
Before I continue my tale of adjoint functors I want to say a little bit about icosahedra, buckyballs, and last letter Galois wrote before his famous duel.... all of which is taken from the following marvelous article:
1) Bertram Kostant, The graph of the truncated icosahedron and the last letter of Galois, Notices of the AMS 42 (September 1995), 959968. Also available at http://www.ams.org/notices/199509/199509toc.html.
When I was a graduate student at MIT I realized that Kostant (who teaches there) was deeply in love with symmetry, and deeply knowledgeable about some of its more mysterious byways. Unfortunately I didn't dig too deeply into group theory at the time, and now I am struggling to catch up.
Let's start with the Platonic solids. Note that the cube and the octahedron are dual  putting a vertex in the center of each of the cube's faces gives you an octahedron, and vice versa. So every rotational symmetry of the cube can be reinterpreted as a symmetry of the octahedron, and vice versa. Similarly, the dodecahedron and the icosahedron are dual, while the tetrahedron is selfdual. So while there are 5 Platonic solids, there are really only 3 different symmetry groups here.
These 3 "Platonic groups" are very interesting. The symmetry group of the tetrahedron is the group A_{4} of all even permutations of 4 things, since by rotating the tetrahedron we can achieve any even permutation of its 4 vertices. The symmetry group of the cube is S_{4}, the group of all permutations of 4 things. What are the 4 things here? Well, we can draw 4 line segments connecting opposite vertices of the cube; these are the 4 things! The symmetry group of the icosahedron is A_{5}, the group of even permutations of 5 things. What are the 5 things? It we take all the line segments connecting opposite vertices we get 6 things, not 5, but we can't get all even permutations of those by rotating the icosahedron. To find the 5 things is a bit trickier; I leave it as a puzzle here. See
2) John Baez, Some thoughts on the number 6, http://math.ucr.edu/home/baez/six.html
for an answer.
Once we convince ourselves that the rotational symmetry group of the icosahedron is A_{5}, it follows that it has 5!/2 = 60 elements. But there is another nice way to see this. Take an icosahedron and chop off all 12 corners, getting a truncated icosahedron with 12 regular pentagonal faces and 20 regular hexagonal faces, with all edges the same length. It looks just like a soccer ball. It's called an Archimedean solid because, while not quite Platonic in its beauty, every face is a regular polygon and every vertex looks alike: two pentagons abutting one hexagon.
The truncated icosahedron has 5 × 12 = 60 vertices. Every symmetry of the icosahedron is a symmetry of the truncated icosahedron, so A_{5} acts to permute these 60 vertices. Moreover, we can find an element of A_{5} that moves a given vertex of the truncated icosahedron to any other one, since "every vertex looks alike". Also, there is a unique element of A_{5} that does the job. So there must be precisely as many elements of A_{5} as there are vertices of the truncated icosahedron, namely 60.
There is a lot of interest in the truncated icosahedron recently, because chemists had speculated for some time that carbon might form C_{60} molecules with the atoms at the vertices of this solid, and a while ago they found this was true. In fact, while C_{60} in this shape took a bit of work to get ahold of at first, it turns out that lowly soot contains lots of this stuff!
Since Buckminster Fuller was fond of using truncated icosahedra in his geodesic domes, C_{60} and its relatives are called fullerenes, and the shape is affectionately called a buckyball. For more about this stuff, try:
3) P. W. Fowler and D. E. Manolpoulos, An Atlas of Fullerenes, Oxford University Press, 1995.
M. S. Dresselhaus, G. Dresselhaus, and P. C. Eklund, Science of Fullerenes and Carbon Nanotubules, Academic Press, New York, 1994.
G. Chung, B. Kostant and S. Sternberg, Groups and the buckyball, in Lie Theory and Geometry, eds. J.L. Brylinski, R. Brylinski, V. Guillemin and V. Kac, Birkhauser, 1994.
In fact, for the person who has everything: you can now buy 99.5% pure C_{60} at the following site:
4) BuckyUSA homepage, http://www.buckyusa.com/Fullerene%20C60.htm
But I digress. Coming back to the 3 Platonic groups... there is much more that's special about them. Most of it requires a little knowledge of group theory to understand. For example, they are the 3 different finite subgroups of SO(3) having irreducible representations on R^{3}. And they are nice examples of finite reflection groups. For more about them from this viewpoint, try "week62" and "week63". Also, via the McKay correspondence they correspond to the exceptional Lie groups E_{6}, E_{7}, and E_{8}  see "week65" for an explanation of this!
