A new class of entries

I think I might like to spend some time thinking about definitions in mathematical physics. What is a quantum field, for instance? Physicists usually give a slightly incoherent answer: a quantum field is a quantum particle at every point, just like a field is a number at every point. You ask them to unpack this a bit, and some might remember that there may be global — what the physicists call "topological" — issues with such a definition, but for now let's only be concerned with the local definition, where a field is a function. So what should a quantum field be?

Conveniently, I'm taking three classes right now on related questions: Differential Geometry, Geometric Methods to ODEs, and Quantum Field Theory. I would like to start a series of entries blogging those classes, and relating it back to such foundational questions. I hope to get to answers involving infinitesimals: Robinson's "Non-standard Analysis", or Kock's "Synthetic Geometry". I don't have the answers yet.

What's most important about fields is their geometric nature. Like the physicists and the classical differential geometers, I may from time to time refer to coordinates, but ultimately I'd like a coordinate-invariant picture — indeed, one without coordinates at all. I also hope to ask and answer issues about how to regularize our fields, by which I mean "how continuous should they be?" This is an extremely non-trivial question: not only is it extremely unclear how to demand that two
"nearby" "quantum particles" be "similar" (we can demand as much of classical fields: for any epsilon, there should be a delta at each point so that within the delta ball at that point the fields don't vary more than epsilon; perhaps we should find the right metric on Schrodinger-quantized particles?), but the physicists don't even want to be stuck with, say, C^\infty fields. They want \delta functions to work within their formalism. And yet they adamantly refuse to consider "pathological" fields that are too "wildly varying".

Eventually, it would be nice also to understand the Lagrangian and Hamiltonian, and this almost-symmetry between position and momentum. For now, I'd like to end this entry with some basic definitions.


Manifolds: There are many equivalent definitions of a manifold. Since the physicists and classical geometers like to work with coordinates (replacing geometry-defined, invariant objects with coordinate-defined, covariant objects), I'll use the definition that mentions coordinates explicitly. A manifold is a (metrizable) topological space M with a maximal atlas — to each "small" open set U in M we assign a module of "coordinate patches" \phi:U\to\R^n, which should be homeomorphisms, subject to some regularity condition: if \phi:U\to\R^n and \psi:V\to\R^n, then \phi\psi^{-1} should be, say, smooth wherever it's defined. In general, modifying the word manifold modifies the condition on \phi\psi^{-1}: a C^\infty manifold has that all the \phi\psi^{-1}'s are C^\infty, for example. I will generally be interested only in C^\infty (aka "smooth") manifolds, although once we understand what kinds of functions the physicists are ok with, we may change that restriction. For a manifold, I demand that the atlas be maximal in the sense that it list all possible coordinatizations consistent with the smoothness condition. It is, of course, sufficient to simply cover our space with (coherent) patches, defining the rest as all other possibilities.

So that we can generalize this definition if we need to, it would be nice to reword this definition in the language of sheaves. The god-given structure on a smooth manifold is exactly enough to tell which functions are differentiable: a sheaf is a topological space along with a ring of "smooth functions" on each open set, so that the function rings align coherently (in full glory, a sheaf is a (contravariant) functor from the category of open sets in the space to the category of commutative \R-algebras whatever your sheaf is of, along with some "local" axioms, which ultimately say that to know a function I need exactly to know it on an open cover). I probably won't use this description, largely because I don't know what other conditions I would want to put on my sheaf in order to make it into something like a smooth manifold. Clearly every manifold generates a sheaf, and I have it on good authority that if two manifolds have the same sheaf, then they are the same manifold.

So what about our most-important of objects: a field? A field is a "section" of a "bundle".

Let's start with the latter of those undefined words. To each point p\in M, we associate a (for now) vector space V_p, called the "fiber at p". And let's (for now) demand some isotropy: V_p should be isomorphic to V_q for any given p and q in M, although not necessarily canonically so. (When we move to the realm of infinite-dimensional fibers, we may demand only that the fibers be somehow "smoothly varying" — I'm not sure yet how to define this. So long as everything is finite-dimensional, the isomorphism class of a fiber is determined by an integer, and integers cannot smoothly vary, so it suffices to consider bundles where the dimension of the fibers is constant.)

