
In this and future issues of This Week's Finds, I'd like to talk a bit more about higherdimensional algebra, and how it should lead to many exciting developments in mathematics and physics in the 21st century. I've talked quite a bit about this already, but I hear from some people that the "big picture" remained rather obscure. The main reason, I suppose, is that I was just barely beginning to see the big picture myself! As Louis Crane noted, in this subject it often feels that we are unearthing the fossilized remains of some enormous prehistoric beast, still unsure of its extent or how it all fits together. Of course that's what makes it so exciting, but I'll try to make sense what we've found so far, and where it may lead. In the Weeks to come, I'll start out describing some basic stuff, and work my way up to some very new ideas.
However, before I get into that, I'd like to say a bit about something completely different: biology.
1) Biological Asymmetry and Handedness, Ciba Foundation Symposium 162, John Wiley and Sons, 1991.
D. K. Kondepudi and D. K. Nelson, Weak neutral currents and the origins of molecular chirality, Nature 314, pp. 438441.
It's always puzzled me how humans and other animals could be consistently asymmetric. A 5050 mix of two mirrorimage forms could easily be explained by "spontaneously broken symmetry", but in fact there are many instances of populations with a uniform handedness. Many examples appear in Weyl's book "Symmetry" (see "week63"). To take an example close to home, the human brain appears to be lateralized in a fairly consistent manner; for example, most people have the speech functions concentrated in the left hemisphere of their cerebrum  even most, though not all, lefthanders.
One might find this unsurprising: it just means that the asymmetry is encoded in the genes. But think about it: how are the genes supposed to tell the embryo to develop in an asymmetric way? How do they explain the difference between right and left? That's what intrigues me.
Of course, genes code for proteins, and most proteins are themselves asymmetric. Presumably the answer lurks somewhere around here. Indeed, even the amino acids of which the proteins are composed are asymmetric, as are many sugars and for that matter, the DNA itself, which is composed of two spirals, each of which has an intrinsic directionality and hence a handedness. The handedness of many of these basic biomolecules is uniform for all life on the globe, as far as I know.
In the conference proceedings on biological asymmetry, there is an interesting article on the development of asymmetry in C. elegans. Ever since the 1960s, this little nematode has been a favorite among biologists because of its simplicity, and because of the advantages understanding one organism thoroughly rather than many organisms in a sketchy way. I'm sure most of you know about the fondness geneticists have for the fruit fly, but Caenorhabditis elegans is a far simpler critter: it only has 959 cells, all of which have been individually named and studied! There are over 1000 people studying it by now, there is a journal devoted to it  The Worm Breeder's Gazette  and it has its own worldwide web server. Moreover, folks are busily sequencing not only the complete human genome but also all 100 million bases of the DNA of C. elegans.
But I digress! The point here is that C. elegans is asymmetric, and exhibits a consistent handedness. And the cool thing is that in the conference proceedings, Wood and Kevshan report on experiments where they artificially changed the handedness of C. elegans embryos when they consisted of only 6 cells! The embryos look symmetric when they have 4 cells; by the time they have 8 cells the asymmetry is marked. By moving some cells around at the 6cell stage, Wood and Kevshan were able to create fully functional C. elegans having opposite the usual handedness.
The question of exactly how the embryo's asymmetry originates from some asymmetry at the molecular still seems shrouded in mystery. And there is another puzzle: how did the biomolecules choose their handedness in the first place? Here spontaneous symmetry breaking  an essentially random choice later amplified by selection  seems a natural hypothesis. But physicists should be interested to note that another alternative has been seriously proposed. Weak interactions violate parity and thus endow the laws of nature with an intrinsic handedness. This means there is a slight difference in energies between any biomolecule and its enantiomer, or mirrorimage version. According to S. F. Mason's article in the conference proceedings, this difference indeed favors the observed forms of amino acids and sugars  the lefthanded or "L" amino acids and the righthanded or "D" sugars. But the difference is is incredibly puny  typically it amounts to 10^{14} joules per mole! How could such a small difference matter? Well, Kondepudi and Nelson have done calculations suggesting that in certain situations where there is both autocatalysis of both L and D forms of these molecules, and also competition between them, random fluctuations can be averaged out, while small energy level differences can make a big difference.
That would be rather satisfying to me: knowing that my heart is where it is for the same reason that neutrinos are lefthanded. But in fact this theory is very controversial.... I mention it only because of its charm.
If we think of the universe as passing through the course of history from simplicity to complexity, from neutrinos to nematodes to humans, it's natural to wonder what's at the bottom, where things get very simple, where physics blurs into pure logic.... far from the "spires of form". Ironically, even the simplest things may be hard to understand, because they are so abstract.
{} {{}} {{{}}} {{},{{}}} {{},{{{}}}} {{},{{}},{{},{{}}}}and the like. But for many purposes, a onetoone and onto function between two sets allows us to treat them as the same. So if necessary, we could actually get by with just one set of each cardinality. For example
{} {{}} {{},{{}}} {{},{{}},{{},{{}}}}and so on. For short, people like to call these
0 1 2 3and so on. We could wonder what comes after all these finite cardinals, and what comes after that, and so on, but let's not. Instead, let's ponder what we've done so far. We started with the universe of sets  not exactly the set of all sets, but pretty close  but very soon we started playing with functions between sets. This is what allowed us to speak of two sets with the same cardinality as being isomorphic.
In short, we are really working with the category of sets. A category is something just as abstract as a set, but a bit more structured. It's not a mere collection of objects; there are also morphisms between objects, in this case the functions between sets.
Some of you might not know the precise definition of a category; let me state it just for completeness. A category consists of a collection of "objects" and a collection of "morphisms". Every morphism f has a "source" object and a "target" object. If the source of f is X and its target is Y, we write f: X → Y. In addition, we have:
1) Given a morphism f: X → Y and a morphism g: Y → Z, there is a morphism fg: X → Z, which we call the "composite" of f and g.
2) Composition is associative: (fg)h = f(gh).
3) For each object X there is a morphism 1_{X}: X → X, called the "identity" of X. For any f: X → Y we have 1_{X} f = f 1_{Y} = f.
That's it.
(Note that we are writing the composite of f: X → Y and g: Y → Z as fg, which is backwards from the usual order. This will make life easier in the long run, though, since fg will mean "first do f, then g".)
Now, there are lots of things one can do with sets, which lead to all sorts of interesting examples of categories, but in a sense the primordial category is Set, the category of sets and functions. (One might try to make this precise, by trying to prove that every category is a subcategory of Set, or something like that. Actually the right way to say how Set is primordial is called the "Yoneda lemma". But to understand this lemma, one needs to understand categories a little bit.)
When we get to thinking about categories a lot, it's natural to think about the "category of all categories". Now just as it's a bit bad to speak of the set of all sets, it's bad to speak of the category of all categories. This is true, not only because Russell's paradox tends to ruin attempts at a consistent theory of the "thing of all things", but because, just as what really counts is the category of all sets, what really counts is the 2category of all categories.
To understand this, note that there is a very sensible notion of a morphism between categories. It's called a "functor", and a functor F: C → D from a category C to a category D is just something that assigns to each object x of C an object F(x) of D, and to each morphism f of C a morphism F(f) of D, in such a way that "all structure in sight is preserved". More precisely, we want:
1) If f: x → y, then F(f): F(x) → F(y).
2) If fg = h, then F(f)F(g) = F(h).
3) If 1_{x} is the identity morphism of x, then F(1_{x}) is the identity morphism of F(x).
It's good to think of a category as a bunch of dots  objects  and arrows going between them  morphisms. I would draw one for you if I could here. Category theorists love drawing these pictures. In these terms, we can think of the functor F: C → D as putting a little picture of the category C inside the category D. Each dot of C gets drawn as a particular dot in D, and each arrow in C gets drawn as a particular arrow in D. (Two dots or arrows in C can get drawn as the same dot or arrow in D, though.)
