtail coherence condition treated by Baez and Langford) were considered by Verity and Street and presented in the Australian Category Seminar, and also (by Verity) at the 1995 Category Theory Conference in Halifax.

It’s maybe a little unseemly for me to say this, but among other things Eugenia in her paper is moving toward a precise comparison between “enrichment-style” notions of weak $n$-category whose first example was given by me, and the class of notions considered by Batanin, by showing precisely how my and similar enrichment-style notions form contractible globular operads. She and Nick Gurski are also working out an enrichment-style notion (modifying my original notion) of cobordism $n$-categories. It seems to me that surface diagrams also fit very naturally in the enrichment approach to $n$-categories, something I hadn’t noticed until I looked at what Eugenia and Nick are doing; in this sense what they are doing is highly relevant to your comment.

Posted by: Todd Trimble on October 16, 2007 4:25 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

David wrote:

What class of surfaces is this that allows three half-planes to meet?

They’re not quite the answer to your question, but I urge you to learn about ‘fake surfaces’.

The reason is that:

1. Fake surfaces look locally just like the surfaces you generically see in soap suds! Three half-planes can meet along an edge, but also four half-planes can meet at a vertex.
2. Every compact 3-manifold contains a ‘special spine’ — a fake surface with some special properties, from which you can reconstruct the 3-manifold. Even better, two fake surfaces give the same 3-manifold if they’re related by some nice moves: the ‘Matveev moves’!
3. Fake surfaces are just what you need for drawing the 2-morphisms in 2Vect that you can build from a ‘semisimple 2-algebra’.

For the definition of ‘fake surface’, ‘special spine’ and ‘Matveev moves’, with pictures, see page 3 in week 4 of the 2005 quantum gravity seminar notes. For more on semisimple 2-algebras, see week 3. Also check out this note on the associator, $6j$ symbols and abstract index notation for categorified tensors.

But, there’s a lot more room for exploring categorified algebraic structures using specific classes of surfaces! I think people should dive in and do this. If I had more time, I’d do it myself.

Posted by: John Baez on October 16, 2007 7:24 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

The drawing at the lower right of the second page of the PDF is isomorphic to a pentatope . The comment on “not true in 4D” relates also to polyhedra able to be triangulated, but not always tetrahedralized.

n-category diagrams drawn on the shimmering iridescent surfaces of soap bubbles. I really want to that in full-color high-res video!

Didn’t you show us the soap bubble 120-cell?

Posted by: Jonathan Vos Post on October 16, 2007 7:49 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Jonathan vos Post wrote:

The comment on “not true in 4D” relates also to polyhedra able to be triangulated, but not always tetrahedralized.

No.

Didn’t you show us the soap bubble 120-cell?

Yes.

Posted by: John Baez on October 17, 2007 6:19 PM | Permalink | Reply to this

### Nontetrahedralizability; Re: This Week’s Finds in Mathematical Physics (Week 257)

I got 50% on this exam from John Baez.

I must have wrongly intuited that nontetrahedralizability was the obstruction to generalizing the fake surfaces to (n+1)-categories.

So is it a mere coincidence that:

I got 50% on my last exam from Lisa Randall, at a Kip Thorne seminar at caltech, when she agreed with my first brane-world hypothesis, but dismissed as mere coincidence the dominance of 3-branes and 7-branes in her model at the time and my pointing out that only 3 and 7 dimensional Euclidean spaces have cross-products.

Posted by: Jonathan Vos Post on October 18, 2007 8:52 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

They’re not quite the answer to your question, but I urge you to learn about ‘fake surfaces’.

It’s an answer to the anything else part of my question. So thanks.

Fake surfaces are just what you need for drawing the 2-morphisms in 2Vect that you can build from a ‘semisimple 2-algebra’.

Is there a more generic way of saying where they’re useful? E.g., just like string diagrams for adjunctions are not just relevant to adjunctions between categories, but also to adjunctions in any bicategory.

