Also available at http://math.ucr.edu/home/baez/week54.html
This Week's Finds in Mathematical Physics (Week 54)
John Baez
I just got back from a quantum gravity conference in Warsaw, and I'm dying
to talk about some of the stuff I heard there, but first I should describe
some work on topology and higher-dimensional algebra that I have been
meaning to discuss for some time now.
1) Timothy Porter, Abstract homotopy theory: the interaction of category
theory and homotopy theory, lectures from the school on "Categories and
Topology", Department of Mathematics, Universita di Genova, report #199,
March 1992.
Timothy Porter is another expert on higher-dimensional algebra whom I
met in Bangor, Wales, where he teaches. As paper 3) below makes
clear, he is very interested in the relationship between traditional
themes in topology and the new-fangled topological quantum field
theories (TQFTs) people have been coming up with these days. The
above paper does not mention TQFTs; instead, it is an overview of
various approaches that people have used to study homotopy theory in
an algebraic way. But anyone seriously interested in the intersection
of physics and topology would do well to get ahold of it, since it's a
pleasant way to get acquainted with some of the beautiful techniques
algebraic topologists have been developing, which many physicists are
just starting to catch up with.
What's homotopy theory? Well, roughly, it's the study of the properties of
spaces that are preserved by a wide class of stretchings and squashings,
called "homotopies".
For example, a closed disc D and a one-point set {p} are quite
different as topological spaces, in that there is no continuous map
from one to the other having a continuous inverse. (This is obvious
because they have a different number of points!) But there is clearly
something similar about them, because you can squash a disc down to a
point without crushing any holes in the process (since the disc has no
holes). To formalize this, note that we can find continuous functions
f: D -> {p}
and
g: {p} -> D
that are inverses "up to homotopy". For example, let f be the only
possible function from D to {p}, taking every point in D to p, and let
g be the map that sends p to the point 0, where we think of D as the
unit disc in the plane. Now if we first do g and then do f we are
back where we started from, so gf is the identity on {p}. But if we
first do f and then g we are NOT necessarily back where we started
from: instead, the function fg takes every point in D to the point 0
in D. So fg is not the identity. But it is "homotopic" to the
identity, by which I mean that there is a continuously varying family
of continuous functions F_t from D to itself, such that F_0 = fg and
F_1 is the identity on D. Simply let F_t be scalar multiplication by
t! As t goes from 1 to 0, we see that F_t squashes the disc down to a
point.
A bit more precisely, and more generally too, if we have two
topological spaces X and Y we say that two continuous functions
f,g: X -> Y are homotopic if there is a continuous function
F: [0,1] x X -> Y
such that
F(0,x) = f(x)
and
F(1,x) = g(x).
Intuitively, this means that f can be "continuously deformed" into g.
Then we say that two spaces X and Y are homotopy equivalent if there
are continuous functions f: X -> Y, g: Y -> X which are inverse up
to homotopy, i.e., such that gf and fg are homotopic to the identity
on X and Y, respectively.
The main goal in homotopy theory is to understand when functions are
homotopic and when spaces are homotopy equivalent. This is incredibly
hard in *general*, but in special cases a huge amount is known. To
take a random (but important) example, people know that all maps from
the sphere to the circle are homotopic. Remember that algebraists
call the sphere S^2 since its surface is 2-dimensional, and call the
circle S^1; in general the unit sphere in R^{n+1} is called S^n. So
for short, one says that all maps from S^2 to S^1 are homotopic. But:
there are infinitely many different nonhomotopic maps from S^3 to S^2!
In fact there is a nice way to label all these "homotopy classes" of
maps by integers. And then: there are only two homotopy classes of
maps from S^4 to S^3. There are also only two homotopy classes of
maps from S^5 to S^4, and from S^6 to S^5, and so on.
Now, the famous topologist J. H. C. Whitehead put forth an important
program in 1950, as follows: "The ultimate aim of *algebraic homotopy*
is to construct a purely algebraic theory, which is equivalent to
homotopy theory in the same way that `analytic' is equivalent to `pure'
projective geometry." Since then a lot of people have approached
this program from various angles, and Porter's paper tours some of the
key ideas involved.
