
For the last year or so I've been really getting interested in ncategories as a possible tool for unifying a lot of strands in mathematics and physics. What's an ncategory? Well, in a sense that's the big question! Roughly speaking, it's a structure where there are a bunch of "objects", and for any pair of objects x,y a bunch of "morphisms" from x to y, written f: x → y, and for any pair of morphisms f, g: x → y a bunch of "2morphisms" from f to g, written F: f => g, and for any pair of 2morphisms F, G: f => g a bunch of "3morphisms" from F to G,... and so on, up to nmorphisms. Ordinary categories, or 1categories, have been studied since the 1940s or so, when they were invented by Eilenberg and Mac Lane. Anyone wanting to get going on those could try:
1) Categories for the Working Mathematician, by S. Mac Lane, Springer, Berlin, 1988.
Roughly, categories give a general framework for dealing with situations where you have "things" and "ways to go between things", like sets and functions, vector spaces and linear maps, points in a space and paths between points, etc. That's a pretty broad territory! ncategories show up when you also have "ways to go between ways", "ways to go between ways to go between ways", etc. That may seem a little weird at first. But in fact they show up in a lot of places if you look for them. Perhaps the most obvious place is topology. If you think of a point in a space as an object, and a path between two points as a morphism:
f x > y
you are easily tempted to think of a "path of paths" as a 2morphism. Here a "path of paths" is just a continuous 1parameter family of paths from x to y, which you can think of as tracing out a 2dimensional surface, as follows:
f > / \ /  \ /  F \ x  y \ V / \ / \ g / >
And one can keep on going and look at "paths of paths of paths", etc. In fact, people in homotopy theory do this all the time.
There is another example, equally primordial, which is a bit more "inbred" in flavor. In other words, if you already know and love ncategories, there is a wonderful example of an (n+1)category which you should know and love too! Now this isn't so bad, actually, because a 0category is basically just a set, namely the set of objects. Since everyone knows and loves sets, everyone can start here! Okay, there is a wonderful 1category called Set, the category of all sets. This has sets as its objects and functions between sets as its morphisms. So now that you know an example of a 1category, you know and love 1categories, right? Well, it turns out there is this wonderful thing called Cat, the 2category of all 1categories. (Usually people restrict to "small" 1categories, which have a mere set of objects, so that set theorists don't start freaking out at a certain point.) To understand why Cat is a 2category is a bit of work, but as objects it has categories, as morphisms it has the usual sort of morphisms between categories, socalled "functors", and as 2morphisms it has the usual sort of things that go between functors, socalled "natural transformations". These are the bread and butter of category theory; just take my word for it if you haven't studied them yet! Okay, so Cat is a 2category, so now you know and love 2categories, right? (Well, I haven't even told you the definitions, but just nod your head.) Guess what: there is this wonderful thing called 2Cat, the 3category of all 2categories! And so on.
So in short, the detailed theory of ncategories at each level automatically leads one to get interested in (n+1)categories. Now for the bad news: so far, people have only figured out the right definition of ncategory for n = 0, 1, 2, and 3. By the "right" definition I mean the ultimate, most general definition, which should be the most useful in many ways. So far people only know about "strict" ncategories for all n, which one can think of as a special case of the ultimate ones; the ultimate 1categories are just categories, the ultimate 2categories are often called bicategories (see the reference to Benabou in week35), and ultimate 3categories are usually called tricategories (see the reference to the paper by Gordon, Power and Street) in week29. Tricategories were just defined last year! They have a whole lot to do with knots, ChernSimons theory, and other 3dimensional phenomena, as one might expect. If we could understand the ultimate 4categories  tetracategories?  it would probably help us with some of the riddles of topology and physics in 4 dimensions. (Indeed, what little we do understand is already helping a bit.)
So anyway, I have been trying to learn about these things, and had the good luck to meet James Dolan via the net, who has helped me immensely, since he eats, lives and breathes category theory, and he is now at Riverside hard at work figuring out the ultimate definition of ncategories for all n. (Although when he reads this, he will not be hard at work; he will be goofing off, reading the news.)
He and I have have written one paper so far espousing our philosophy concerning ncategories, topology, and physics:
2) Higherdimensional algebra and topological quantum field theory, by John Baez and James Dolan, Jour. Math. Phys. 36 (1995), 60736105. Also available as arXiv:qalg/9503002.