How could we know that there isn’t a further interesting class of surface, called perhaps ‘pseudo-fake surfaces’, corresponding to algebraically interesting things occurring?

Will novel things emerge in higher dimensions, or will it just be about coherence relations of constructions from lower dimensions?

Posted by: David Corfield on October 17, 2007 12:25 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

John wrote:

Fake surfaces are just what you need for drawing the 2-morphisms in 2Vect that you can build from a ‘semisimple 2-algebra’.

David wrote:

Is there a more generic way of saying where they’re useful?

Good point — why limit ourselves to the world of linear algebra?

Probably fake surfaces are 2-morphisms in something like the ‘free symmetric monoidal 2-category on a symmetric Frobenius 2-algebra’.

A Frobenius algebra sounds like something in the world of linear algebra, but it’s actually something we can define in any monoidal category: we get an ordinary Frobenius algebra when we take that monoidal category to be Vect. You’ve seen what kind of pictures correspond to morphisms in the ‘free monoidal category on a Frobenius algebra’ — they’re in week224.

But this is just part of a big periodic table of Frobenius gadgets!

The best introduction so far may be this:

Aaron Lauda, Frobenius algebras and planar open string topological field theories, available as
math.QA/0508349.

After considering some baby cases, he draws the pictures for 2-morphisms in the ‘free monoidal 2-category on a Frobenius 2-algebra’. (He doesn’t use quite the same jargon, but never mind.)

Going symmetric would add extra wiggle room.

I’m not sure I’ve got the details exactly right, but fake surfaces will certainly be the 2-morphisms in a nice ‘free blah-di-blah on a blee-di-blee’, and I think I’m pretty close.

The main thing is that the tensor product in our 2-algebra gives us surfaces where 3 half-planes meet along a line, while the associator gives us surfaces where 4 half-planes meet at a vertex. The ‘Frobeniusness’ means we can freely switch inputs and outputs — it’s a way of getting our 2-algebra to be its own dual.

Posted by: John Baez on October 17, 2007 6:01 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

I am sorry for a little offtopic but maybe you know some good online category theory tutorials (including the introduction to applications of category theory) for a begginer (who has some mathematical and philosophical background)?

Posted by: modest_philosopher on October 16, 2007 11:13 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

I like the online book by Goldblatt as an introduction. Don’t be scared by the title. It’s also available quite cheap from Dover Publications.

There are also a bunch of texts available online here. Of these, the book by Adamek, Herrlich and Strecher may be easiest, followed by the lecture notes by Barr and Wells, followed by the book by Turi. Your opinion may differ!

If all these are too hard, get ahold of Lawvere and Schanuel’s Conceptual Mathematics: a First Introduction to Categories.

Posted by: John Baez on October 17, 2007 8:27 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Is it just me, or are other people finding that the fascinating video on turning a sphere inside out always freezes at about six and a half minutes, just when it’s starting to get interesting?

Posted by: Greg Egan on October 17, 2007 3:42 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Greg wrote:

Is it just me, or are other people finding that the fascinating video on turning a sphere inside out always freezes at about six and a half minutes, just when it’s starting to get interesting?

This never happens to me — I just watched it again, and nothing bad happened.

There’s a slider underneath the video screen. What happens if you pause the movie, slide the slider past the 6:30 point, and restart the movie?

(We seem to be talking a lot about the difficulties of watching videos these days. I guess that’s a good thing — it means we’re going through yet another technology change. In a few years we’ll look back on this and laugh… I hope. I remember the issue of This Week’s Finds where I first talked about the World Wide Web. And the one where I first talked about search engines.)

This movie has some nice examples of ‘proof through discussion’ — not rigorous formal proofs, but people talking about a fact that seems surprising at first, analyzing it until it seems undeniably true. It reminds me a bit of Yatima learning the Gauss–Bonnet theorem in Diaspora.

So, I hope you get to see the whole thing.

Posted by: John Baez on October 17, 2007 6:25 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Thanks, John, your suggestion worked! Well, I also had to re-boot a few times when the Flash plug-in seemed to corrupt the OS at various points later on, but I managed to see the whole thing by skipping past the trouble spots each time I started again.

Anyway, it was definitely worth the effort!

Posted by: Greg Egan on October 17, 2007 8:40 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

I’m a bit surprised everyone doesn’t already know about Outside In. If you didn’t know about it, you might like the other of the Geometry Center’s videos, Not Knot (and part 2 of same).

Posted by: John Armstrong on October 17, 2007 12:50 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

John A. wrote:

I’m a bit surprised everyone doesn’t already know about $n$-categories! But as long as there are guys holed up near the Afghanistan–Pakistan border who don’t know and love everything about math and physics that I do, I’m gonna keep writing This Week’s Finds and dropping it in leaflet form out of helicopter windows worldwide. I’ll get you eventually, Osama!

Greg writes:

Thanks, John, your suggestion worked! Well, I also had to re-boot a few times when the Flash plug-in seemed to corrupt the OS at various points later on, but I managed to see the whole thing by skipping past the trouble spots each time I started again.

Anyway, it was definitely worth the effort!

Wow, that’s quite a compliment — it must have been good if it was worth that much effort. Corrupting the operating system?? What Flash plug-in are you talking about? I didn’t even know Flash was needed to watch this stuff. I’d like to make the technical prerequisites as low as possible.

Maybe I should learn to convert .mov files to some other formats…

Posted by: John Baez on October 17, 2007 7:30 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Greg: do you think your problems could have been caused by bit errors in the downloaded file?

Posted by: John Baez on October 17, 2007 7:32 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

John wrote:

What Flash plug-in are you talking about? I didn’t even know Flash was needed to watch this stuff.

The google video page you linked to contains an embed element, with the media type “application/x-shockwave-flash”. There was a link there also for downloading the file in “Video iPod” format; I don’t have an iPod, but that format turned out to be MP4. However, when I downloaded the file it turned out to be much smaller than the download dialogue initially suggested, which implied that the server had truncated the download prematurely, so I didn’t make the effort of hunting down software for playing MPEG4 files to see if that would work. (The standard QuickTime player doesn’t.)

Greg: do you think your problems could have been caused by bit errors in the downloaded file?

I don’t know. The freezes and OS corruption weren’t exactly repeatable; there was no precise image in the Outside In movie that would always cause problems. But if I started watching from the beginning, something would always go wrong at about 6 m 30 s.

Maybe I should learn to convert .mov files to some other formats…

Well from my point of your view, your files are just fine. When I download the .mov files from the UCR server, QuickTime player plays them very nicely.

Posted by: Greg Egan on October 18, 2007 12:04 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Greg Egan wrote:

Well from my point of your view, your files are just fine…

Aargh, I was getting them mixed up just then — I could blame it on waking up too early this morning, but it’s probably just incipient senility. Anyway, I hadn’t known those Google video pages invoked ‘x-shockwave-flash’. That’s annoying. But, I’m glad my files are working fine. I’m still wondering if I could compress them a bunch while maintaining readability…

Posted by: John Baez on October 18, 2007 1:10 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Here are some interesting comments emailed to me by Jordan Ellenberg, who has given me permission to append them to week257:

As always, an enjoyable This Week’s Finds!

1. In the viewpoint of Deninger, very badly oversimplified, $Spec \mathbb{Z}$ is to be thought of not just as a 3-manifold but as a 3-manifold with a flow, in which the primes are not just knots, but are precisely the closed orbits of the flow!

2. One thing to keep in mind about the analogy is that “the complement of a knot or link in a 3-manifold” and “the complement of a prime or composite integer in $Spec \mathbb{Z}$” (which is to say $Spec \mathbb{Z}[1/N]$ ) are both “things which have fundamental groups,” thanks to Grothendieck in the latter case. And much of the concrete part of the analogy (like the stuff about linking numbers) follows from this fact.

