Also available at http://math.ucr.edu/home/baez/week74.html
March 5, 1996
This Week's Finds in Mathematical Physics - Week 74
John Baez
Before continuing my story about higher-dimensional algebra, let me say
a bit about gravity. Probably far fewer people study general relativity
than quantum mechanics, which is partially because quantum mechanics is
more practical, but also because general relativity is mathematically
more sophisticated. This is a pity, because general relativity is so
beautiful!
Recently, I have been spending time on sci.physics leading an informal
(nay, chaotic) "general relativity tutorial". The goal is to explain
the subject with a minimum of complicated equations, while still getting
to the mathematical heart of the subject. For example, what does
Einstein's equation REALLY MEAN? It's been a lot of fun and I've
learned a lot! Now I've gathered up some of the posts and put them on a
web site:
1) John Baez et al, General relativity tutorial,
http://math.ucr.edu/home/baez/gr/gr.html
I hope to improve this as time goes by, but it should already be fun to
look at.
Let me also list a couple new papers on the loop representation of quantum
gravity, dealing with ways to make volume and area into observables in
quantum gravity:
2) Abhay Ashtekar and Jerzy Lewandowski, Quantum Theory of Geometry I:
Area Operators, 31 pages in LaTeX format, to appear in Classical and
Quantum Gravity, preprint available as gr-qc/9602046.
Jerzy Lewandowski, Volume and Quantizations, preprint available as
gr-qc/9602035.
Roberto De Pietri and Carlo Rovelli, Geometry Eigenvalues and Scalar
Product from Recoupling Theory in Loop Quantum Gravity, 38 pages, 5
Postscript figures, uses RevTeX 3.0 and epsfig.sty, preprint available
as gr/qc-9602023.
I won't say anything about these now, but see "week55" for some
information on area operators.
Okay, where were we? We had started messing around with sets, and we
noted that sets and functions between sets form a category, called Set.
Then we started messing around with categories, and we noted that not
only are there "functors" between categories, there are things that ply
their trade between functors, called "natural transformations". I then
said that categories, functors, and natural transformations form a
2-category, called Cat. I didn't really say what a 2-category is,
except to say that it has objects, morphisms between objects, and
2-morphisms between morphisms. Finally, I said that this pattern
continues: nCat forms an (n+1)-category.
By the way, I said last time that Set was "the primordial category".
Keith Ramsay reminded me by email that this can be misleading. There
are other categories that act a whole lot like Set and can serve equally
well as "the primordial category". These are called topoi. Poetically
speaking, we can think of these as alternate universes in which to do
mathematics. For more on topoi, see "week68". All I meant by saying
that Set was "the primordial category" is that, if we start from Set and
various categories of structures built using sets --- groups, rings,
vector spaces, topological spaces, manifolds, and so on --- we can then
abstract the notion of "category", and thus obtain Cat. In the same
sense, Cat is the primordial 2-category, and so on.
I mention this because it is part of a very important broad pattern in
higher-dimensional algebra. For example, we will see that the complex
numbers are the primordial Hilbert space, and that the category of
Hilbert spaces is the primordial "2-Hilbert space", and that the
2-category of 2-Hilbert spaces is the primordial "3-Hilbert space", and
so on. This leads to a quantum-theoretic analog of the hierarchy of
n-categories, which plays an important role in mathematical physics.
But I'm getting ahead of myself!
Let's start by considering a few examples of categories. I want to pick
some examples that will lead us naturally to the main themes of
higher-dimensional algebra. Beware: it will take us a while to get
rolling. For a while --- maybe a few issues of This Week's Finds ---
everything may seem somewhat dry, pointless and abstract, except for
those of you who are already clued in. It has the flavor of
"foundations of mathematics," but eventually we'll see these new
foundations reveal topology, representation theory, logic, and quantum
theory to be much more tightly interknit than we might have thought. So
hang in there.
For starters, let's keep the idea of "symmetry" in mind. The typical
way to think about symmetry is with the concept of a "group". But
to get a concept of symmetry that's really up to the demands put on it
by modern mathematics and physics, we need --- at the very least --- to
work with a *category* of symmetries, rather than a group of symmetries.
To see this, first ask: what is a category with one object? It is a
"monoid". The *usual* definition of a monoid is this: a set M with an
associative binary product and a unit element 1 such that a1 = 1a = a
for all a in S. Monoids abound in mathematics; they are in a sense the
most primitive interesting algebraic structures.
To check that a category with one object is "essentially just a monoid",
note that if our category C has one object x, the set hom(x,x) of all
morphisms from x to x is indeed a set with an associative binary
product, namely composition, and a unit element, namely 1_x. (Actually,
in an arbitrary category hom(x,y) could be a class rather than a set.
But let's not worry about such nuances.) Conversely, if you hand me a
monoid M in the traditional sense, I can easily cook up a category with
one object x and hom(x,x) = M.
How about categories in which every morphism is invertible? We
say a morphism f: x -> y in a category has inverse g: y -> x if fg = 1_x
and gf = 1_y. Well, a category in which every morphism is invertible is
called a "groupoid".
Finally, a group is a category with one object in which every morphism
is invertible. It's both a monoid and a groupoid!
When we use groups in physics to describe symmetry, we think of each
element g of the group G as a "process". The element 1 corresponds to the
"process of doing nothing at all". We can compose processes g and h ---
do h and then g -- and get the product gh. Crucially, every process g
can be "undone" using its inverse g^{-1}.
We tend to think of this ability to "undo" any process as a key aspect
of symmetry. I.e., if we rotate a beer bottle, we can rotate it back so
it was just as it was before. We don't tend to think of SMASHING the
beer bottle as a symmetry, because it can't be undone. But while
processes that can be undone are especially interesting, it's also nice
to consider other ones... so for a full understanding of symmetry we
should really study monoids as well as groups.
But we also should be interested in "partially defined" processes,
processes that can be done only if the initial conditions are right.
This is where categories come in! Suppose that we have a bunch of
boxes, and a bunch of processes we can do to a bottle in one box to turn
it into a bottle in another box: for example, "take the bottle out of
box x, rotate it 90 degrees clockwise, and put it in box y". We can
then think of the boxes as objects and the process as morphisms: a
process that turns a bottle in box x to a bottle in box y is a morphism
f: x -> y. We can only do a morphism f: x -> y to a bottle in box x,
not to a bottle in any other box, so f is a "partially defined" process.
This implies we can only compose f: x -> y and g: u -> v to get fg:
x -> v if y = u.
So: a monoid is like a group, but the "symmetries" no longer need be
invertible; a category is like a monoid, but the "symmetries" no longer
need to be composable!
Note for physicists: the operation of "evolving initial data from one
spacelike slice to another" is a good example of a "partially defined"
process: it only applies to initial data on that particular spacelike
slice. So dynamics in special relativity is most naturally described
using groupoids. Only after pretending that all the spacelike slices
are the same can we pretend we are using a group. It is very common to
pretend that groupoids are groups, since groups are more familiar, but
often insight is lost in the process. Also, one can only pretend a
groupoid is a group if all its objects are isomorphic. Groupoids really
are more general.
Physicists wanting to learn more about groupoids might try:
3) Alan Weinstein, Groupoids: unifying internal and external symmetry,
available as http://math.berkeley.edu/~alanw/Groupoids.ps or
http://www.ams.org/notices/199607/weinstein.pdf
So: in contrast to a set, which consists of a static collection of
"things", a category consists not only of objects or "things" but also
morphisms which can viewed as "processes" transforming one thing into
another. Similarly, in a 2-category, the 2-morphisms can be regarded as
"processes between processes", and so on. The eventual goal of basing
mathematics upon omega-categories is thus to allow us the freedom to
think of any process as the sort of thing higher-level processes can go
between. By the way, it should also be very interesting to consider
"Z-categories" (where Z denotes the integers), having j-morphisms not
only for j = 0,1,2,... but also for negative j. Then we may also think
of any thing as a kind of process.
How do the above remarks about groups, monoids, groupoids and categories
generalize to the n-categorical context? Well, all we did was start
with the notion of category and consider two sorts of requirement: that
the category have just one object, or that all morphisms be invertible.
A category with just one object --- a monoid --- could also be seen as a
set with extra algebraic structure, namely a product and unit. Suppose
we look at an n-category with just one object? Well, it's very similar:
then we get a special sort of (n-1)-category, one with a product and
unit! We call this a "monoidal (n-1)-category". I will explain this
more thoroughly later, but let me just note that we can keep playing
this game, and consider a monoidal (n-1)-category with just one object,
which is a special sort of (n-2)-category, which we could call a "doubly
monoidal (n-2)-category", and so on. This game must seem very abstract
and mysterious when one first hears of it. But it turns out to yield a
remarkable set of concepts, some already very familiar in mathematics,
and it turns out to greatly deepen our notion of "commutativity". For
now, let me simply display a chart of "k-tuply monoidal n-categories"
for certain low values of n and k:
k-tuply monoidal n-categories
n = 0 n = 1 n = 2
k = 0 sets categories 2-categories
k = 1 monoids monoidal monoidal
categories 2-categories
k = 2 commutative braided braided
monoids monoidal monoidal
categories 2-categories
k = 3 " " symmetric weakly
monoidal involutory
categories monoidal
2-categories
k = 4 " " " " strongly
involutory
monoidal
2-categories
k = 5 " " " " " "
The quotes indicate that each column "stabilizes" past a certain
point. If you can't wait to read more about this, you might try
"week49" for more, but I will explain it all in more detail in future
issues.
What if we take an n-category and demand that all j-morphisms (j > 0) be
invertible? Well, then we get something we could call an "n-groupoid".
However, there are some important subtle issues about the precise sense
in which we might want all j-morphisms to be invertible. I will have to
explain that, too.
Let me conclude, though, by mentioning something the experts should
enjoy. If we define n-groupoids correctly, and then figure out how to
define omega-groupoids correctly, the homotopy category of
omega-groupoids turns out to be equivalent to the homotopy category of
topological spaces. The latter category is something algebraic
topologists have spent decades studying. This is one of the main ways
n-categories are important in topology. Using this correspondence
between n-groupoid theory and homotopy theory, the "stabilization"
property described above is then related to a subject called "stable
homotopy theory", and "Z-groupoids" are a way of talking about "spectra"
--- another important tool in homotopy theory.
The above paragraph is overly erudite and obscure, so let me explain the
gist: there is a way to think of a topological space as giving us an
omega-groupoid, and the omega-groupoid then captures all the information
about its topology that homotopy theorists find interesting. (I will
explain in more detail how this works later.) If this is *all*
n-category theory did, it would simply be an interesting language for
doing topology. But as we shall see, it does a lot more. One reason is
that, not only can we use n-categories to think about spaces, we can
also use them to think about symmetries, as described above. Of course,
physicists are very interested in space and also symmetry. So from the
viewpoint of a mathematical physicist, one interesting thing about
n-categories is that they *unify* the study of space (or spacetime) with
the study of symmetry.
I will continue along these lines next time and try to fill in some of
the big gaping holes.
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