a           a          a
a + b  =   +    =    +   =    +    =  b + a
             b       b      b

This is all very simple. Historically, however, it took people a long to really understand. It's one of those things that's too simple to take seriously until it turns out to have complicated ramifications. Now it goes by the name of the "Eckmann-Hilton theorem", which says that "a monoid object in the category of monoids is a commutative monoid". You practically need a PhD in math to understand that! However, lest you think that Eckmann and Hilton were merely dressing up the obvious in fancy jargon, it's important to note that what they did was to figure out a framework that turns the above "picture proof" that a+b = b+a into an actual rigorous proof! This is one of the goals of higher-dimensional algebra.

The above proof that a+b = b+a uses 2-dimensional space, but if you really think about it also uses a 3rd dimension, namely time: the time that passes as you move "a" around "b". If we draw this 3rd dimension as space rather than time we can visualize the process of moving a around b as follows:

             a       b
              \     /
               \   /
                \ /
                 /
                / \
               /   \
              /     \
             b       a

             a       b
              \     /
               \   /
                \ /
                 \
                / \
               /   \
              /     \
             b       a

In fact, the mathematics of braids, and related things like knots, is crucially important for understanding quantum gravity in 3-dimensional spacetime. Spacetime is really 4-dimensional, of course, but quantum gravity in 4-dimensional spacetime is awfully difficult, so in the late 1980s people got serious about studying 3-dimensional quantum gravity as a kind of warmup exercise. It turned out that the math required was closely related to some mysterious new mathematics related to knots and "braidings". At first this must sound bizarre: a deep relationship between knots and 3-dimensional quantum gravity! However, after you fight your way through the sophisticated mathematical physics that's involved, it becomes clear why they are related: both rely crucially on "3-dimensional algebra", the algebra describing how you can move things around in 3-dimensional spacetime.

However, there is more to the story, because knot theory also seems deeply related to 4-dimensional quantum gravity. Here the knots arise as "flux tubes of area" living in 3-dimensional space at a given time. Recent work on quantum gravity suggests that as time passes these knots (or more generally, "spin networks") move around and change topology as time passes.

To really understand this, we probably need to understand "4-dimensional algebra". Unfortunately, not enough is known about 4-dimensional algebra. The problem is that we don't know much about 4-categories! To do n-dimensional algebra in a really nice way, you need to know about n-categories. Roughly speaking, an n-category is an algebraic structure that has a bunch of things called "objects", a bunch of things called "morphisms" that go between objects, and similarly 2-morphisms going between morphisms, 3-morphisms going between 2-morphisms, and so on up to the number n. You can think of the objects as "things" of whatever sort you like, the morphisms as processes going from one thing to another, the 2-morphisms as meta-processes going from one process to another, and so on. Depending on how you play the n-category game, there are either no morphisms after level n, or only simple and bland ones playing the role of "equations". The idea is that in the world of n-categories, one keeps track of things, processes, meta-processes, and so on to the nth level, but after that one calls it quits and uses equations.

So what is the definition of 4-categories? Well, Eilenberg and Mac Lane defined 1-categories, or simply "categories", in a paper that was published in 1945:

1) S. Eilenberg and S. Mac Lane, General theory of natural equivalences, Trans. Amer. Math. Soc. 58 (1945), 231-294.

Benabou defined 2-categories - though actually he called them "bicategories" - in a 1967 paper:

2) J. Benabou, Introduction to bicategories, Springer Lecture Notes in Mathematics 47, New York, 1967, pp. 1-77.

Gordon, Power, and Street defined 3-categories - or actually "tricategories" - in a paper that came out in 1995:

3) R. Gordon, A. J. Power, and R. Street, Coherence for tricategories, Memoirs Amer. Math. Soc. 117 (1995) Number 558.

This step took a long time in part because it took a long time for people to understand deeply where braidings fit into the picture.

But what about 4-categories and higher n? Well, the history is complicated and I won't get it right, but let me say a bit anyway. First of all, there are some things called "strict n-categories" that people have known how to define for arbitrarily high n for quite a while. In fact, people know how to go up to infinity and define "strict ω-categories"; see for example:

4) S. E. Crans, On combinatorial models for higher dimensional homotopies, Ph.D. thesis, University of Utrecht, Utrecht, 1991.

Strict n-categories are quite interesting and important, but I'm mainly mentioning them here to emphasize that they are not what I'm talking about. People sometimes often call strict n-categories simply "n-categories", and call the more general n-categories I'm talking about above "weak n-categories". However, I think the weak n-categories will will eventually be called simply "n-categories", because they are far more interesting and important than the strict ones. Anyway, that's what I'm doing here.

Secondly, when you define n-categories you have to make some choice about the "shapes" of your j-morphisms. In general they should be some j-dimensional things, but they could be simplices, or cubes, or other shapes. In some ways the simplest shapes are "globes", a j-dimensional globe being a j-dimensional ball with its boundary divided into two hemispheres, the "inface" and "outface", which are themselves (j-1)-dimensional globes. This corresponds to a picture where each "process" has one input and one output, which are themselves processes having the same input and output. The definitions of category, bicategory, and tricategory work this way. In fact, Ross Street came up with a very nice definition of n-categories for all n using simplices in 1987:

5) Ross Street, The algebra of oriented simplexes, Jour. Pure Appl. Alg. 49 (1987), 283-335.

Since then, however, he and his students and collaborators seem to have been working to translate this definition into the "globular" formalism... while also making some other important adjustments too technical to discuss here. In particular, Dominic Verity and Todd Trimble have done a lot of work on getting the definition of n-category worked out, and a while ago I learned that Trimble came up with a definition of "tetracategory" (or what I'm calling simply "4-category") in August of 1995. I don't think this has been published, however.

James Dolan came to U. C. Riverside in the fall of 1993, and ever since then, he and have been talking about n-categories and their role in physics. Most of the category theory I know, I learned in this process. It soon became clear that we needed a nice definition of n-category for all n in order to turn our hopes and dreams into theorems. After a while we started working pretty hard on this. His job was to come up with all the bright ideas, and mine was to get him to explain them, to try to poke holes in them, and to figure out rigorous proofs of all the things that were so obvious to him that he couldn't figure out how (or why) to prove them. We sent a summarized version of our definition to Ross Street at the end of 1995: