June 2, 1995

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 F₀ = fg and F₁ 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 S² since its surface is 2-dimensional, and call the circle S¹; in general the unit sphere in Rⁿ⁺¹ is called Sⁿ. So for short, one says that all maps from S² to S¹ are homotopic. But: there are infinitely many different nonhomotopic maps from S³ to S²! 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 S⁴ to S³. There are also only two homotopy classes of maps from S⁵ to S⁴, and from S⁶ to S⁵, 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 S⁹ to S⁶, 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", Catⁿ 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 Catⁿ 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, there things form a category, say Cat²(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 Cat³(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....

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 g₁ g₂ g₃ = 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.

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 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 2-gerbes and 2-stacks - which are roughly like sheaves of 2-categories.

© 1995 John Baez
baez@math.removethis.ucr.andthis.edu