3. On a similar note, a recent paper of Dunfield and Thurston which I like a lot, “Finite covers of random 3-manifolds,” develops a model of “random 3-manifold” and shows that the behavior of the first homology of a random 3-manifold mod $p$ is exactly the same as the predicted behavior of the mod $p$ class group of a random number field under the Cohen–Lenstra heuristics. In other words, you should not think of $Spec \mathbb{Z}$ or $Spec \mathbb{Z}[1/N]$ as being anything like a particular 3-manifold – better to think of the class of 3-manifolds as being like the class of number fields.

Posted by: John Baez on October 17, 2007 4:37 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

John, you said:

This is true for each prime $p$. But the integers, $\mathbf{Z},$ are more complicated than any of these $\mathbf{Z}/p$’s. To be precise, we have maps

$\mathbf{Z} \to \mathbf{Z}/p$

for each $p$. So, if we think of $\mathbf{Z}$ as a kind of space, it’s a big space that contains all the “planes” corresponding to the $\mathbf{Z}/p$’s. So, it’s 3-dimensional!

In short: from the viewpoint of ï¿½tale topology, the integers have one dimension that says which prime you’re at, and two more coming from the plane-like nature of each individual $\mathbf{Z}/p$.

I’m going to be a bit abrupt here and say that I think this point of view is wrong. Or at least it is so different to what I’m used to thinking that I would need much more detail to be convinced there is something to it.

Let me explain what I think is wrong and what I think the right way of thinking about it is.

First of all, you’ve used $\mathrm{Z}/p$ to mean two different things, and I think you’ve conflated them. There is $\mathrm{Spec}(\mathbf{Z}/p)$ and then there is the set of points of the affine line $\mathrm{Spec}(\mathbf{Z}/p[x])$ with coordinates in $\mathbf{Z}/p$. This is like confusing a point, whose function algebra is $\mathbf{C}$, with the space $\mathbf{C}$, whose function algebra is way bigger.

But you’re interested in $\mathrm{Spec}(\mathbf{Z})$. As you say, there are closed subspaces coming from the Spec$(\mathbf{Z}/p)$’s. Note that the affine line is nowhere in sight.

What about each $\mathrm{Spec}(\mathbf{Z}/p)$? They are each 1-dimensional from the point of view of the etale topology. Why? It is a basic fact, almost by definition, that a sheaf on $\mathrm{Spec}$ of a field is the same thing as a $G$-set, where $G$ is the absolute Galois group of the field. Therefore the etale cohomology is just the group cohomology of $G$. Therefore the etale cohomolical dimension of $\mathrm{Spec}$ of a field is the same thing as the cohomological dimension of its absolute Galois group. What is the abolute Galois group of a finite field? It’s the cyclic group freely generated by the Frobenius operator. And free cyclic groups have cohomological dimension 1. (I’m being a bit loose: there is more than one notion of cohomological dimension, and the absolute Galois group of a finite field is actually the free pro-cyclic group on one generator, but since we always look at cohomology with finite coefficients, we can ignore this.)

In other words, the $\mathbf{Z}/p$’s that cover $\mathbf{Z}$ are each one-dimensional, not two-dimensional.

So then why should there be the two dimensions of primes needed to make $\mathrm{Spec}(\mathbf{Z})$ three-dimensional? I don’t think there is a pure-thought answer to this question. As you wrote, there is a scientific answer in terms of Artin-Verdier duality, which is pretty much the same as class field theory. There is also a pure-thought answer to an analogous question. Let me try to explain that.