There should be some sort of association between nearby fibers: locally (on small neighborhoods U) the bundle should look like U\times V. So I ought to demand that the bundle be equipped with a manifold structure, which aligns coherently with M: a bundle E is a manifold along with a projection map \pi_E : E\to M, such that the inverse image of each point is a vector space. This is the same as saying that among the coordinate patches in E's atlas, there are some of the form \Phi: \pi^{-1}(U) \to \R^(n+k), (where, of course, n is the dimension of M and k is the dimension of each fiber) so that \Phi = (\phi,\alpha), where \phi is a coordinate patch on M and \alpha is linear on each fiber. We can naturally embed M\into E by identifying each point p\in M with (p,0) in E (where 0 is the origin of the fiber at p).

I will soon make like a physicist and forget about global issues, but I do want to provide one example of why global issues are important: the cylinder and the mobius strip are both one-dimensional ("line") bundles over the circle. The latter has a "twist" in it: as you go around the circle, you come back with an extra factor of -1.

So what's a section of a bundle? A (global) section is a map s:M\to E so that \pi s:M\to M is the identity, i.e. a section picks out one vector from each fiber. We will for now think of our sections as being C^\infty.


The most important kinds of fields are "scalar" fields, by which I exactly mean a function, i.e. a number at every point. I want to do this because I want to consider other spaces of fields as modules over the ring of scalar fields, so I need to be able to multiply. Of course, there are many times when I don't want a full-fledged scalar field. The potential energy, for instance, is only defined up to a constant: I will eventually need my formalism to accommodate objects that have fields as derivatives, but aren't fields themselves. Since potentials don't care about constants, we could imagine that after going around a circle we measure a different potential energy than we had to begin with, but that we never picked up any force. The string theorists, in fact, need similar objects: locally, string theory looks like (conformal) field theory on the string's worldsheet. But perhaps the string wraps around a small extra dimension? This is why in the previous paragraph I refer to "global" sections: I really ought to allow myself a whole sheaf of fields, understanding that sometimes I want to work with fields that are only defined in a local area. But the physicists are generally clever about this type of problem, so, at the risk of saying things that we might think generalize but actually don't, I'm going to restrict my attention to scalar fields.

In which case, yes, by "scalar field" I mean "function from M \to \R". "A section of M\times\R". "A number at each point". Those who prefer to start with the sheaf of scalar fields will be happy to know that, when I define tangent vectors and their relatives in the next entry, I will start with these scalar fields.

a statement of belief

Pacifism is something I've struggled with since at least mid high school. When the President started making waves about Iraq, the American Left moved strongly towards an isolationist/pacifist stance, and although I was nervous about the occasional paleoconservative philosophy, I was already on the bandwagon, having felt that the President's hasty response in Afghanistan was poorly executed, hasty, and morally questionable. At the same time, however, I was reading A Problem From Hell: America and the Age of Genocide by Samantha Power, a fantastic book by a New York Times writer that lays the blame for the 20th Century's genocides squarely at the feet of this country and its reluctance to involve itself militarily in foreign affairs.

I did, at the time, describe myself as "trying to move towards pacifism". In my case, it wasn't a question of will power, but of wrestling with the morally ambiguous issue of military humanitarian intervention.

Having grown up in a Christian society, immersed in "turn the other cheek" rhetoric, I definitely understand the appeal. It is the noble thing to do for the resource-rich. For the resource-poor, "turning the other cheek" effectively means not responding to oppression, and it is totally not clear to me, in instances of direct physical threat, when the switch from resource rich to resource poor happens.

Were I attacked, would I be able to kill someone? No. Of course not. Do I think it would be moral to do so? Probably not. Were I to watch someone rape and murder my sister, I still would probably be unable to kill them; were the choice between killing them or having them rape and murder my sister, I think that I would not be able to bring myself to killing someone. But the moral action? Probably, yes, to prevent imminent harm murder might be valid.