Yet another interesting fact about these groups is buried in Galois' last letter, written to the mathematician Chevalier on the night before Galois' fatal duel. He was thinking about some groups we'd now call PSL(2,F). Here F is a field (for example, the real numbers, the complex numbers, or Z_{p}, the integers mod p where p is prime). PSL(2,F) is a "projective special linear group over F." What does that mean? Well, first of all, SL(2,F) is the 2×2 matrices with entries in F having determinant equal to 1. These form a group under good old matrix multiplication. The matrices in SL(2,F) that are scalar multiples of the identity matrix form the "center" Z of SL(2,F)  the group of guys who commute with everyone else. We can form the quotient group SL(2,F)/Z, and get a new group called PSL(2,F).
Now Galois was thinking about PSL(2,Z_{p}) where p is prime. There's an obvious way to get this group to act as permutations of p+1 things. Here's how! For any field F, the group SL(2,F) acts as linear transformations of the 2dimensional vector space over F, and it thus acts on the set of lines through the origin in this vector space... which is called the "projective line" over F. But anything in SL(2,F) that's a scalar multiple of the identity doesn't move lines around, so we can mod out by the center and think of the quotient group PSL(2,F) as acting on projective line. (By the way, this explains the point of working with PSL instead of plain old SL.)
Now, an element of the projective line is just a line through the origin in F^{2}. We can specify such a line by taking any nonzero vector (x,y) in F^{2} and drawing the line through the origin and this vector. However, (x',y') and (x,y) determine the same line if (x',y') is a scalar multiple of (x,y). Thus lines are in 11 correspondence with vectors of the form (1,y) or (x,1). When our field F is Z_{p}, there are just p+1 of these. So PSL(2,Z_{p}) acts naturally on a set of p+1 things.
What Galois told Chevalier is that PSL(2,Z_{p}) doesn't act nontrivially as permutation of any set with fewer than p+1 elements if p > 11. This presumably means he knew that PSL(2,Z_{p}) does act nontrivially on a set with only p elements if p = 5,7, or 11. For example, PSL(2,5) turns out to be isomorphic to A_{5}, which acts on a set of 5 elements in an obvious way. PSL(2,7) and PSL(2,11) act on a 7element set and an 11element set, respectively, in sneaky ways which Kostant describes.
These cases, p = 5, 7 and 11, are the the only cases where this happens and PSL(2,Z_{p}) is simple. (See "week66" if you don't know what "simple" means.) In each case it is very amusing to look at how PSL(2,Z_{p}) acts nontrivially on a set with p elements and consider the subgroup that doesn't move a particular element of this set. For example, when p = 5 we have PSL(2,5) = A_{5}, and if we look at the subgroup of even permutations of 5 things that leaves a particular thing alone, we get A_{4}. Kostant explains how if we play this game with PSL(2,7) we get S_{4}, and if we play this game with PSL(2,11) we get A_{5}. These are the 3 Platonic groups again!!
But notice an extra curious coincidence. A_{5} is both PSL(2,5) and the subgroup of PSL(2,11) that fixes a point of an 11element set. This gives a lot of relationships between A_{5}, PSL(2,5), and PSL(2,11). What Kostant does is take this and milk it for all it's worth! In particular, it turns out that one can think of A_{5} as the vertices of the buckyball, and describe which vertices are connected by an edge using the embedding of A_{5} in PSL(2,11). I won't say how this goes... read his paper!
This may even have some applications for fullerene spectroscopy, since one can use symmetry to help understand spectra of compounds. (Indeed, this is one way group theory entered chemistry in the first place.)
In "week76" we talked about an "equivalence" of categories. We can summarize it this way: an "equivalence" of the categories C and D is a pair of functors F: C → D and G: D → C together with natural transformations e: FG => 1_{D} and i: 1_{C} => GF that are themselves invertible. (Note that we are now writing products of functors in the order that ordinary mortals typically do, instead of the backwards way we introduced in "week73". Sorry! It just happens to be better to write it this way now.) Now, the concept of "adjunction" is a cousin of the concept of "equivalence", and it's nice to have a definition of adjunction that brings out this relationship.
