Categorifying Fractional Euler Characteristics
Posted by John Baez
Mathematicians should know how to count. Most of us do. But still there are some mysteries left when we try to count things in a way that gives a negative, fractional, irrational or complex answer.
Luckily, we’ve been making lots of progress. The Euler characteristic of a space can be a negative integer. The cardinality of a groupoid can be a nonnegative real number. The Euler characteristic of a category can even be a negative real number! And here’s yet another approach:
- Igor Frenkel, Catharina Stroppel, Joshua Sussan, Categorifying fractional Euler characteristics, the Jones-Wenzl projector and 3j-symbols.
The fun starts around page 17, when they look for a kind of entity whose Euler characteristic can be a function like this:
$\frac{1}{1 + q^2}$
Their first trick is to expand it as a power series:
$\frac{1}{1 + q^2} = 1 - q^2 + q^4 - \cdots$
Then they use a standard trick in this game, namely to define the ‘$q$-dimension’ of a graded vector space in such a way that the $q$-dimension of a $d$-dimensional space of grade $k$ is $d q^k$. This allows them to define the Euler characteristic of a chain complex of graded vector spaces: it’s the alternating sum of their $q$-dimensions.
So, suppose we have a chain complex of graded vector spaces
$V_0 \leftarrow V_1 \leftarrow V_2 \leftarrow \cdots$
where $V_k$ is 1-dimensional and of grade $2k$. Then its Euler characteristic is
$1 - q^2 + q^4 - \cdots$
as desired!
This is just the beginning of the game; it quickly gets a lot more deep. But this is nice to see.
(By the way, I have tried to give their paper the title it deserves, not the one it has.)
Re: Categorifying Fractional Euler Characteristics
This is the part that drives me crazy — not because I don’t like it, but because I don’t feel like I understand it fully. This is clearly a great trick, but it always seems totally ad hoc. Is there some abstract context that it comes out of?
In particular, there is a general definition of the dimension/Euler-characteristic of a dualizable object in a symmetric monoidal category: the trace of its identity map. I would really like to know whether this sort of ‘dimension’ be realized in such a way.
Perhaps there is an answer to this in the paper; I didn’t see it, but I only read the small portion that looked like it would be comprehensible to me. (-: