**Stephen P. King** (*stephenk1@home.com*)

*Wed, 08 Sep 1999 22:49:25 -0400*

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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.)

**Next message:**Stephen P. King: "[time 719] [Fwd: Does a fundamental time exist in GR and QM?]"**Previous message:**Stephen P. King: "[time 717] Re: [time 716] Time operator => Ensembles of clocks?"

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