First, consider what happens in the definition of adjunction if we take c = Rd. Then we have a natural isomorphism between hom(LRd,d) and hom(Rd,Rd). Now there is a special element of hom(Rd,Rd), namely the identity 1_{Rd}. This gives us a special element of hom(LRd,d). Let's call this
e_{d}: LRd → d
What is this morphism like in an example? Say L: Set → Grp takes each set to the free group on that set, and R: Grp → Set takes each group to its underlying set. Then if d is a group, LRd is the free group on the underlying set of d. There's an obvious homomorphism from LRd to d, taking each word of elements in d and their inverses to their product in d. That's e_{d}. It goes from the free thing on the underlying thing of d to the thing d itself!
In fact, since everything in sight is natural, whenever we have an adjunction the morphisms e_{d} define a natural transformation
e: LR => 1_{D}
Next, consider what happens in the definition of adjunction if we take d = Lc. Then we have a natural isomorphism between hom(c,RLc) and hom(Lc,Lc). Now there is a special element in hom(Lc,Lc), namely the identity 1_{Lc}. This gives us a special element in hom(c,RLc). Let's call this
i_{c}: c → RLc
Again, it's good to consider the example of sets and groups: if c is a set, RLc is the underlying set of the free group on c. There is an obvious way to map c into RLc. That's i_{c}. It goes from the thing c to the underlying thing of the free thing on c.
As before, we get a natural transformation
i: 1_{C} => RL
So, as in an equivalence, when we have an adjunction we have natural transformations e: LR => 1_{D} and i: 1_{C} => RL. Unlike in an equivalence, they needn't be natural isomorphisms, as the example of sets and groups shows. But they do have some cool properties, which are nice to draw using pictures.
First, we draw e as a Vshaped thing:
L R \ / \ / \/The idea here is that e goes from LR down to the identity 1_{D}, which we draw as "nothing". We can think of L and R as processes, and the Vshaped thing as the metaprocess of L and R "colliding into each other and cancelling out", like a particle and antiparticle. (Lest you think that's just purple prose, wait and see! Eventually I'll explain what all this has to do with antiparticles!) Similarly, we draw i as an upsidedownVshaped thing:
/\ / \ / \ R LIn other words, i goes from the identity 1_{C} to RL.
We can also use this sort of notation to talk about identity natural transformations. For example, if we have any old functor F, there is an identity natural transformation 1_{F}: F => F, which we can draw as follows:
F    FWe draw it as a boring vertical line because "nothing is happening" as we go from F to F.
Now, I haven't talked much about the ways one can compose natural transformations like i and e, but remember that they are 2morphisms, or morphismsbetweenmorphisms, in Cat (the 2category of all categories). This means that they are inherently 2dimensional, and in particular, one can compose them both "horizontally" and "vertically". I'll explain this more next time, but for now please take my word for it! Using these composition operations, one can make sense of the following equations involving i and e:
R R /\   / \   / \    \ / =   \ /   \/  R Rand
L L  /\   / \   / \  \ /  =  \ /   \/   L LIn the first equation we are asserting that a certain way of sticking together i and e and some identity natural transformations gives 1_{R}: R => R. In the second we are asserting that some other way gives 1_{L}: L => L.
I will explain these more carefully next time, but for now I mainly want to state that we can also define an adjunction to be a pair of functors L: C → D and R: D → C together with natural transformations e: LR => 1_{D} and i: 1_{C} => RL making the above 2 equations hold! This is the definition of "adjunction" that is the most similar to the definition of "equivalence".
Now, topologically, these 2 equations simply say that if you have a wiggly curve like
/\  / \  / \   \ /  \ /  \/or
 /\  / \  / \ \ /  \ /  \/ you can pull it tight to get
     Thus, while
\ / \ / \/and
/\ / \ / \are not exactly "inverses", there is some subtler sense in which they "cancel out". This corresponds to the notion that while adjoint functors are not inverses, not even up to a natural isomorphism, they still are "like inverses" in a subtler sense.
Now this may seem like a silly game, drawing natural transformations as "string diagrams" and interpreting adjoint functors as wiggles in the string. But in fact this is part of a very big, very important, and very fun game: the relation between ncategory theory and the topology of submanifolds of R^{n}. Right now we are dealing with Cat, which is a 2category, so we are getting into 2dimensional pictures. But when we get into 3categories we will get into 3dimensional pictures, and knot theory... and what got me into this whole business in the first place: the relation between knots and physics. In higher dimensions it gets even fancier.
So I will continue next time and explain the recipes for composing natural transformations, and the associated string diagrams, more carefully.
To continue reading the `Tale of nCategories', click here.
© 1996 John Baez
baez@math.removethis.ucr.andthis.edu
