Also available at http://math.ucr.edu/home/baez/week78.html
March 28, 1996
This Week's Finds in Mathematical Physics - Week 78
John Baez
Last Week I began explaining the concept of "adjoint functor".
This Week I want to finish explaining it --- or at least finish
one round of explanation! Then we'll begin to be able to
see the unity of category theory, topology, and quantum theory.
These may seem rather distinct subjects, but they have
an interesting tendency to blur together when one is doing topological
quantum field theory. Part of the point of higher-dimensional
algebra is to explain this.
So, remember the idea of adjoint functors. Say we have categories C and D
and functors L: C -> D and R: D -> C. Then we say L is the "left adjoint" of
R, or that R is the "right adjoint" of L, if for any object c of C and object
d of D, there is a natural one-to-one correspondence between hom(Lc,d) and
hom(c,Rd). An example to keep in mind is when C is the category of sets
and D is the category of groups. Then L turns any set into the free group
on that set, while R turns any group into the underlying set of that group.
All sorts of other "free" and "underlying" constructions are also left and
right adjoints, respectively.
Now the only thing I didn't make very precise is what I mean by "natural"
in the above paragraph. Informally, the idea of a "natural" one-to-one
correspondence is that doesn't depend on any arbitrary choices. The famous
example is that if we have a finite-dimensional vector space V, it's always
isomorphic to its dual V*, but not naturally so: to set up an isomorphism
we need to pick a basis e_i of V, and this gives a dual basis f^i of V*, and
then we get an isomorphism sending e_i to f^i, but this isomorphism depends
on our choice of basis. But V is *naturally* isomorphic to its double dual
V**.
Now, it's hard to formalize the idea of "not depending on any arbitrary
choices" directly, so one needs to reflect on why it's bad for an
isomorphism to depend on arbitrary choices. The main reason
is that the arbitrariness may break a useful symmetry.
In fact, Eilenberg and Mac Lane invented category theory in order to
formalize this idea of "naturality as absence of symmetry-breaking".
Of course, they needed the notion of category to get a sufficiently general
concept of "symmetry". They realized that a nice way
to turn things in the category C into things in the category D is typically
a functor F: C -> D. And then, if we have two functors F,G: C -> D, they
defined a "natural transformation" from F to G to be a bunch of morphisms
T_c : F(c) -> G(c),
one for each object c of C, such that the following diagram
commutes for every morphism f: c -> c' in C:
F(f)
F(c) -------> F(c')
| |
T_c | | T_{c'}
| |
V V
G(c) -------> G(c')
G(f)
This condition says that the transformation T gets along with all the
"symmetries", or more precisely morphisms f, in the category C. We
can do it either before or after applying one of these symmetries, and
we get the same result. For example, a vector space construction which
depends crucially on a choice of basis will fail this condition if we take
f to be a linear transformation corresponding to a change of basis.
A "natural isomorphism" is then just a natural transformation that's
invertible, or in other words, one for which all the morphisms T_c
are isomorphisms.
Okay. Hopefully that explains the idea of "naturality" a bit better.
But right now we are trying to figure out what we mean by
saying that hom(Lc,d) and hom(c,Rd) are naturally isomorphic. To do this,
we need to introduce a couple more ideas: the product of categories,
and the opposite of a category.
First, just as you can take the Cartesian product of two sets, you can take
the Cartesian product of two categories, say C and D.
It's not hard. An object of C x D is just a pair of objects,
one from C and one from D. A morphism in C x D is just a pair of morphisms,
one from C and one from D. And everything works sort of the way you'd expect.
Second, if you have a category C, you can form a new category C^{op},
the opposite of C, which has the same objects as C, and has the arrows
in C turned around backwards. In other words, a morphism f: x -> y in
C^{op} is defined to be a morphism f: y -> x in C, and the composite fg
of morphisms in C^{op} is defined to be their composite gf in C. C^{op}
is like a through-the-looking-glass version of C where they do
everything backwards. A functor F: C^{op} -> D is also called a
"contravariant" functor from C to D, since we can think of it as a
screwy functor from C to D with F(fg) = F(g)F(f) instead of the usual
F(fg) = F(f)F(g). Whenever you see this perverse contravariant behavior
going on, you should suspect an opposite category is to blame.
Now, it turns out that we can think of the "hom" in a category C as a
functor
hom(-,-): C^{op} x C -> Set
Here the -'s denote blanks to be filled in. Obviously, for any object
(x,x') in C^{op} x C, there is a nice juicy set hom(x,x'), the set of
all morphisms from x to x'. But what if we have a morphism
(f,f'): (x,x') -> (y,y')
in C^{op} x C? For hom(-,-) to be a functor, we should get a
nice juicy function
hom(f,f'): hom(x,x') -> hom(y,y').
How does this work? Well, remember that a morphism (f,f') as above
is really just a pair consisting of a morphism f: x -> y in C^{op}
and a morphism f': x' -> y' in C. A morphism f: x -> y in C^{op} is
just a morphism f: y -> x in C. Now say we have an unsuspecting
element g of hom(x,x') and we want to hit it with hom(f,f') to get
something in hom(y,y'). Here's how we do it:
hom(f,f'): g |-> f'gf
We compose it with f' on the left and f on the right! Composing on
the left is a nice covariant thing to do, but composing on the right
is contravariant, which is why we needed the opposite category C^{op}.
Okay, now back to our adjoint functors L: C -> D and R: D -> C.
Now we are ready to say what we mean by hom(Lc,d) and hom(c,Rd) being
naturally isomorphic. Using the stuff we have set up, we can define
two functors
hom(L-,-): C^{op} x D -> Sets
and
hom(-,R-): C^{op} x D -> Sets
and we are simply saying that for L and R to be adjoints, we demand
the existence of a natural isomorphism between these functors!