More generally, I simply do not believe that retributive justice is ethical. And since it is unethical to deprive you of the right to make personal decisions about life and death (you, for instance, have the right, in my mind, to suicide), it is certainly unethical to do so as punishment. But incarceration has four uses (and, since I don't value life per se the way many people do, I see murder as essentially a complete and violent form of incarceration) --- as retributive justice, as a way of bettering people, as deterrent, and to prevent other harm --- and the last is potentially ethical (the second would be if it were effective, but it is not). I can morally justify murder for, and only for, the purpose of preventing future harm, only as a last resort, and only when "turn the other cheek" is not the correct response. Ultimately, ethical decisions do involve balancing acts.

So what about the utilitarian test of the tourist, who may either kill one captured Indian (setting the rest free), or allow all twenty to be killed? I think that either choice must be allowed as an ethical choice; I myself would be entirely unable to fire the gun. But ultimately the answer is not really either: the completely ethical action is to consult first with the Indians and ask what they want. What's unethical about the situation is that the tourist is ultimately one of the oppressors, making life and death decisions for the oppressed people. Perhaps one Indian is willing to sacrifice themselves. Perhaps they decide to draw straws. Or perhaps they decide that they would all be happier dying than knowing that they lived only because someone else died for them. It should be their decision to make.

Similarly in international affairs, if we see endemic oppression, we may, and indeed we must, involve ourselves to help the oppressed. We must be careful to do the most effective things, and this is rarely, I believe, militaristic, and we must base our decisions strongly on what the oppressed people would like us to do. (This is, of course, hard. The most oppressed people are often sub-altern.) But, when we have the resources to just stand in the way of oppression, and absorb the onslaught of violent attempts to maintain the oppression while "turning the other cheek", then we cannot justify engaging ourselves in violence.

And, yet, we often find that we do not have such resources. And then is humanitarian military aid ethical? It helped, Power says, in Kosovo. I don't know.

A Categorical Definition

Categories, best described with commutative diagrams, allow for truly non-linear thinking, and yet they are usually defined linearly, similar to the way groups are usually introduced. This almost makes sense: morphisms, as one-dimensional objects, are about the most natural thing to compose linearly. And yet the power of category theory comes from the non-linear diagrams people draw, showing non-linear relationships between objects. The snake lemma, for instance, is poorly expressed and even more poorly understood if you are limited to writing words on a page.

I am, on this blog, constrained to poor, linear writing. Nevertheless, I would like to provide a definition of "category" using only visual, diagramatic ideas. Perhaps I will succeed in ASCIIing the diagrams. This definition, I hope, will ultimately be seen as providing a more basic understanding of these powerful creatures.


To begin with, a diagram is a (labeled) directed graph: it's composed of (labeled) vertices ("objects") connected by (labeled) directed edges ("arrows"). There is a natural notion of "subdiagram", formed by deleting any collection of edges, and any collection of vertices and all their edges. A category is a collection of diagrams, subject to certain rules, to be enunciated lower down. The diagrams in a category are said to commute. I will sometimes leave off labels, in which case I generally mean to refer to all (commutative) diagrams of that "shape".

Rule 0: A diagram commutes if and only if all its finite subdiagrams commute. (Thus any subdiagram of a commutative diagram commutes. In any category, the empty diagram commutes.)

One advantage of this construction is that I don't need any of that junk about "a collection of objects, and between each pair A and B of objects a set Hom(A,B) of arrows...". Instead, I can say simply that a morphism is a commutative diagram A ---> B, where I have left of the label on the arrow: I will write "f:A->B" for the very simple diagram of an arrow from A to B labeled by f, but only because I have to make it fit in this constrained formatting.