Part of the reason for pursuing this program is simply to get good at
computing things, in a manner similar to how analytic geometry helps
you solve problems in `pure' geometry. This is not my main interest;
if I want to know how many homotopy classes of maps there are from S^9
to S^6, or something, I know where to look it up, or whom to ask -
which is infinitely more efficient than trying to figure it out
myself! And indeed, there is a formidable collection of tools out
there for solving various sorts of specific homotopy-theoretic
problems, not all of which rely crucially on a *general* purely
algebraic theory of homotopy.
I'm more interested in this program for another reason, which is
simply to find an algebraic language for talking about things being
true "up to homotopy". As I've tried to explain in recent "weeks",
there are many situations where equations should be replaced by some
weaker form of equivalence. Taking this seriously leads naturally to
the study of n-categories, in which equations between j-morphisms can
be replaced by specified (j+1)-morphisms. But Porter describes a host
of different (though related) formalisms set up to handle this sort of
issue. A few of the main ones are: simplicial sets, simplicial
objects in more general categories, Kan complexes, Quillen's "model
categories", Cat^n groups, and homotopy coherent diagrams.
Understanding how all these formalisms are related and what they are
good for is quite a job, but this paper helps one get started.
So far everything I've actually said is quite elementary - I've made
reference to some impressive buzzwords without explaining them, but that
doesn't count. So I should put in something for the folks who want
more! Let me say a word or two about Cat^n groups. The definition of
these is a typical mind-blowing piece of higher-dimensional algebra, so
I can't resist explaining it. (After a while these definitions stop
seeming so mind-boggling, and then one is presumably beginning
understand the point of the subject!) In "week53" I gave a definition
of a category using category theory. This might seem completely
circular and useless, but of course I was illustrating quite generally
how one could define a "model" of a "finite limit theory" using category
theory. The idea was that a category is a *set* of objects, a *set* of
morphisms, together with various *functions* like the source and target
functions which assign to any morphism (or "arrow") its source and
target (or "tail" and "tip"). These sets and functions needed to
satisfy various axioms, of course.
Now *sets* and *functions* are the objects and morphisms in the
category of sets, which folks call Set. So in "week53" I cooked up a
little category Th called "the theory of categories", which has
objects called "ob" and "mor", morphisms called "s" and "t", etc..
These were completely abstract gizmos, not actual sets and functions.
But we required them to satisfy the exact same laws that the sets of
objects and morphisms, and the source and target functions, and so on,
satisfy in an actual category. Then a functor from Th to Set which
preserves finite limits is called a "model" of the theory of
categories, because it assigns to the completely abstract gizmos
actual sets and functions satisfying the same laws. In other words,
if we have a functor F: Th -> Set, we have an actual set F(ob) of
objects, an actual set F(mor) of morphisms, an actual function F(s)
from F(ob) to F(mor), and so on. In short, we have an actual
category!
Now to get this trick to work we didn't need much to be true about the
category Set: all we needed was that it had finite limits. (Ignore
this technical stuff about limits if you don't get it; you can still
get the basic idea here.) And there are lots of categories with this
property, like the category Grp of groups. So we can also talk about
a model of the theory of categories in the category of groups! What
is this? Well, it's just a functor from Th to Grp that preserves
finite limits. More concretely, it's exactly like a category, except
everywhere in the definition of category where you see the word "set",
scratch that out and write in "group", and everywhere you see the word
"function", scratch that out and write in "homomorphism". So you have
a *group* of objects, a *group* of morphisms, together with various
*homomorphisms* like the source and target, and so on... satisfying
laws perfectly analogous to those in the definition of a category!
Folks call this kind of thing a "categorical group", a "category
object in Grp" or an "internal category in Grp". From the point of
view of sheer audacity alone, it's a wonderful concept: we have taken
the definition of a category and transplanted it from the soil in
which it was born, namely the category Set, into new soil, namely the
category Grp! But more remarkably still, the study of these
"categorical groups" is equivalent to the study of "homotopy 2-types"
- that is, topological spaces, but where you only care about them up
to homotopy, and even more drastically, where nothing above dimension
2 concerns you. This result is due (as far as I can tell) to Ronnie
Brown and C. B. Spencer, building on earlier work of Mac Lane and
Whitehead.
But why stop here? Consider the category Cat(Grp) of these category objects
in Grp. Take my word for it, such a thing exists and it has finite limits.
That means we can look for models of the theory of categories in Cat(Grp)
- i.e., functors from Th to Cat(Grp), preserving finite limits. In fact,
THESE things form a category, say Cat^2(Grp), and THIS category has
finite limits, so we can look for models of the theory of categories in
THIS category, and THESE form a category Cat^3(Grp), which also has
finite limits... etc. So we can construct an insanely recursive hierarchy:
groups
category objects in the the category of groups
category objects in the category of (category objects in the category of groups)
etc....
Now, truly wonderfully, L. Loday showed that the study of Cat^n(Grp) is
equivalent (in a certain precise sense) to the study of homotopy
n-types... i.e., homotopy theory where you don't care about phenomena
above dimension n:
2) L. Loday, Spaces with finitely many non-trivial homotopy groups, Jour.
Pure Appl. Algebra 24 (1982), 179-202.
Subsequently, Ronnie Brown, Loday and others have done some interesting
topology using this fact. But the most remarkable thing, in a way, is how
taking a perfectly basic concept, the concept of GROUP, and then doing
category theory "internally" in the category of groups in an iterated
fashion, winds up being very much the same as doing topology - or at least
homotopy theory. This suggests that there is something deeply algebraic
about homotopy theory in the first place.
3) Timothy Porter, Interpretations of Yetter's notion of G-coloring:
simplicial fibre bundles and non-abelian cohomology, available at
http://citeseer.ist.psu.edu/100965.html
Physicists know and love the Dijkgraaf-Witten model, a 2+1-dimensional
TQFT that depends on a finite group G. It's easy to compute the
Hilbert space of states for any compact oriented 2-manifold in this
TQFT. Just triangulate your 2-manifold and let your Hilbert space
have as a basis the set of all possible ways of labelling the edges
with elements of G such that g_1 g_2 g_3 = 1 whenever we have 3 edges
going counterclockwise around any triangle. Yetter figured out how to
generalize this to get an interesting TQFT from any finite categorical
group:
4) David N. Yetter, Topological quantum field theories associated to
finite groups and crossed G-sets, Journal of Knot Theory and its
Ramifications 1 (1992), 1-20.
TQFTs from homotopy 2-types, Journal of Knot Theory and its Ramifications 2
(1993), 113-123.
This should be the beginning of some bigger pattern relating homotopy
theory and TQFTs. Jim Dolan and I have our own theories as to how
this pattern should work (see "week49") but they are still a rather
long ways from being theorems. Porter, who is an expert in simplicial
methods, makes the relationship (or ONE of the relationships) very
clear in this special case.
5) Justin Roberts, Skein theory and Turaev-Viro invariants, preprint.
(Justin Roberts can be reached via email at roberts@math.berkeley.edu)
Refined state-sum invariants of 3- and 4-manifolds, preprint.
Skeins and mapping class groups, Math. Proc. Camb. Phil. Soc. 115 (1994),
53-77.
G. Masbaum and Justin Roberts, On central extensions of mapping class
groups, Mathematica Gottingensis, Schriftenreihe des
Sonderforschungsbereichs Geometrie und Analysis, Heft 42 (1993).
The first two papers here might be the most interesting for physicists.
They both deal with 3d and 4d TQFTs constructed using quantum SU(2):
in particular, the Turaev-Viro theory in dimension 3, and the
Crane-Yetter-Broda theory in dimension 4. The first theory is interesting
physically because it corresponds to 3d Euclidean quantum gravity with
cosmological constant. The second theory is interesting mainly because
it's one of the few 4d TQFTs for which the Atiyah axioms have been shown;
sometime I will explain the Lagrangian for this theory, which seems not to
be well-known.
In Roberts' first paper, which was already discussed in "week14", he
gave a simple proof that the partition function for the Turaev-Viro
theory was the absolute value squared of that for Chern-Simons theory,
perhaps the most famous of TQFTs. He also showed that the partition
function for the Crane-Yetter-Broda theory was a function of the
signature and Euler characteristic (classical invariants of
4-manifolds). In the second paper, he considers observables for the
TV and CYB theories depending on cohomology classes in the manifold.
I wish I had energy now to explain a bit more about these observables,
since they are very interesting, but I don't!
6) Lawrence Breen, On the Classification of 2-Gerbes and 2-Stacks,
Asterisque 225, 1994.
Suffice it to say that if gerbs and stacks - which are, very roughly,
sort of like sheaves of categories - are too simple to interest you,
it's time to read about 2-gerbs and 2-stackes - which are roughly like
sheaves of 2-categories.
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