Math Forum Discussions

John Baez

Posts: 542
Registered: 12/6/04

Re: This Week's Finds in Mathematical Physics (Week 209)
Posted: Feb 1, 2005 2:31 PM

Plain Text

In article <coisoa$207$1@news2.netvision.net.il>,
Squark <fiis5d@yahoo.com> wrote:

>John Baez wrote:

>> The n-categories called "n-groupoids" magically know
>> everything there is to know about homotopy theory,

>That part I know...

>> while those called
>> "n-categories with duals" know everything there is to know about the
>> topology of manifolds.

>But how would this work? What are n-categories with duals anyway?

We don't know a general definition, but we have a good idea
in low-dimensional cases. Consider the category Hilb, where the
objects are finite-dimensional complex Hilbert spaces and
the morphisms are linear operators. This category has a tensor
product so we call it a monoidal category. In any monoidal category,
we can say what it means for an object to have a "dual". In the
case of Hilb, every object H has a dual H* - the dual Hilbert space.
We can also say what it means for a morphism to have a "dual". In
the case of Hilb, every morphism f: H -> H' has a dual f*: H' -> H -
the adjoint operator. So, we say our category has duals for objects
and morphisms.

Interestingly, the main source of monoidal categories with duals
is topology: categories of cobordisms. This suggests an interesting
relation between quantum mechanics (which uses Hilb) and topology.
Indeed, this relation is precisely what we exploit in topological
quantum field theory! In fact, I claim it will eventually make a
lot of mysteries in quantum theory seem less mysterious. The
reason we find quantum theory mysterious is that it's based on
monoidal categories with duals, and these are fairly unfamiliar -
UNTIL we start drawing pictures of cobordisms and see that they
formalize simple ideas about spacetime topology!

So, I hope that quantum theory will become a lot clearer when we
unify it with a theory of spacetime. I explain this hope in vastly
more detail here:

http://math.ucr.edu/home/baez/quantum/

Now, to make a little more progress in explaining n-categories
with duals, I should explain why they are a bit like n-groupoids,
which are n-categories with inverses.

To start, let me explain why the dual of a finite-dimensional
vector space, is similar to the inverse of a number.

I assume you know that multiplying numbers is a lot like tensoring vector
spaces. For example, just as multiplication distributes over addition,
tensoring distributes over direct sums. Also, just as there is a number
called 1 which is the unit for multiplication, there is a 1-dimensional vector
space, the ground field itself, which is the unit for tensoring. Let me
take the unusual liberty of writing tensor products by juxtaposition, so
that xy is the tensor product of the vector space x and the vector space
y, and let me call the 1-dimensional vector space that's the unit for
tensoring simply "1".

Now, if a number x has an inverse y, we have

yx = 1

and

1 = xy.

Similarly, if a vector space x has a dual y, we have linear maps

e: yx -> 1

and

i: 1 -> xy

What are these linear maps? Well, the whole point of the dual vector
space y is that a vector in y is a linear functional from x to 1.
This "dual pairing" between vectors in y and those in x defines a
linear map e: yx -> 1, often called the "counit". On the other hand,
the space xy can be thought of as the space of linear transformations of
x. The linear map i: 1 -> xy sends any scalar (i.e., any vector in 1)
to the corresponding scalar multiple of the identity transformation of
x.

So we see that dual vector spaces are a bit like inverse numbers, except
that we don't have yx = 1 and 1 = xy, and we don't even have that yx is
*isomorphic* to 1 and 1 is *isomorphic* to xy. We just have some maps
going from yx to 1, and from 1 to xy.

These maps satisfy two equations, though. Here's the first. We start
with x, use the obvious isomorphism to map to 1x, then use i: 1 -> xy to
map this to xyx, then use e: yx -> 1 to map this to x1, and then use the
other obvious isomorphism to map back to x. This composite of maps
should be the identity on x. What this says is that the identity linear
transformation of x really acts as the identity!

We can draw this as follows. Draw the counit e: yx -> 1 as follows:

y x
\ /
\ /
\/

and draw the unit i: 1 -> xy as follows:

/\
/ \
/ \
x y

Then the above equation says that

x x
/\ | |
/ \ | |
/ \ | |
x| y\ x/ = |
| \ / |
| \/ |
x x

Here the left side, which we read from top to bottom, corresponds to the
composite x -> 1x -> xyx -> x1 -> x. (The factors of 1 are invisible in the
picture, since they don't do much.) The left side corresponds to the
identity map x -> x.

The second equation goes like this. We start with y, use the obvious
isomorphism to map to y1, then use the unit to map this to yxy, then use
the counit to map this to 1y, and then use the other obvious isomorphism
to map back to y. This composite should be the identity on y. What
this says is that the identity linear transformation of x also acts
dually as the identity on y! We can draw this as follows:

y y
| /\ |
| / \ |
| / \ |
y\ x/ y| = |
\ / | |
\/ | |
y y

In general, whenever we have a monoidal category, we say an object
x has a dual y when it's equipped with morphisms i: 1 -> xy, e: yx -> 1
satisfying these two equations - which are called the "zig-zag identities".

You can see already that curves in the plane give us an example of
a monoidal category where all the objects have duals.

So, you can begin to see the relation between monoidal categories
with duals and the topology of 1-dimensional manifolds embedded in
2-dimensional space. This is the tip of an enormous iceberg called
the "tangle hypothesis", which is explained here:

http://www.arxiv.org/abs/q-alg/9503002

In particular, if this hypothesis could be made precise, we would
have a completely algebraic description of smooth manifolds in terms
of n-categories with duals. For some progress see this:

http://www.arxiv.org/abs/math.QA/9811139

which gives an algebraic description of 2-dimensional manifolds
embedded in 4-dimensional space in terms of braided monoidal
2-categories with duals.

(Maybe you can already guess the pattern!)

There's a lot more to say but maybe this is a start.