One of the main themes of this paper is what I sometimes jokingly call the "periodic table". Say you have an (n+k)category with only one object, one morphism, one 2morphism, ... and only one (k1)morphism. Then all the interest lies in the kmorphisms, the (k+1)morphisms, and so on up to the (k+n)morphisms. So there are n interesting levels of morphism, and we can actually think of our (n+k)category as an ncategory of a special sort. Let's call this kind a "ktuply monoidal ncategory". Now we can make a chart of these:
ktuply monoidal ncategories n = 0 n = 1 n = 2 k = 0 sets categories 2categories k = 1 monoids monoidal monoidal categories 2categories k = 2 commutative braided braided monoids monoidal monoidal categories 2categories k = 3 " " symmetric weakly monoidal involutory categories monoidal 2categories k = 4 " " " " strongly involutory monoidal 2categories k = 5 " " " " " "
First, I should emphasize that some parts of the chart as I've drawn it here are a bit conjectural; since we don't know what the most general 7categories are like, for example, we don't really know for sure what 5tuply monoidal 2categories are like. The exact status of all the entries on the table is made more clear in the paper. For now, let me just say, first, that these various flavors of ncategories turn out to be of great interest in topology  some have already been used a lot in topological quantum field theory and knot theory, other less, so far, but they all seem to have lot to do with generalizations of knot theory to different dimensions. Second, it seems that the nth column "stabilizes" by the time you get down to the (n+2)nd row. This very interesting pattern turns out also to have a lot to do with knots and their generalizations, and also to a subject called stable homotopy theory.
Now it also appears that there is a nice recipe for hopping down the columns. (Again, we only understand this perfectly in certain cases, but the pattern seems pretty clear.) In other words, there's a nice recipe to get a (k+1)tuply monoidal ncategory from a ktuply monoidal one. It goes like this. Hang on to your seat. You start with a ktuply monoidal ncategory C. It's a special sort of (n+k)category, so its an object in (n+k)Cat. But (n+k)Cat, remember, is an (n+k+1)category. Now look at the largest sub(n+k+1)category of (n+k)Cat which has C as its only object, 1_{C} (the identity of C) as its only morphism, 1_{1C} as its only 2morphism, 1_{11C} as its only 3morphism, and so on, up to 1_{111…} as its only kmorphism. Let's call this C'. If one keeps count, this should be a (k+1)tuply monoidal ncategory. That's how it goes.
Now say we do this to an example. Say we do it to the category C of all representations of a finite group G. This is in fact a monoidal category, so the result C' is a braided monoidal category. It is, in fact, just the category of representations of the "quantum double" of G, which is an example of what one might call a "finite quantum group". These play a big role in the study of ChernSimons theory with finite gauge group (see the papers by Freed and Quinn in week48). One can also get the other quantum groups with the aid of this "quantum double" trick. A good description of this case appears in:
3) Double construction for monoidal categories, by Christian Kassel and Vladimir Turaev, Publication de l'Institute de Recherche Mathematique Avancee, 1992.
So this is rather remarkable: starting from a finite group, and all this ncategorical abstract nonsense, out pops precisely the raw ingredients for a perfectly respectable 3dimensional topological quantum field theory! Understanding why this kind of thing works is part of the aim of Dolan's and my paper, though there are some important pieces of the puzzle that we don't get around to mentioning there.
Right now I'm busily working out the details of how to get braided monoidal 2categories from monoidal 2categories by the same trick, with the aid of Martin Neuchl and Frank Halanke here. These should have a lot to do with 4dimensional topological quantum field theories (see e.g. the paper by Crane and Yetter cited in week46). And here I can't resist mentioning a very nice paper by Neuchl and Schauenburg,
4) Reconstruction in braided categories and a notion of commutative bialgebra, Martin Neuchl and Peter Schauenburg, Mathematisches Institut, Theresienstr. 39, 80333 Muenchen, Feb. 20, 1995.
Let me conclude by describing this. I always let myself get a bit more technical at the end of each issue, so I'll do that now. The relationship between Hopf algebras and monoidal categories is given by "TannakaKrein reconstruction theorems", which give conditions under which a monoidal category is equivalent to the category of representations of a Hopf algebra, and actually constructs the Hopf algebra for you. In physics people use related but fancier "DoplicherHaagRoberts" theorems to reconstruct the gauge group of a quantum field theory. This paper starts with the beautiful TannakaKrein theorem in
5) Tannaka duality for arbitrary Hopf algebras, by Peter Schauenburg, AlgebraBerichte 66 (1992).
Leaving out a bunch of technical conditions that make the theorem actually TRUE, it says roughly that when you have a braided monoidal category B, a category, and a functor f: C → B, there is a coalgebra object a in B, the universal one for which f factors through the forgetful functor from aComod (the category of acomodule objects in B) to B. The point is that the ordinary TannakaKrein theorem is a special case of this one where B is the category of vector spaces. The point of the new paper is as follows. Suppose C is actually braided monoidal and f preserves the braiding and monoidal structure. Then we expect a to actually be something like commutative bialgebra object in B. The paper makes this precise. There are actually some sneaky issues involved in doing so. In particular, the "quantum double" trick for categories makes an appearance here. I guess I'll leave it at that!
© 1995 John Baez
baez@math.removethis.ucr.andthis.edu
