The Wiz said, "Okay, maybe today I'll finally get around to defining a topological quantum field theory, or TQFT. I've told you a TQFT is a functor, which basically says that sticking spacetimes together like this:
| x v | / \ | f | \_/ | y v | / \ | g | \_/ | z v |corresponds to composing linear operators. That's the 1-dimensional aspect of a TQFT. And I've told you that a TQFT is a monoidal functor, which basically says that sticking together spacetimes like this:
| | x v x' v | | / \ / \ | f | | f'| \_/ \_/ | | y v y' v | |corresponds to tensoring linear operators. That's the 2-dimensional aspect of a TQFT. And now comes...." Here he paused, waiting for someone to make the obvious guess.
"The 3-dimensional aspect?" asked Oz, nervously expecting a trap of some sort.
"Right! And what's that?"
Toby's eyes glinted. "A TQFT is a braided monoidal functor."
"Exactly! The third dimension lets us switch things. A braided monoidal category is simply a context where we can both set things side by side and talk about the process of switching them! We saw an example in Week 3. Given two vector spaces V and W, we defined the braiding
BV,W: V (x) W -> W (x) Vto be the linear map that switches vectors:
BV,W(v (x) w) = w (x) v.We drew it like this:
| | | | V v v W \ / \ / / / \ / \ W v v V | | | |and drew its inverse like this:
| | | | W v v V \ / \ / \ / \ / \ V v v W | | | |They satisfy a few obvious rules... and by a process of abstraction, these rules become the definition of a braided monoidal category.
First let me give the definition of a strict braided monoidal category. At this point, it becomes a real pain to write down the laws without using our diagrammatic notation, so I'll use that!
A strict braided monoidal category is a strict monoidal category C equipped with:
1) for every pair of objects u and v, an isomorphism Bu,v: u (x) v -> v (x) u, called the "braiding".such that these laws hold:
Now I bet you're going to ask...."
for every object u and morphism f: x -> y we have| | | | u v v x u v v x | | | | | / \ \ / | | f | \ / | \ / / | | / \ | v y / \ | | = | | \ / x v | \ / | | / / \ | / \ | f | | / \ \ / | | | | | y v v u y v v u | | | |and| | | | x v v u x v v u | | | | / \ | \ / | f | | \ / \ / | / | | / \ y v | / \ | | = | | \ / | v x \ / | | / | / \ / \ | | f | / \ | \ / | | | | y v v u y v v y | | | |2) "the triangle equations":
for any objects x, y and z we have| || | | | x v vv y (x) z x v y v v z | || | | | \ // \ / / \ // \ / / \ // / / // = / \ / // \ / \ / // \ / / // \ / / \ || | | | | y (x) z vv v x y v z v v x || | | | |and|| | | | | x (x) y vv v z x v y v v z || | | | | \\ / \ \ / \\ / \ \ / \\ / \ / \\ \ / \ / \\ \ / \ / \\ / \ / \\ / \ \ | || | | | z v vv x (x) y z v v x v y | || | | |
"Why are those called the triangle equations?" asked Jay.
"And what do they mean, exactly?" added Oz. "Why did you draw two strands really close together in those last pictures?"
"Luckily," said the Wiz, "I can answer all 3 questions simultaneously. Since the two triangle equations are similar, I'll only talk about the first one. This one says it doesn't matter whether we move an object past two others in two separate steps, or all at once. In other words, it says this triangular diagram commutes:
Bx, y (x) z x (x) y (x) z ---------------> y (x) z (x) x \ ^ \ / \ / Bx,y (x) 1z \ / 1y (x) Bx,z \ / \ / \ / v / y (x) x (x) zThat's why it's called a triangle equation! When we write it like this:
| || | | | x v vv y (x) z x v y v v z | || | | | \ // \ / / \ // \ / / \ // / / // = / \ / // \ / \ / // \ / / // \ / / \ || | | | | y (x) z vv v x y v z v v x || | | | |we draw the strands for y and z close to each other when we're treating y (x) z as a single object and braiding x past this object in one single step. We draw them far apart when we're doing it in two steps. Of course, the equation says it doesn't really matter how we draw it!
Last week I said that monoidal categories come in two flavors, "strict" and "weak", depending on whether we impose the laws for the tensor product as equations or, more wisely, as isomorphisms. Braided monoidal categories also come in these two flavors. I just defined the strict ones. Similarly, a weak braided monoidal category is a weak monoidal category with a braiding satisfying all the above laws... except that the triangle equations become hexagons!
