Stephen P. King (stephenk1@home.com)
Wed, 08 Sep 1999 22:49:25 -0400
Hi All,
Here is John Baez talking about something that I think relates to our
discussion! :-)
Stephen
attached mail follows:
>One might say the category *is* not Set with extra structure,
>but it may be *interpeted* as Set with extra structure.
>(Ultimately, category theory rejects that distinction as meaningless.)
>In fact, james said the Yoneda lemma ensures that
>every category may be so interpreted, modulo set/class distinctions.
>Since I never got the hang of Yoneda, I'll believe him.
It took me ages to get the hang of the Yoneda lemms, so I
sympathize with you. Let me just say a few things that might
eventually help. One way to think of the Yoneda lemma is
precisely this: that the objects of any category can be
interpreted as sets with extra structure. Think about this
a minute. We have an abstract category C and we wish to
associate to each object of C some set equipped with extra
structure. Moreover, we want to do this in a way which
completely records everything there is to know about this
object. How can we do it?
Well, the only interesting thing about an object in a category
is its morphisms to and from other objects, and how these compose
with *other* morphisms. This principle should be our guide.
So, what should we do? Simple: associate to the object c the
set of all morphisms from c to other objects in C! Let's call
this set hom(c,-).
Of course, this is more than a mere set: it's a set with extra
structure. First of all, it's a set made of lots of little
subsets for each object c' in C, we get a subset hom(c,c'),
consisting of all morphisms from c to c'. Second of all, it's
a set with an "action of C". In other words, given an element
f in hom(c,c'), and a morphism g: c' -> c'', we get an element
fg in hom(c,c''), just by composing f and g.
A set with all this structure has a name: it's called a "functor
from C to Set".
The Yoneda lemma says that this "set with extra structure" knows
everything you'd ever want to know about the object c. You might
enjoy making this precise and proving it. (The only way to understand
certain theorems, it seems, is to prove them yourself - sometimes
repeatedly.)
Note that I got away with less than you might have thought I'd
need! I only considered the morphisms *from* c, not the morphisms
*to* c. In fact there is another version of the Yoneda lemma
that uses the morphisms *to* c instead. I believe this is
the one people usually talk about - but of course it doesn't
really matter.
The importance of all this for physics is as follows. Lots of
people working on quantum gravity like to stress the importance
of "relationalism" - the idea that physical things only have properties
by virtue of their relation with other physical things. For example,
it only makes sense to say how something is moving *relative* to
other things. This idea is an old one, going back at least to
Leibniz, and attaining a certain prominence with Mach (who primarily
applied it to position and velocity, rather than other properties).
Relationalism is appealing, at least to certain kinds of people, but
it's a bit dizzying: if all properties of a thing make sense only in
relation to other things, how do we get started in the job of describing
anything at all? The danger of "infinite regress" has traditionally
made certain other kinds of people recoil from relationalism; they
urge that one posit of something "absolute" to get started.
Category theory provides a nice simple context to see relationalism
in action, in a completely rigorous and precise form. In a category,
objects do not have "innards" - viewed in isolation, they are all
just featureless dots. It's only by virtue of their morphisms to and
from other objects (and themselves) that they acquire distinct
personalities. This is why an isomorphism between objects allows
us to treat them as "the same": it establishes a 1-1 correspondence
between their morphisms to, or from, other objects. (Moreover, this
correspondence preserves the extra structure described above.)
This suggests that a truly relational theory of physics should
take advantage of category theory.
Of course, when people say something like the previous sentence,
we usually don't take them very seriously: philosophically, it
may seem like a great idea to base one's theory of physics on this
or that principle, but the test of a theory of physics is the
predictions it makes, not how cool its underlying philosophy may
be.
Thus, to prove that I am not merely engaged in idle chit-chat, I
should really exhibit a theory of physics, based on category theory,
which makes new predictions, and ultimately I should test these
predictions and make sure they're right.
Unfortunately I can't do this yet.
What I *can* do, though, is to note that various "toy models" -
simplified theories of physics - can be formulated very neatly
using category theory. The neatest example is 3d Riemmannian
quantum gravity, possibly coupled to Chern-Simons gauge fields.
A more ambitious example is the Barrett-Crane model of 4d Lorentzian
quantum gravity. It's incredibly elegant, but it's too soon to
say it's "right": since it has local degrees of freedom, unlike
the 3d example, it is not exactly soluble, so we don't yet know
how to calculate with it, other than by throwing a supercomputer
at it (which none of us category-theoretic physicists can afford).
And, of course, by "right" I don't mean that the Barrett-Crane
models stands a chance of describing *our* universe, since it
doesn't include matter, only gravity. What I mean is simply that
it reduces to the vacuum Einstein equations in a suitable low-
energy limit.
(I should be careful: it's possible that wormhole ends in pure
quantum gravity manage to act like spin-1/2 particles - thus giving
us "matter without matter". I actually hope this is true. However,
I find it rather unlikely that the Standard Model gauge group
could pop out of pure quantum gravity, even with wormholes around.
So my most optimistic realistic hope is that a souped-up Barrett-
Crane model with a bigger gauge group could give real-world physics.)
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