Rule 1: If two diagrams commute, then their disjoint union commutes. If diagram D contains an object label A, and E contains B, and if A ---> B commutes (morphism called f), then in the disjoint union we can connect A and B via f so that the diagram still commutes:

DDD      EEE
D    f     E
D A ---> B E

Often we will draw a dotted arrow in a diagram. In these notes, creating dotted arrows is too hard; I will use equal signs instead, as in ===>. A diagram with a dotted arrow means "If the diagram without the dotted arrow commutes, then there is exactly one diagram with an (labeled) arrow in place of the dotted one." Sometimes folks will write a label on the arrow, in which they mean the name (label) that the (unique) arrow which replaces it should have.

Rule 2: For any diagram E containing a finite chain (in the picture, I draw E "surrounding" the chain, to suggest that various objects in the chain might have other arrows to and from the rest of E), we have

EEEEEEEEEEEEEEEEEEEEEEEEE
E    __ B -...-> C      E
E    /|           \     E
E   /              _|   E
E  A =============>  D  E

So, in particular, some corollaries:

• "composition"
  

  A -> B -> C
   ========>

  
  
• "associativitiy"
  

  if  B -> C  and  B -> C  commute
      ^   >         \   |
      | /             > v
      A                D
  
  then so do     C  and  B
               > |       ^            /   v       |   >
            A -> D       A -> D

  
  (of course, with all edges (consistently) labeled.)
  
• "identity"
  

   A <==||    \\\    //
   \====/

  normally called "\mathbb{1}_A", I will just call this morphism "1_A". I will let you write out the "left and right identity laws" in this language; they follow from Rule 2.

However, one more rule concerning the various 1_A morphisms is necessary.

Rule 3If a commutative diagram contains 1_A:A->A for some A, then we can maintain commutativity by replacing this diagram with A: all arrows into and out of either A in the original diagram now go into and out of the single A in the new diagram. Conversely, any object A in a diagram may be replaced by 1_A:A->A, where all arrows on A are now duplicated, and placed once on each A. I will not try to draw this, but you should.

And that, folks, is it. Rules 0 through 3 suffice to define a category.


Of course, we should say some more, to convince you that pure diagrammatic thinking is useful. For instance, an isomorphism is a commutative diagram of the form

 --->
A    B.
 <---

With our "dotted arrow" (except I'm using equal signs) notation, we can go on to define "universal properties". It seems to me that there should really be "universal properties" and "co-universal properties", depending on which direction the arrows go. To define these, we introduce another new symbol [], which is kindof like ==>; whereas ==> defined an arrow uniquely, [] defines objects up to isomorphism. How? Well, say we have a diagram with a box. Then we're saying that (if the rest of the diagram commutes), then there's some object X which can fill the box (i.e. a labeling for that object that makes the diagram commute) s.t. if any other Y can also fill the box, then there's a dotted arrow from Y to X. I.e.:

  DD                  DD
DDDDDD              DDDDDD
[]  DD  means that  X   DD
DDDDDD              DDDDDD
  DD                  DD

                    DD
                  DDDDDD
and if any other  Y   DD, then
                  DDDDDD
                    DD
 DDDDDDDD
DD   DD DD
Y => X  DD
DD   DD DD
 DDDDDDDD

where the idea in the last picture is that Y and X each have all the arrows going from and to them and D that the original diagram says they should have. The dual notion to a universal property is a "co-universal property", in which Y => X is replaced by "Y <= X" in the definition. If needed, I will write that as {}.

Now, I don't actually know of any useful (co-)universal properties that are not (co-)limits — well, I think tensor products might be one, but I don't remember how to define that — so I really ought to just define limit. And maybe I should have just started with them, I dunno.

Anyhoo, given a (commutative) diagram D, the limit of the diagram lim D is the (universal property) diagram given by D with a box added, and arrows from the added box to each object in D. Co-limits are the dual notion. Limits, like anything defined via universal property, need not exist. Some examples:

A x B  =  lim  A  B

                      B
A x_C B  =  lim       |
                      v
                 A -> C

terminal object = lim (empty diagram)

          1_A
lim A = A ---> A, which we're considering to be equivalent to A.