You see, in a weak monoidal category, we have to use the associator to reparenthesize the triple tensor products we're playing around with here. So the triangle equations I wrote down turn into hexagons, like this:
Bx, yz x(yz) ---------------> (yz)x ^ | ax,y,z | | ay,z,x | v (xy)z y(zx) | ^ Bx,y (x) 1z | | 1y (x) Bx,z v | ay,x,z (yx)z ----------------> y(xz)and
Bxy, z (xy)z ---------------> z(xy) | ^ ax,y,z | | az,x,y v | x(yz) (zx)y | ^ 1x (x) By,z | | Bx,z (x) 1y v | ay,x,z x(zy) <---------------- (xz)yHere I've suppressed tensor product symbols between objects, to reduce clutter."
"Hey! That looks like a rectangle, not a hexagon!" said Oz.
The Wizard sighed. "I know, I know... but in this darned ASCII environment, there are limits to how nice these pictures can look. Someday all this will be written up in LaTeX. Then it will be beautiful." He looked wistfully out the window.
"Not that it really matters", he continued, turning back to the class. "Anyway: if you wander around looking for braided monoidal categories, you'll find lots. For example, nCob is a braided monoidal category where the braiding looks like this:
____ ____ / \ / \ \ S /\ | T \ \__/ \ / \_____/ \ \ / / \ \ / / \ / / \ / ` / \ / ` / / ` / / ` / / ` / \ / ` / \ / / \ /... / \ ...\ / ` / \/ \ | T `/ \ S / \_____/ \__/Also, Hilb is a braided monoidal category where the braiding is the linear map that switches vectors in a tensor product
BH,H' (h (x) h') = h' (x) h.If you stare closely at the definitions, you'll see that most of these braided monoidal categories are weak, not strict. Luckily, Mac Lane proved that weak monoidal categories are all equivalent (in a certain precise sense) to strict ones. So often we can get away with using the strict ones... and that's what we'll do for now.
A braided monoidal functors is just one that preserves the tensor product and braiding. More precisely:
If C and D are strict braided monoidal categories, a strict braided monoidal functor F: C -> D is a monoidal functor such that for all objects x and y in C, F(Bx,y) = BF(x), F(y).
As Toby noted, a TQFT is, among other things, a braided monoidal functor from nCob to Hilb!
Okay, so we've done the 3-dimensional aspect of a TQFT. Next comes...?"
"The 4-dimensional aspect?" said Oz.
"Right! And what's that?"
Toby said, "A TQFT is a symmetric monoidal functor."
"Exactly," said the Wiz. "First of all, we say a braided monoidal category is symmetric if for all objects x and y,
Bx,y = By,x-1In pictures, this says:
| | | | x v v y x v v y | | | | \ / \ / \ / \ / / = \ / \ / \ / \ / \ | | | | y v v x x v v y | | | |In other words, it doesn't matter which way we switch one object past another: over, or under! This rule makes sense when the pictures are in 4 dimensions. Both nCob and Hilb satisfy this rule, so they're symmetric monoidal categories. So is the category of sets with the Cartesian product as its product, and the obvious braiding.
A symmetric monoidal functor is just a braided monoidal functor between symmetric monoidal categories. There are no extra laws that need to hold, this time.
Great. So much for the 4-dimensional aspect of a TQFT. Next comes...?"
"The 5-dimensional aspect!" said Oz, confidently.
"No!" shouted the Wiz. "There is no 5-dimensional aspect!" And he hurled a fireball straight at Oz:
It did not skim over Oz's head this time, nor did it merely singe his ear -- it hit him straight-on, with effects too devastating to describe here.
"Ha!" said the Wiz. "Fools 'em every time. It's a fascinating fact about category theory that this hierarchy stabilizes at dimension 4:
dimension 1 -- category dimension 2 -- monoidal category dimension 3 -- braided monoidal category dimension 4 -- symmetric monoidal category dimension 5 -- symmetric monoidal category dimension 6 -- symmetric monoidal category . . . . . .There's a lot more to say about what this means, but it would be too much of a digression right now... since we've finally reached our goal!"
"Our goal?" groggily asked Oz, who was lying flat on the floor, still recovering from that last fireball.
"Yes, our goal! I can finally tell you the complete definition of a TQFT! Here it is:
An n-dimensional TQFT is a symmetric monoidal functor from nCob to Hilb.
That's all! But as I hinted, the nicest TQFTs are the "unitary" ones. Here's how we define those:
A unitary n-dimensional TQFT is a symmetric monoidal *-functor from nCob to Hilb.
Okay, let's take a break. Next time we'll classify all 1-dimensional TQFTs.
That should give you more of a feeling for what these definitions amount
to. After that we'll classify 2d TQFTs, and then we'll begin the long march
towards constructing 3d and 4d TQFTs. And ultimately... 4d quantum gravity!
For more details on the definition of "symmetric monoidal functor" and Mac Lane's theorem, click here!