I am always trying to figure out what n-categories have to do with topology and physics, especially quantum gravity. I am also always trying to explain this, in many different ways, at different levels. I first got excited about this stuff when I learned about the category of tangles, so that's probably a good place to start learning about it... it's really beautiful stuff.
I'm not going to be very formal, so anyone who wants the rigorous details should take a look at these references:
I will just say that a tangle is a bunch of strands connecting n points on the ceiling to m points on the floor, possibly with a bunch of knots thrown in the middle:
| | \ / \ / \ /\ / \ / \ / \ / \ / \ \ / / \ | \ / \ | | / \ /\ / / \ \ \ / | / \/ \ / | \____/ | | | |
For us two tangles will be the same (technically, "isotopic") if one can be deformed into the other; i.e., we think of the strands as being infinitely flexible and are allowed to wiggle them around but not move them over the ceiling or under the floor; we aren't allowed to move the places where the strands touch the ceiling and floor, though.
Note that knots, links and braids are all special cases of tangles. Tangles are great because the provide a nice algebraic structure to study all of these things.
We say the above tangle is in Hom(2,2) because there are 2 points on the ceiling and 2 on the floor. Here is an element of Hom(2,4):
| | \ / \ / \ /\ / \ / \ | \ / \ | \ | | / \ |
and here is an element of Hom(4,0):
| | | | \ / \ /\ / \/ \ \ / \/ \ \____/
Note that we can "compose" these tangles to get one in Hom(2,0):
| | \ / \ / \ /\ / \ / \ / \ / \ | \ \ | / \ | | | | | \ / \ /\ / \/ \ \ / \/ \ \____/
In Hom(n,n) there is an "identity" tangle which is just a bunch of vertical strands:
| | | | | | | | | | | | | | | | | |
and if you compose any tangle T with the identity on the right or left you get T back again. This, together with the associativity of composition, is all we mean by saying that tangles form a category. (If you'd feel happier seeing a formal definition of "categories", you might want to click here.)
But we can also take the "tensor product" of two tangles, by setting them side by side. The tensor product of
| | \ / \ / \ /\ / \ / \ | \ / \ | \ | | / \ |
and
| | | | \ / \ /\ / \/ \ \ / \/ \ \____/
is
| | | | | | \ / \ / \ /\ / \ / \/ \ \ \ /\ / \/ \ / \ / \ \____/ | \ / \ | \ | | / \ |
Notice that the tensor product is associative and that composition and tensor product satisfy certain obvious identities (just like the identities that hold for tensor product and composition of linear maps between vector spaces).
By now the physicists must be wondering how mathematicians get paid to play around with this sort of thing. I will try to head off such rude remarks by noting that another name for tangles would be "Feynman diagrams". Of course Feynman diagrams are labeled tangles - the strands carry spin, momentum, and other quantum numbers. Also Feynman diagrams have vertices, which tangles don't. Also Feynman diagrams don't really live in 3-dimensional space. Nonetheless, there is a real relationship. For starters, if we were going to talk about labeled tangles with vertices we would be working with a generalization that has been studied by Reshetikhin and Turaev in their paper:
In physics these are also called `spin networks', and they play an important role in quantum gravity:
Anyway, here's the question of the day: how do we describe the category of tangles. Well, Turaev and Yetter both showed that it can be described by generators and relations almost like a group can. The generators are as follows. First, the identity 1 in Hom(1,1):
| | | | |
Second, the basic "right-handed crossing" r in Hom(2,2):
\ / \ / / / \ / \
and the left-handed crossing r^{-1} in Hom(2,2):
\ / \ / \ / \ / \
(Note that rr^{-1} and r^{-1}r are both the identity in Hom(2,2), so the names are appropriate.)
Third, the "cup" in Hom(2,0):
\ / \ / \ / \/
and the "cap" in Hom(0,2):
/\ / \ / \ / \
These are really primordial things! In what sense do they "generate" the category of tangles? Well, any tangle can be formed from these guys by taking tensor products and composites. For example, this guy
| | \ / \ / \ /\ / \ / \ | \ / \ | / | | / \ |
can be written as
(1 x r x 1)(r^{-1} x cap)
where I'm using x for tensor product and juxtaposition for composition.
Okay, so those are the generators - if you don't believe me, prove it! What are the relations? This is the cool part. Of course, there are a bunch of relations that just come from the properties of the tensor product and composition. Tensor product and composition satisfy these relations in any "monoidal category", but what we want are the relations special to the category of tangles. They were figured out by Turaev and Yetter, who actually came up with slightly different, but equivalent, sets of relations. As I like them, they are as follows... I'll draw them rather than write them as formulae.
| /\ | | /\ | / \ | | / \ \ / \ | \ / \ \ / | | \ / | \ | = | = / | / \ | | / \ | / \ / | / \ / | \ / | | \ / | \/ | | \/ | | |It says you can straighten out a 360-degree twist. If we dropped this rule, we'd be dealing with "framed tangles", which are more like ribbons than pieces of string. (No Möbius strips allowed, though!)
\ / | | \ / \ / | | \ / / | | \ / \ | | / \ / \ = | | = / \ \ / | | \ / \ / | | \ / \ | | / / \ | | / \ / \ | | / \Algebraically, it just says that the inverse braiding r^{-1} lives up to its name.
| / | | \ | \ / | | \ / \ | | \ / \ | | / \ | \ / \ / | | \ / \ / | | \ = \ | | / \ / \ | | / \ / \ | \ / | / \ / \ | | \ / \ | | / \ | \ | | / | | \ | | / |It's also called the "Yang-Baxter equation".
| /\ /\ | | | / \ / \ | | | / \ / \ | | \ / | = | \ / = | \ / | | \ / | \/ | | \/ |
| \ / \ / | | \ / \ / | | \ / | | / \ / \ | | / \ = / \ | \ / | | \ / \ / | | \ / \/ | | \/ | \ / \ / | | \ / \ / | | / \ | | / \ / \ | | / \ = / \ | \ / | | \ / \ / | | \ / \/ | | \/ /\ | | /\ / \ | | / \ / \ | | / \ | \ / \ / | | \ / \ / | | / = \ | | / \ / \ | | / \ / \ | /\ | | /\ / \ | | / \ / \ | | / \ | \ / \ / | | \ / \ / | | \ = / | | / \ / \ | | / \ / \ |
To see the power of these identities (together with the rules satisfied by tensor product and composition), use them to deduce this equation:
| /\ /\ | | / \ / \ | \ / \ / \ / \ / | | \ / \ | = | \ / \ | | / \ / \ / \ / \ | \ / \ / | | \/ \/ | | |
There are endless fun and games to be had with these rules, which encode a lot of the topology of 3-dimensional space. Try this book for some of this fun:
If you read this paper of mine, you can get a feeling for why I think algebraic descriptions of knots, tangles, spin networks and so on are so important in physics:
You may wonder where the identities for the category of tangles "come from" - in other words, why they are exactly what they are! James Dolan and I have a theory about this which we call the "tangle hypothesis". For more details, try reading these references: