This Week's Finds in Mathematical Physics (Week 76)

John Baez

Yesterday I went to the oral exam of Hong Xiang, a student of Richard Seto who is looking for evidence of quark-gluon plasma at Brookhaven. The basic particles interacting via the strong force are quarks and gluons; these have an associated kind of "charge" known as color. Under normal conditions, quarks and gluons are confined to lie within particles with zero total color, such as protons and neutrons, and more generally the baryons and mesons seen in particle acccelerators - and possibly glueballs, as well. (See "week68" for more on glueballs.)

However, the current theory of the strong force - quantum chromodynamics - predicts that at sufficiently high densities and/or pressures, a plasma of protons and neutrons should undergo a phase transition called "deconfinement", past which the quarks and gluons will roam freely. At low densities, this is expected to happen at a temperature corresponding to about 200 MeV per nucleon (i.e., proton or neutron). If my calculation is right, this is about 2 trillion Kelvin! At low temperatures, it's expected to happen at about 5 to 20 times the density of an atomic nucleus. (Normal nuclear matter has about .16 nucleons per femtometer cubed.) For more on this, check out these:

1) Relativistic Heavy Ion Collider homepage, http://www.bnl.gov/RHIC/

CERN Courier, Phase diagram of nuclear matter, http://www.cerncourier.com/main/article/40/5/17/1/cernquarks1_6-00

The folks at Brookhaven are attempting to get high densities and temperatures by slamming two gold nuclei together. They are getting densities of about 9 times that of a nucleus... and I forget what sort of temperature, but there is reason to hope that the combined high density and pressure might be enough to cause deconfinement and create a "quark-gluon plasma". Colliding gold on gold at high energies produces a enormous spray of particles, but amidst this they are looking for a particular signal of deconfinement. They are looking for φ mesons and looking to see if their lifetime is modified. A φ meson is a spin-1 particle made of a strange quark / strange antiquark pair; strange quarks and antiquarks are supposed to be common in the quark-gluon plasma formed by the collision. Folks think the lifetime of a φ will be affected by the medium it finds itself in, and that this should serve as a signature of deconfinement. In fact, they may have already seen this!

People might also enjoy looking at this review article:

2) Adriano Di Giacomo, Mechanisms of colour confinement, preprint available as hep-th/9603029.

Okay, let me continue the tale of n-categories. I want to lead up to their role in physics, but to do it well, there are quite a few things I need to explain first. One of the important things about n-category theory is that they allow a much more fine-grained approach to the notion of "sameness" than we would otherwise be able to achieve.

In a bare set, two elements x and y are either equal or not equal; there is nothing much more to say.

For example, in the category Set an isomorphism is just a one-to-one and onto function. If we know two sets x and y are isomorphic we know that they are "the same in a way", even if they are not equal. But specifying an isomorphism f: x → y does more than say x and y are the same in a way; it specifies a particular way to regard x and y as the same.

In short, while equality is a yes-or-no matter, a mere property, an isomorphism is a structure. It is quite typical, as we climb the categorical ladder (here from elements of a set to objects of a category) for properties to be reinterpreted as structures, or sometimes vice-versa.

What about in a 2-category? Here the notion of equality sprouts still further nuances. Since I haven't defined 2-categories in general, let me work with an example, Cat. This has categories as its objects, functors as its morphisms, and natural transformations as its 2-morphisms.

So... we can certainly speak, as before, of the equality of categories. We can also speak of the isomorphism of categories: an isomorphism between C and D is a functor F: C → D for which there is an inverse functor G: D → C. I.e., FG is the identity functor on C and GF is the identity on D, where we define the composition of functors in the obvious way. But because we also have natural transformations, we can also define a subtler notion, the equivalence of categories. An equivalence is a functor F: C → D together with a functor G: D → C and natural isomorphisms a: FG → 1C and b: GF → 1D. A "natural isomorphism" is a natural transformation which has an inverse.

Abstractly, I hope you can see the pattern here: just as we can "relax" the notion of equality to the notion of isomorphism when we pass from sets to categories, we can relax the condition that FG and GF equal identity functors to the condition that they be isomorphic to identity functors when we pass from categories to the 2-category Cat. We need to have the natural transformations to be able to speak of functors being isomorphic, just as we needed functions to be able to speak of sets being isomorphic. In fact, with each extra level in the theory of n-categories, we will be able to come up with a still more refined notion of "n-equivalence" in this way. That's what "processes between processes between processes..." allow us to do.

