
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 higherdimensional 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 higherdimensional 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 newfangled 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 onepoint 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 → 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 homotopic 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 homotopic. 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 2dimensional, 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 homotopytheoretic 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 ncategories, in which equations between jmorphisms 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 mindblowing piece of higherdimensional algebra, so I can't resist explaining it. (After a while these definitions stop seeming so mindboggling, 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 2types"  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, there 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 ntypes  i.e., homotopy theory where you don't care about phenomena above dimension n:
2) L. Loday, Spaces with finitely many nontrivial homotopy groups, Jour. Pure Appl. Algebra 24 (1982), 179202.
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 Gcoloring: simplicial fibre bundles and nonabelian cohomology, available at http://citeseer.ist.psu.edu/100965.html
Physicists know and love the DijkgraafWitten model, a 2+1dimensional TQFT that depends on a finite group G. It's easy to compute the Hilbert space of states for any compact oriented 2manifold in this TQFT. Just triangulate your 2manifold 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 Gsets, Journal of Knot Theory and its Ramifications 1 (1992), 120.
TQFTs from homotopy 2types, Journal of Knot Theory and its Ramifications 2 (1993), 113123.
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 TuraevViro invariants, preprint.
Refined statesum invariants of 3 and 4manifolds, preprint.
Skeins and mapping class groups, Math. Proc. Camb. Phil. Soc. 115 (1994), 5377.
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 TuraevViro theory in dimension 3, and the CraneYetterBroda 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 wellknown.
In Roberts' first paper, which was already discussed in "week14", he gave a simple proof that the partition function for the TuraevViro theory was the absolute value squared of that for ChernSimons theory, perhaps the most famous of TQFTs. He also showed that the partition function for the CraneYetterBroda theory was a function of the signature and Euler characteristic (classical invariants of 4manifolds). 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 2Gerbes and 2Stacks, Asterisque 225, 1994.
Suffice it to say that if gerbes and stacks  which are, very roughly, sort of like sheaves of categories  are too simple to interest you, it's time to read about 2gerbes and 2stacks  which are roughly like sheaves of 2categories.
© 1995 John Baez
baez@math.removethis.ucr.andthis.edu