Instead of considering $\mathbf{Z}$, let’s consider $F[x]$, where $F$ is a finite field. They are both principal ideal domains with finite residue fields, and this makes them behave very similarly, even on a deep level. I’ll explain why $F[x]$ is three-dimensional, and then by analogy we can hope $\mathbf{Z}$ is, too. Now $F[x]$ is an $F$-algebra. In other words, $X=\mathrm{Spec}(F[x])$ is a space mapping to $S=\mathrm{Spec}(F)$. I already explained why $S$ is a circle from the point of view of the etale topology. So, if $X$ is supposed to be three-dimensional, the fibers of this map better be two-dimensional. What are the fibers of this map? Well, what are the points of $S$? A point in the etale topology is $\mathrm{Spec}$ of some field with a trivial absolute Galois group, or in other words, an algebraically closed field (even better, a separably closed one). Therefore a etale point of $S$ is the same thing as $\mathrm{Spec}$ of an algebraic closure $\bar{F}$ of $F$. What then is the fiber of $X$ over this point? It’s $\mathrm{Spec}$ of the ring $\bar{F}[x]$. Now, *this* is just the affine line over an algebraically closed field, so we can figure out its cohomological dimension. The affine line over the complex numbers, another algebraically closed field, is a plane and therefore has cohomological dimension 2. Since etale cohomology is kind of the same as usual singular cohomology, the etale cohomological dimension of Spec$(\bar{F}[x])$ ought to be 2.

Therefore $X$ looks like a 3-manifold fibered in 2-manifolds over Spec$(F)$, which looks like a circle. Back to Spec$(\mathbf{Z})$, we analogously expect it to look like a 3-manifold, but absent a (non-formal) theory of the field with one element, $\mathbf{Z}$ is not an algebra over anything. Therefore we expect Spec$(\mathbf{Z})$ to be a 3-manifold, but not fibered over anything.

Posted by: James on October 17, 2007 2:00 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

In case anyone out there is wondering, there are three guys named James who post to the $n$-Café: Jim Stasheff, James Dolan, and this James, who I plan to pester with questions about number theory / algebraic geometry.

James: thanks for these corrections and clarifications! None of my errors and oversimplifications should be blamed on Minhyong Kim.

It will take me time to ask really good questions, so for now I’ll just say some random crud…

First of all, you’ve used $\mathbb{Z}/p$ to mean two different things, and I think you’ve conflated them.

You’re right, I completely screwed up! I was desperate to get that third dimension. So, I started trying to imagine each $Spec \mathbb{Z}/p)$ as a punctured surface containing a highly metaphorical ‘circle’ generating its nontrivial $\pi_1$ (which I know is really best understood in terms of deck transformations of covers, or more precisely still, the Frobenius). But you’re right: Riemann surfaces are analogous to some things in number theory, but not to $\mathbb{Z}/p$. $\mathbb{Z}/p$ is more like a circle itself than a surface containing a circle.

So then why should there be the two dimensions of primes needed to make $Spec(\mathbb{Z})$ three-dimensional? I don’t think there is a pure-thought answer to this question.

This is the only point where I feel compelled to disagree. There may not be a pure-thought answer yet, but the integers are too important for us to rest content until we have a simple story about why their spectrum is 3-dimensional. As long as we say ‘oh, it’s just a complicated calculation’, we won’t get the gut-level intuitive understanding needed to make certain kinds of progress.

I’m probably too stupid to discover this simple story myself, but I hope that by continuing to annoy experts such as yourself, I’ll eventually get one of you to cough it up.

As you wrote, there is a scientific answer in terms of Artin-Verdier duality, which is pretty much the same as class field theory.

I guess for now I should start learning Artin–Verdier duality. I didn’t know it was pretty much the same as class field theory, so that’s a very helpful hint — thanks! Class field theory is right about where my self-education in number theory starts fizzling out, but James Dolan spent about a year teaching it to me, and we developed some nice intuitive ways to think about it, so you’re actually making me optimistic that I can not only learn but grok Artin–Verdier duality, and then maybe comprehend the 3rd cohomology of $Spec(\mathbb{Z})$.