Of course, this seems abstract, but if you work it out in some of
the examples of adjoint functors given in "week76" you'll see it all
makes good sense.
Now let me start explaining what this all has to do with quantum theory.
(I'll put off the topology until next Week.) First of all, the "hom functor"
we introduced,
hom(-,-): C^{op} x C -> Set
should remind you a whole lot of the inner product on a Hilbert space
H. The inner product is linear in one slot and conjugate-linear in the
other, just like hom is covariant in one slot and contravariant in the other.
In fact, the inner product can be thought of as a bilinear map
<-,->: H^{op} x H -> C
where H^{op}, the "opposite" Hilbert space, is like H but with a complex
conjugate thrown into the definition of scalar multiplication, and here
C denotes the complex numbers!
Second of all, the definition of adjoint functor, with hom(Lc,d) and
hom(c,Rd) being naturally isomorphic, should remind you of adjoint linear
operators on Hilbert spaces. If we have a linear operator L: H -> K from
a Hilbert space H to a Hilbert space K, its adjoint R: K -> H is given
by
=
for all h in H and k in K.
In fact, the whole situation with adjoint functors is a kind of "categorified"
version of the situation with adjoint linear operators. Everything has been
boosted up one notch on the n-categorical ladder. What I mean is this:
the Hilbert spaces H and K above are *sets*, with *elements* h and k, while
the categories C and D are *categories*, with *objects* c and d. The
inner product of two elements of a Hilbert space is a *number*, while the
hom of two objects in a category is a *set*. Most interesting, the definition
of adjoint operators requires that and be *equal*, while the
definition of adjoint functors requires only that and be
*naturally isomorphic*.
So we can think of adjoints in category theory as a boosted-up version
of the adjoints in quantum theory. But these days, I prefer to think of
the adjoints in quantum theory as a watered-down or "decategorified" version
of the adjoints in category theory. The reason is that categorification ---
as noted by Louis Crane, who I believe invented the term --- is a risky,
hit-or-miss business, while decategorification is much more systematic.
Decategorification is the simply the process of neglecting the difference
between isomorphism and equality. If we start with an n-category and
then get lazy and decide to think of invertible n-morphisms as *equations*
between the (n-1)-morphisms, we get an (n-1)-category. If we keep slacking
off like this, before you know it we're doing set theory! The final stage
of decategorification is when we get sloppy and instead of keeping track
of *set*, we merely record the *number* of its elements.
It's amusing to imagine this process of decategorification as
one of those elaborate Gnostic myths about the Fall. We start in
the paradise of omega-categories (or perhaps even higher up), but
by the repeated sin of confusing equality with isomorphism we fall all
the way down the n-categorical ladder to the crude world of sets, or
worse, simply numbers. But all this happened a long time ago: now
we need to work our butt off to climb back up! In other words,
historically our early ancestors dealt with finite sets by replacing
them with something cruder: their numbers of elements. Counting is
actually very handy, of course, but it can only tell if the
cardinalities of two sets are *equal*; it doesn't address the problem of
specific *isomorphisms* between sets. To climb back up the
n-categorical ladder, we needed to start with the set N of natural numbers
0, 1, 2, 3, ...
and by dint of strenous mental effort realize that this is just the
decategorification of the category FinSet of finite sets. (In fact,
category-theorists routinely use 2 to stand for the 2-element set in the
skeletal category equivalent to FinSet, and so on --- see "week76".)
Now, you are certainly entitled to wonder if this elaborate mathematical-
theological fantasy is just a joke or if it has some practical spinoffs.
For example, is there anything we can *do* with the analogy between adjoint
operators and adjoint functors? As it turns out, there is. The point
is that the analogy is not quite precise. For example, every linear
operator has an adjoint, but not every functor has an adjoint --- nor
need it be "linear" in any sense. If we endeavor to make the analogy
precise, we will invent a special sort of category called a "2-Hilbert
space" which is the precise categorified analog of a Hilbert space. And
we will invent a nice sort of "linear" functor between these, and all
such functors will have adjoints. Furthermore, in this situation all
left adjoints will also be right adjoints... fixing another funny
discrepancy. And these 2-Hilbert spaces turn out to be closely related
to 2-dimensional topological quantum field theories (in general, n-Hilbert
spaces appear to be related to n-dimensional TQFTs), as well as some
interesting aspects of group representation theory.
I'm busily writing a paper on exactly this stuff, but I have not
explained enough category theory here to describe it in detail yet. For now,
let me just make the connection between the hom(-,-) of category theory
and the <-,-> of quantum theory more clear, and hopefully more plausible.
If we have states h and h' in a Hilbert space, keeps track of the
*amplitude* of getting from h to h'. (Often people will say "from h' to h",
but here I think I really want to go the other way.) This is a mere *number*.
If we have objects c and c' in a category, hom(c,c') is the actual *set*
of ways to get from c to c', that is, the set of morphisms from c to c'.
When one computes transition amplitudes by summing over paths, as in
Feynman path integrals, one is in a sense decategorifying, that is, turning
a set of ways to get from here to there into a number, the transition
amplitude. However, one is not just counting the ways, one is counting
them "with phase".... and I must admit that the role of the *complex
numbers* in quantum theory is still puzzling from this viewpoint. For
some food for thought, you might want to check out Dan Freed's work on
torsors, which are a categorified version of phases:
1) Higher algebraic structures and quantization, by Dan Freed,
Commun. Math. Phys. 159 (1994), 343-398, also available as
hep-th/9212115.
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