I will stop here. Many category theory books from around here and onward start using diagrammatic reasoning and definitions more frequently, so I refer you to them for other definitions of other objects. My goal was to give diagrammatic definitions of the most basic elements of category theory, and to suggest that categories are best thought of not as collections of objects and morphisms, but simply as collections of diagrams.

Today's News

Today's headlines in the New York Times:

• Lockheed Martin got another government contract.
  
• Bush said something he said last week too.
  
• Folks post stuff online.
  
• Someone in Chicago wants to be mayor.

No news is good news?

Young boys and a man

While looking out the window at a rainy Newark Airport and waiting for a very delayed flight, I found myself standing next to a young boy — perhaps five or six — eating a large roll of bread. I struck up a conversation, and we were soon joined by his older brother — six or seven. I let the conversation go wherever it wandered, and learned quite a lot: that their father is a pilot; that the Yankees are the best baseball team, pitching is the best position, and next year they won't use the tee until you get six strikes; that the bushes below the hotel in Hawaii with the big rooms (three balconies in the suite!) now house a favorite action figure; that the police climbing the stairs into the jet-way were probably entering the airplane, because if there were a bad guy in the terminal, the security would have caught him in the initial screening (in fact, they were there to escort a very drunk passenger, who had repeatedly opened an alarmed door, from the terminal to the hospital).

After a while, their farther joined us at the window. "Tell the man next to you" — me — "what the kind of plane with the bump on top is," he asked his younger son. "I'll give you a hint: it starts Seven...."
"Um, Seven Seven?"
"No, Seven Forty-Seven."
"Seven Forty-Seven."
"And if there are [a particular kind of wing flaps]" — here my memory of the technical terms, which he used, has gone — "then it's a 747-400."


What I found most memorable about this discussion was not the ease with which we changed topics — an ease I normally associate with the uniformly brilliant kids at Mathcamp; an ease often pathologized as ADHD and ruined with drugs such as speed ritalin — nor the freedom with which these kids would talk to a complete stranger. What stuck with me was one particular piece of language: "Tell the man next to you..."

Those who've known me for a while may remember previous discussions I've had (though I think not here) about the different words "boy", "man", "kid", etc., which I find fascinating. I've intentionally used some throughout this entry: Mathcamp students and five-year-olds I've both described as "kids," for instance, whereas my first companion was a "young boy." I generally insist that periodicals refer to high school, and certainly college, students as "men" and "women": my freshman roommate was on the men's swim team, and in my brother's CS class there are only six women, as opposed to "boys'" and "girls." Mathcampers, on the other hand, and even my housemates, I often think of as "boys and girls". Not "children," perhaps, but "kids."

What's hardest, though, is self-identity — I'm good at holding multiple contradictory beliefs about the external realty — I had never before defined myself as someone who could be a "man [standing] next to you." Perhaps, when discussing sexual and gender politics, I've identified myself as a "(suitably adjectived) man," but more often as a "male." Categories like "men who have sex with men" are so entirely foreign and don't seem to apply to me or any of my peers. People in my socioeconomic class don't become "adults" until closer to 26, but I'm definitely no longer a "young adult." I'm a "student" or a "guy," not a "man."

One reason for my sojourn to New York was to attend a ninetieth birthday party and family reunion, where I spent some time chatting with various second cousins whom I haven't seen in ten years. My father, an older brother, is younger than his cousins, so while I played cards and board games with my fourteen-year-old cousin, the majority of "my generation" were three to ten years older than me. One announced the wonderful news of her pregnancy, making the matriarch whose birthday we were celebrating extremely happy. I'm used to my peers consisting of younger siblings and students exactly my age; I'm used to understanding those classmates only a few years older than me as significantly closer to adult, since they tend to be grad students when I'm an undergrad, or undergrads when I'm in high school.

But I'll be graduating in four months, and dreaming of my own apartment, and, eventually, house and family. I watch my fresh-out-of-college friends with their jobs in Silicon Valley, and can't help but think how similar that life is to college — they have roommates, come to campus, go on dates. They're no more "adults" than I am.