But I take your point that I should try to learn about $Spec(F_q[x])$ in parallel, and maybe first.

…absent a (non-formal) theory of the field with one element, $\mathbb{Z}$ is not an algebra over anything.

Hmm, so there it is again: the mystical field with one element! I should try to apply Durov’s theory of generalized rings, which is his attempt to handle the field with one element, to see if I can do anything here.

Anyway, thanks a million, and I hope you continue to correct me as I continue to blunder like a drunken bull into this beautiful china shop.

Posted by: John Baez on October 17, 2007 7:15 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

John said:

Anyway, thanks a million, and I hope you continue to correct me as I continue to blunder like a drunken bull into this beautiful china shop.

Well, I had to redeem myself after that embarrassment about the non-naturality of the map from direct sums to direct products.

This is the only point where I feel compelled to disagree. There may not be a pure-thought answer yet, but the integers are too important for us to rest content until we have a simple story about why their spectrum is 3-dimensional. As long as we say ‘oh, it’s just a complicated calculation’, we won’t get the gut-level intuitive understanding needed to make certain kinds of progress.

I agree completely. And it’s good to see you’re turning into a *real* number theorist who’s interested in $\mathbf{Z}$, and not one of those wimpy guys like Drinfeld and Lafforgue who study function fields. :)

Here is a reason on a very gut level. First consider a manifestly geometric example. A point has dimension 0, a punctured disk has dimension 1, and a Riemann surface has dimension two. Going up in dimension comes first from local winding and then globalization.

Similarly, a finite field has dimension one. A local field, such as $\mathbf{Q}_p$ or Laurent series $\mathbf{F}_q((x))$ has one higher dimension, the extra dimension coming from local winding, or ”ramification” as those in the business call it. A ring of integers in a global field, for example $\mathbf{Z}$ or $\mathbf{F}_q[x]$, has one more dimension coming from globalization.

So if you allow yourself to think of $\mathbf{Z}$ as being the ring of functions on something like a Riemann surface, then it’s all the usual stuff except that now your points are 1-dimensional, so everything else gets bumped up by one.

I don’t know if you like that better (or worse, for that matter).

Posted by: James on October 18, 2007 8:44 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Conc.: “absent a (non-formal) theory of the field with one element,”

Harans article looks very interesting.

Posted by: T.R. on October 18, 2007 12:20 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

For anyone who is interested in converting arithmetic into geometry, I highly recommend reading the intro of this paper. The rest of the paper is quite dense and I can’t understand it due to a lack of both expertise and time (but perhaps James could understand it).

Posted by: Charlie Stromeyer Jr on October 18, 2007 5:19 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

My friend the number theorist Minhyong Kim has a new blog where he answers math questions, so I posted a question about why Spec($\mathbb{Z}$) is 3-dimensional, and he posted an interesting reply.

David Corfield (and others) might also like Minhyong’s essay on Mathematical Vistas.

Posted by: John Baez on December 16, 2007 1:55 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

In case some other people actually read that reply, I’ll add two quick comments: I would not like to give the impression that the sentiment on F_1 towards the end is original to me. The consequences of `differentiating integers’ or having a diagonal are well-known among arithmetic geometers. Also, I myself do not make a fetish of the Riemann hypothesis, although over the years, I have come to recognize its importance. The reason I recommended focusing on definite problems was exactly as I wrote: The problems should help with the construction of a good theory.

MK

Posted by: Minhyong Kim on December 16, 2007 3:32 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

See, for example:

Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

and

A099379 The real part of n’, the arithmetic derivative for Gaussian integers.

Posted by: Jonathan Vos Post on December 18, 2007 8:43 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Minhyong wrote:

The consequences of ‘differentiating integers’ or having a diagonal are well-known among arithmetic geometers.

Since they’re completely unknown to me, could you maybe say a brief word about each? Nothing too technical, just the rough idea?