But let me attempt to bring this notion of equivalence of categories down to earth with some examples. Consider first a little category C with only one object x and one morphism, the identity morphism 1x: x → x. We can draw C as follows:

x

where we don't bother drawing the identity morphism 1x. This category, by the way, is called the "terminal category". Next consider a little category D with two objects y and z and only four morphisms: the identities 1y and 1z, and two morphisms f: y → z and g: z → y which are inverse to each other. We can draw D as follows:
g
<----------
y               z
---------->
f

where again we don't draw identities.

So: C is a little world with only one object, while D is a little world with only two isomorphic objects... that are isomorphic in precisely one way! C and D are clearly not isomorphic, because for a functor F: C → D to be invertible it would need to be one-to-one and onto on objects, and also on morphisms.

However, C and D are equivalent. For example, we can let F: C → D be the unique functor with F(x) = y, and let G: D → C be the unique functor from D to C. (There is only one functor from any category to C, since C has only one object and one morphism; this is why we call C the terminal category.) Now, if we look at the functor FG: C → C, it's not hard to see that this is the identity functor on C. But the composite GF: D → D is not the identity functor on D. Instead, it sends both y and z to y, and sends all the morphisms in D to 1y. But while not equal to the identity functor on D, the functor GF is naturally isomorphic to it. We can define a natural transformation b: GF → 1D by setting by = 1y and bz = f. Here some folks may want to refresh themselves on the definition of natural transformation, given in "week75", and check that b is really one of these, and that b is a natural isomorphism because it has an inverse.

The point is, basically, that having two uniquely isomorphic things with no morphisms other than the isomorphisms between them and the identity morphisms isn't really all that different from having one thing with only the identity morphism. Category theorists generally regard equivalent categories as "the same for all practical purposes". For example, given any category we can find an equivalent category in which any two isomorphic objects are equal. We call these "skeletal" categories because all the fat is gone and just the essential bones are left. For example, the category FinSet of finite sets, with functions between them as morphisms, is equivalent to the category with just the sets

0 = {}

1 = {0}

2 = {0,1}

3 = {0,1,2}

etc., and functions between them as morphisms (see "week73"). Essentially all the mathematics that can be done in FinSet can be done in this skeletal category. This may seem shocking, but it's true.... Similarly, the category Set is equivalent to the category Card having one set of each cardinality. Also, the category Vect of complex finite- dimensional vector spaces, with linear functions between them as morphisms, is equivalent to a skeletal category where the only objects are those of the form Cn. This example should not seem shocking; it's this fact which allows unsophisticated people to do linear algebra under the impression that all finite-dimensional vector spaces are of the form Cn, and still manage to do all the practical computations that more sophisticated people can do, who know the abstract definition of vector space and thus know of lots more finite-dimensional vector spaces.

However, there is another thing we can do in Cat, another refinement of the notion of isomorphism, which I alluded to in "week75". This is the notion of "adjoint functor". Let me mention a few examples (in addition to the example given in "week75") and let the reader ponder them before giving the definition. Let Grp denote the category with groups as objects and homomorphisms as morphisms, a homomorphism f: G → H between groups being a function with f(1) = 1 and f(gh) = f(g)f(h) for all g, h in G. Then there is a nice functor

L: Set → Grp

which takes any set S to the free group on S: this is the group L(S) formed by all formal products of elements in S and inverses thereof, with no relations other than those in the definition of a group. For example, a typical element of the free group on {x,y,z} is xyzy-1xxy.

(It's easy to see that f: S → T is a function between sets, there is a unique homomorphism L(f): L(S) → L(T) with L(f)(x) = f(x) for all x in S, and that this makes L into a functor.)

There is also a nice functor

R: Grp → Set

taking any group to its underlying set, and taking any homomorphism to its underlying function. We call this a "forgetful" functor since it simply amounts to forgetting that we are working with groups and just thinking of them as sets.

Now there is a sense in which L and R are reverse processes, but it is delicate. They certainly aren't inverses, and they aren't even part of an equivalence between Set and Grp. Nonetheless they are "adjoints". If the reader hasn't thought about this, she may enjoy figuring out what this might mean... perhaps keeping the adjoint operators mentioned in "week75" in mind.