In addition, however, there is a very sensible notion of a "2morphism", that is, a morphism between morphisms between categories! It's called a "natural transformation". The idea is this. Suppose we have two functors F: C → D and G: C → D. We can think of these as giving two pictures of C inside D. So for example, if we have any object x in C, we get two objects in D, F(x) and G(x). A "natural transformation" is then a gadget that draws an arrow from each dot like F(x) to the dot like G(x). In other words, for each x, the natural transformation T gives a morphism T_{x}: F(x) → G(x). But we want a kind of compatibility to occur: if we have a morphism f: x → y in C, we want
F(f) F(x) > F(y) _{ }   T_{x} T_{y} v v G(x) > G(y) G(f)to commute; in other words, we want T_{x} G(f) = F(f) T_{y}.
This must seem very boring to the people who understand it and very mystifying to those who don't. I'll need to explain it more later. For now, let me just say a bit about what's going on. Sets are "zerodimensional" in that they only consist of objects, or "dots". There is no way to "go from one dot to another" within a set. Nonetheless, we can go from one set to another using a function. So the category of all sets is "onedimensional": it has both objects or "dots" and morphisms or "arrows between dots". In general, categories are "onedimensional" in this sense. But this in turn makes the collection of all categories into a "twodimensional" structure, a 2category having objects, morphisms between objects, and 2morphisms between morphisms.
This process never stops. The collection of all ncategories is an (n+1)category, a thing with objects, morphisms, 2morphisms, and so on all the way up to nmorphisms. To study sets carefully we need categories, to study categories well we need 2categories, to study 2categories well we need 3categories, and so on... so "higher dimensional algebra", as this subject is called, is automatically generated in a recursive process starting with a careful study of set theory.
If you want to show off, you can call the 2category of all categories Cat, and more generally, you can call the (n+1)category of all ncategories nCat. nCat is the primordial example of an (n+1)category!
Now, just as you might wonder what comes after 0,1,2,3,..., you might wonder what comes after all these ncategories. The answer is "ωcategories".
What comes after these? Well, let us leave that for another time. I'd rather conclude by mentioning the part that's the most fascinating to me as a mathematical physicist. Namely, the various dimensions of category turn out to correspond in a very beautiful  but still incompletely understood  way to the various dimensions of spacetime. In other words, the study of physics in imaginary 2dimensional spacetimes uses lots of 2categories, the study of physics in a 3d spacetimes uses 3categories, the study of physics in 4d spacetimes appears to use 4categories, and so on. It's very surprising at first that something so simple and abstract as the process of starting with sets and recursively being led to study the (n+1)category of all ncategories could be related to the dimensionality of spacetime. In particular, what could possibly be special about 4 dimensions?
Well, it turns out that there are some special things about 4 dimensions. But more on that later.
To continue reading the `Tale of nCategories', click here.
Addendum: Long after writing the above, I just saw an interesting article on chirality in biology:
2) N. Hirokawa, Y. Tanaka, Y. Okada and S. Takeda, Nodal flow and the generation of leftright asymmetry, Cell 125 1 (2006), 3345.
It reports on detailed studies of how leftright asymmetry first shows in the development of animal embryos. It turns out this asymmetry is linked to certain genes with names like Lefty1, Lefty2, Nodal and Pitx2. About half of the people with a genetic disorder called Kartagener's Syndrome have their organs in the reversed orientation. These people also have immotile sperm and defective cilia in their airway. This suggests that the genes controlling leftright asymmetry also affect the development of cilia! And the link has recently been understood...
The first visible sign of leftright asymmetry in mammal embryos is the formation of a structure called the "ventral node" after the frontback (dorsalventral) and topbottom (anteriorposterior) symmetries have been broken. This node is a small bump on the front of the embryo.
It has recently been found that cilia on this bump wiggle in a way that makes the fluid the embryo is floating in flow towards the left. It seems to be this leftward flow that generates many of the more fancy leftright asymmetries that come later.
How do these cilia generate a leftward flow? It seems they spin around clockwise, and are tilted in such a way that they make a leftward swing when they are near the surface of the embryo, and a rightward swing when they are far away. This manages to do the job... the article discusses the hydrodynamics involved.
I guess now the question becomes: why do these cilia spin clockwise?
© 1996 John Baez
baez@math.removethis.ucr.andthis.edu