I’m wondering, for example, whether ‘differentiating integers’ has anything to do with the concept of ‘number derivative’ mentioned in — for example — this paper by a student of mine:

Maybe so, since he cites this paper:

in which the authors claim to be on the trail of something big, related to the field with one element:

Just as the function ring case we expect the existence of the coefficient field for the integer ring. Using the notion of one element field in place of such a coefficient field, we calculate absolute derivations of arithmetic rings. Notable examples are the matrix rings over the integer ring, where we obtain some absolute rigidity. Knitting up prime numbers via absolute derivations we speculate the arithmetic landscape. Our result is only a trial to a proper foundation of arithmetic.

Posted by: John Baez on December 19, 2007 10:19 PM | Permalink | Reply to this
Read the post A Topos for Algebraic Quantum Theory
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### Re: This Week’s Finds in Mathematical Physics (Week 257)

Read Lieven on 3-dimensional Spec($\mathbb{Z}$).

Posted by: David Corfield on December 27, 2008 1:17 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Isn’t it the case that every ring of algebraic integers is a homomorphic image of $\mathbb{Z}[X]$? If so, is there an object in arithmetic topology that corresponds to the ring of polynomials over $\mathbb{Z}$?

Posted by: Scott McKuen on December 27, 2008 11:12 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

From what I remember, this is not true, though I don’t have a ready counter-example. But it is true locally: the ring of integers in any finite extension of the field of $p$-adic numbers is a homomorphic image of $Z_p[x]$, where $Z_p$ denotes the ring of $p$-adic integers and where $p$ is a prime number.

Picturing the affine line over $Z$ (i.e. Spec $Z[x]$) from this topological point of view might be pushing the analogy beyond its breaking point. It’s tempting to view it as a family of affine lines parametrized by Spec $Z$—in other words, by our 3-manifold. But the topological affine line is just the usual space $C$ of complex numbers, so this fibration would have the same homotopy type as the 3-manifold. While I don’t know for a fact that $Z[x]$ and $Z$ don’t have the same etale homotopy type, it seems a bit unlikely to me. For instance for any finite field $F$, the rings $F[x]$ and $F$ don’t have the same etale homotopy type: because of wild ramification, the affine line isn’t simply connected in characteristic $p$!

On the other hand, I would bet that for the projective line $P^1$ over $Z$, the analogy is as good as the original one was. Then $P^1$ would be interpreted as a space fibered in 2-spheres (=topological projective lines) over the 3-manifold that is supposed to be Spec $Z$.

Posted by: James on December 28, 2008 5:03 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

James wrote:

While I don’t know for a fact that $\mathbb{Z}[x]$ and $\mathbb{Z}$ don’t have the same etale homotopy type, it seems a bit unlikely to me.

I don’t know anything about this stuff, but could this issue be related to $\mathrm{A}^1$ homotopy theory, where people try to use the affine line as the parameter space for a new kind of ‘homotopy’, applicable to algebraic geometry?

A bit more precisely: could it be that $\mathbb{Z}$ and $\mathbb{Z}[x]$ fail to be homotopy equivalent in etale homotopy theory, but become so in $\mathrm{A}^1$ homotopy theory?

Posted by: John Baez on December 28, 2008 5:49 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

I don’t really know anything about $A^1$-homotopy theory either, but vector bundles being homotopically equivalent to their zero sections sounds familiar, so I wouldn’t be surprised if what you said is true.

Posted by: James on December 29, 2008 3:33 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

Thomas Riepe, who is trapped someplace where he can’t post to the $n$-Café writes:

As someone not much interested in algebraic topology, I found chapter 5 of these lectures a very readable introduction in what $\mathrm{A}^1$ homotopy theory shall be.

Posted by: John Baez on December 31, 2008 7:32 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 257)

I found Sikora’s article interesting.

Posted by: Thomas on December 28, 2008 1:57 PM | Permalink | Reply to this

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