Also available at http://math.ucr.edu/home/baez/week79.html
April 1, 1996
This Week's Finds in Mathematical Physics  Week 79
John Baez
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 x 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 any vertex of the truncated icosahedron to any other, since "every
vertex looks alike". 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 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 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 2x2 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.)
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 => 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 L
In 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



F
We 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 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
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.

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