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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), 959-968. Also available at http://www.ams.org/notices/199509/199509-toc.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 self-dual. 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 A4 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 S4, 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 A5, 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 A5, 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 A5 acts to permute these 60 vertices. Moreover, we can find an element of A5 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 A5 that does the job. So there must be precisely as many elements of A5 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 C60 molecules with the atoms at the vertices of this solid, and a while ago they found this was true. In fact, while C60 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, C60 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 C60 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 R3. 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 E6, E7, and E8 - 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 Zp, 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,Zp) 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 2-dimensional 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 F2. We can specify such a line by taking any nonzero vector (x,y) in F2 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 1-1 correspondence with vectors of the form (1,y) or (x,1). When our field F is Zp, there are just p+1 of these. So PSL(2,Zp) acts naturally on a set of p+1 things.
What Galois told Chevalier is that PSL(2,Zp) 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,Zp) 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 A5, which acts on a set of 5 elements in an obvious way. PSL(2,7) and PSL(2,11) act on a 7-element set and an 11-element 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,Zp) 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,Zp) 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) = A5, and if we look at the subgroup of even permutations of 5 things that leaves a particular thing alone, we get A4. Kostant explains how if we play this game with PSL(2,7) we get S4, and if we play this game with PSL(2,11) we get A5. These are the 3 Platonic groups again!!
But notice an extra curious coincidence. A5 is both PSL(2,5) and the subgroup of PSL(2,11) that fixes a point of an 11-element set. This gives a lot of relationships between A5, 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 A5 as the vertices of the buckyball, and describe which vertices are connected by an edge using the embedding of A5 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.)
Now let me return to the tale of adjoint functors! I have been
stressing the fact that two functors L: C → D and R: D → C are
adjoint if there is a natural isomorphism between hom(Lc,d) and
hom(c,Rd). We can say that an "adjunction" is a pair of
functors L: C → D and R: D → C together with a natural
isomorphism between hom(Lc,d) and hom(c,Rd). But there is another way
to think about adjunctions which is also good.
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 => 1D and i: 1C => 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 1Rd. This gives us a special element of hom(LRd,d). Let's call
this
ed: 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 ed. 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 ed define a natural transformation
e: LR => 1D
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 1Lc. This gives us a special element in hom(c,RLc). Let's
call this
ic: 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 ic. It goes from the thing c
to the underlying thing of the free thing on c.
As before, we get a natural transformation
i: 1C => RL
So, as in an equivalence, when we have an adjunction we have natural
transformations e: LR => 1D and i: 1C => 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 V-shaped thing:
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 1F: F => F, which we can draw as
follows:
Now, I haven't talked much about the ways one can compose natural
transformations like i and e, but remember that they are 2-morphisms,
or morphisms-between-morphisms, in Cat (the 2-category of all
categories). This means that they are inherently 2-dimensional, 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:
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 => 1D and i: 1C => 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
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 n-category theory and the
topology of submanifolds of Rn. Right now we are dealing
with Cat, which is a 2-category, so we are getting into 2-dimensional
pictures. But when we get into 3-categories we will get into
3-dimensional 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
n-Categories', click here.
© 1996 John Baez
L R
\ /
\ /
\/
The idea here is that e goes from LR down to the identity 1D, which we
draw as "nothing". We can think of L and R as processes, and the
V-shaped thing as the meta-process 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
upside-down-V-shaped thing:
/\
/ \
/ \
R L
In other words, i goes from the identity 1C to RL.
F
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F
We draw it as a boring vertical line because "nothing is happening" as
we go from F to F.
R R
/\ | |
/ \ | |
/ \ | |
| \ / = |
| \ / |
| \/ |
R R
and
L L
| /\ |
| / \ |
| / \ |
\ / | = |
\ / | |
\/ | |
L L
In the first equation we are asserting that a certain way of sticking
together i and e and some identity natural transformations gives
1R: R => R. In the second we are asserting that some
other way gives 1L: L => L.
/\ |
/ \ |
/ \ |
| \ /
| \ /
| \/
or
| /\
| / \
| / \
\ / |
\ / |
\/ |
you can pull it tight to get
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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.
baez@math.removethis.ucr.andthis.edu
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