We seem to still get a lot of google searches for this post. Jacob has an expository article out now that does a much better job of addressing this material than my liveblogging. You should read that paper instead.
Jacob Lurie is in town giving two topology talks. The first one is on classifying 2-d extended TQFT (a topic near and dear to my heart), and the second is a more leisurely introduction to extended 2-dimensional TQFTs . As is often the case when Jacob is in town, the room is rather packed.
At the moment I’m liveblogging the second talk, for the first talk go past the flip.
Jacobs 2nd talk is starting now, and since Peter Teichner just described it as “the talk where you start from the beginning” I’m going to try to continue the liveblogging, and hopefully it’ll make the earlier talk make more sense.
In this talk, Jacob is describing his joint work with Mike Hopkins on extended TQFT inspired by Kevin Costello’s papers.
Jacob starts off recalling Atiyah’s celebrated definition that an n-dimensional TQFT is a tensor functor from nCob to complex vector spaces. The “functor” part here means that gluing cobordisms corresponds to composition of linear maps. The “tensor” part says that 
.
Then he recalls the well-known result that 2-dimensional TQFTs are classified by Frobenius algebras. To see this, you first consider the vector space assigned to a circle. Then a pair of pants gives a multiplication on this space, and a disc gives a trace. Using the relations between cobordisms you can see that these algebraic structures fit together to make a Frobenius algebra.
The moral of this story is that we should understand n-dimensional TQFT you want to understand it on some simple pieces, and then take your manifold and chop it up into those simple pieces. This is nice, but unfortunately you can’t chop things too finely. You aren’t allowed to chop it up in ways that have corners. This suggests another definition.
Definition: An extended TQFT (in dimension n) is a rule
- closed n-manifold –> complex number
- closed (n-1)-manifold –> vector space
- bordism of (n-1)-manfiold –> map of vector spaces
- closed (n-2)-manifold –> linear category
- bordism (n-2)-manifold –> linear functor
- …
The “…” is not intended to mean that it is easy to keep going, only that you’re meant to try. But since we’re only talking about low-dimensional topology and “here low means 
” we don’t really need to understand the “…”.
This definition can be summarized as “An extended TQFT is a functor between n-categories.”
At this point there’s a bit of a digression in which Rob Kirby wants to know why we should think about this hard problem of what an n-category is when we don’t have any examples in dimensions above 3. Jacob says “I’m the wrong man to ask, I only understand what’s going on in dimension less than 2.”
After that digression he moves on to describe the Baez-Dolan Cobordism Hypothesis (paraphrased by Jacob): “Extended TQFTs are “easy to describe/construct.” Elaborating a bit further he says that you only need to describe the TQFT on very small building blocks, and then n-category theory will do all the work for you. Rather than making the conjecture more precise he’s going to give examples where the conjecture is known to hold.
(non-)example (n=2): We restrict our attention to a smaller category where we only allow certain bordisms allowed by string topology based on some manifold M. To a circle we assign the homology of the loop space on M. To a pair of pants we assign the Chas-Sullivan product on homology. (To a disc we don’t get anything, since that’s a bordism that isn’t allowed.)
But rather than just assigning homology, we’d rather assign the chain complex itself. Unfortunately given a bordism you only get a chain homotopy between the corresponding complexes. Nonetheless we can cook up out of this more operations on F(circle) associated to higher homology of Bord(M,N).
A better way to restate this is that Bord(M,N) = Map(F(M), F(N)) where the latter space of chain complexes is thought of as a topological space. So our TQFT here is actually a functor of 
-categories! That is the 2-morphism spaces aren’t just a set, they’re actually topological spaces, and the functor respects this topological structure.
Now we get down to the question at hand. Define the monoidal -category 2Bord defined by
- The objects of 2Bord are oriented (compact) 0-manifolds
- The morphisms of 2Bord are bordisms between 0-manifolds
- The space of 2-morphisms from f to g is the classifying space of bordisms from f to g which are trivial on their boundary
We want to classify tensor functors from this -category to other -categories. By Baez-Dolan we should expect this question to have an easy answer: all we need to know is where a point goes!
A point corresponds to some object C. The point with the opposite orientation corresponds to a dual to C (using a line segment as the map), so we need to require that C be dualizable. Then we can figure out where a circle goes just by making the circle out of two segments. So the circle goes to the “dimension” of C, which is an element of End(C).
This is already enough to classify 1-dimensional extended TQFTs! Exciting. Now we need to figure out how to promote 1-dim extended TQFTs to 2-dimensional ones.
So where is a disc going to go? Well, it must land in 
. Using the circle action on dim C (given by the circle action on the circle) we know that the disc lands in the circle fixed points of .
The punchline is that this is all that you need to know. The only data is a dualizable object and a circle fixed point in . You may need to check lots of relations, but you don’t need any more data than that.
This fact allows Jacob to give a quick description of string topology, and a proof that it is homotopy invariant. Since I don’t understand string topology, I’ll stop here.
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