I have no trouble being "mature", or "old", or even relatively "grown up". But I'm twenty-one years old, and have a hard time thinking of myself as an "adult". Identifying as a "man" is impossible, and it is my current self-descriptor.

Angst and graduate school

I'm taking the GREs tomorrow (today), so instead of sleeping I'm avoiding looking up the rules and instructions. To do well on tests, it's best to go in knowing the structure of both the individual questions and the test as a whole. I don't yet, because I've been procrastinating with such useful time-sinks as listening to all of these pieces (link from the most excellent TWF234, about math and music). Oh, and actually getting things done --- I've written thank-you cards, answered e-mails --- there's no better way to be truly productive than to avoid something you really, really have to do.

One of my major accomplishments was writing back to a mathematical physicists from whom I had asked advice about grad schools. My e-mail ended up doing a decent job of outlining both some of my angst and some of my intellectual excitement; I thought you might enjoy it, and I'd gladly here your advice as well:


I'm at what I imagine to be the hardest part of grad school applications (and academic life? I'm sure there are harder things that my fantasy of the "easy life after grad school" leaves out) before actually writing them: figuring out what I want to do. This seems to come in two parts: 1. what am I interested in (and how to formulate it, and how much to formulate it or leave interests undecided as yet)? 2. and what people and departments are right for me given my interests?

I know I want to go into a career in mathematics; I want to teach calculus, and, though I enjoy the amorphous lands between math and physics, I've been ultimately happiest in math departments. At the same time, I know that the mathematics I want to study should have obvious connections with the physics: I'm happiest when I can use language and ways of thinking from physical theories, and when it's clear how the mathematical objects I'm playing with are connected to various attempts at fundamental theories. I want an anthropologist to conclude that my epistemology involves a real world that I'm studying (as opposed to those mathematicians who study platonic, nonexistent ideals). I assume that such is "mathematical physics" --- I definitely enjoy the material in John Baez's This Week's Finds. I would not be interested in studying interesting applied math such as fluid mechanics (or cryptography).

More precisely? I've been devouring This Week's Finds recently, so have been enjoying Baez's fascination with n-categories. I could happily study those for a while. Mostly as a way for me to record and inspire my own thoughts, I've been working on defining linear algebra entirely in terms of Penrose's tangle notation for tensors. I generally feel like algebraic notations, in which ideas are strung in lines, is restrictive and doesn't take advantage of the page.

My favorite toy is the hyperreal numbers, invented more or less by Abraham Robinson. These beasties have the power to do all of calculus, and provide actual interpretations for divergent sums, concepts like "much smaller than", and other important tools that are generally treated with intuition rather than rigor in most of math and physics. I would love to work on various projects to interpret and rigorize the mathematical footing of modern physics with this type of under-used tool. I hold that Robinson's calculus is more powerful than Cauchy's --- it's a conservative extension, so it can't prove anything that Cauchy can't, but it provides much more elementary meanings to a lot of the intuition. Vector fields really are infinitesimal, etc. The problem is that Robinson didn't get very far in constructing a user interface for his operating system. He can do Leibniz calculus, but no better than Cauchy can, and he didn't go farther. Cauchy is Windows to Robinson's Unix; I want to write Macintosh, incorporating QFT and the like.

What interests me most, beyond the actual mathematics, is the methods and institutions of mathematics and physics, and bettering those. I'm fascinated by the ways people think about math and physics, and the language they use, in a normative way: I want to find ways of understanding objects that get at their meanings, and specifically by combining math and physics intuition. It seems that the physicists are much more willing to take a cavalier attitude towards rigor, instead inventing formalisms that _might_ work in order to answer hard, hands-on questions. Whereas mathematicians may be better able to think extremely abstractly and provide the rigor, thereby arriving at a deeper meaning for the physicists' doodles. I want to help the mathematicians think in terms of particles, local processes, and effective theories (not to mention in terms of two-dimensional diagrams rather than "linear" equations). So what I would actually like to do is act as a translator.

So I've gotten some of the way towards an answer to my first question. Of course my interests will change as I continue to learn more math and physics. But my second question? I need all the advice I can get.

I think that I'm a strong applicant. I've taken the undergrad Intro to String Theory, and I'll be taking QFT this year. In math, I've taken a fair amount of algebra and analysis; most of my math knowledge comes from reading (including lots of math and physics blogs) and attending (now as a counselor) Canada/USA Mathcamp, which tries to expose its students to a wide variety of graduate-level math. So I'm looking at the top: strong departments, with people doing what I'm interest in.

But which are those? And who are the people?

Are there mathematical physics journals I should be reading or glancing at, because they're interesting or because they will suggest people and places I should pursue?

If you have any other advise for an aspiring (and presumably as-yet naive) mathematician with physics envy, please do share. Thank you so much.

A conventional question

I'm in the progress of writing up what I understand about tensors, defining them from scratch, using only intuition and Penrose's graphical notation. Eventually, perhaps I will write a version of my notes for Wikipedia, since their current article on the subject is laughably bad. I first read about them in this post by jao at physics musings; I had started reading Penrose's most recent book, The Road to Reality: A Complete Guide to the Laws of the Universe, which explores them in some depth. I am going to shamelessly reproduce jao's picture of such diagrams, so that you have some idea what I'm talking about:



Incidentally, I wonder what the history of such doodles really is. I hear talk of "einbeins", "zweibeins" and "dreibeins", lit. one-leg, two-leg, and three-leg, and if I knew more German, multi-legs ("mehrbeins"?), which sound like these tensorial pictures. Based on skimming the discussion here, it looks like einbeins are related, but not fully formulated. I wonder why someone would refer to an objects legs, though, unless it had legs.

Anyway, the notational question I wanted to ask was this:

We generally write "vectors" (as opposed to covectors) with raised indices, and covectors with lowered indices. This has physical significance: the Poincare group acts differently depending on whether the index is raised or lowered: on lowered indices, symmetries act by the adjoint, and so it's really a "dual" action, in the sense that it happens in the backwards order. So, although in some sense vectors and covectors are interchangeable, interpretations of diagrams are not.

Since vectors' indices are raised, Penrose proposes that a vector ought to have one "arm" (an edge coming out of the top), whereas a covector ought to have one leg. This makes sense, and closely matches how he thinks of contractions: contracting indices corresponds to drawing curves from the tops of the vectors to the bottoms of the covectors.

On the other hand, as soon as you start playing around with Penrose's diagrams — well, as soon as Josh H. started playing with them, when I introduced them to him over IM — you notice the connection between these diagrams and various ideas from quantum topology. In particular, diagrams like this look an awful lot like tangles.

This is actually no surprise. A n,m-tensor (one with n arms and m legs, so e.g. a vector is a 1,0-tensor), by definition, is a map from V tensored with itself m times to V tensored with itself n times. (By convention, V tensored with itself 0 times is the ground field — no, not a generic one-dimensional vector space, because I do in fact need the special number "1". This is so that there is a natural isomorphism between "V^0 tensor W" and W.)

But this, then, is a problem, because this commits me to reading my morphisms as going up. But my friends who study TQFTs think of their cobordisms as going down (see, for example, the many This Week's Finds starting at Week 73, in which Baez gives a mini course on n-categories).

So clearly one of us is right, and the other is wrong. Either we should think of vectors as having a head and a leg (more than half the time I catch myself drawing them this way anyway), or we should think of cobordisms, tangles, and their cousins as transforming the bottom of the page into the top of the page.

I'm leaning towards the latter, but only because there's one more, very established, case in which this matters. Diagrams of Minkowski space, and more generally of, for example, light cones in curved space, the positive time dimension is always drawn going up the page. And if our conventions are to have any sensible physical meaning, morphisms must correspond to forward time evolution.

So only typesetters and screen-renderers (and English language readers), who insist and putting (0,0) in the upper left corner of the page, have it backwards. Then again, mathematicians have known that for years.

But, of course, we do live in a democracy. If you tell me that infinitely more people and publications think time flows down, then I'll happily switch my conventions.