David wrote:

I’m not sure we should be asking Tom to summarise his own paper…

No, but nonetheless he should do it if we ask him.

I remember during a discussion of my book on the Foundations of Mathematics list, which was sparked off by someone reporting John’s review, feeling a wee bit annoyed being asked to summarise what I’d written by a person who felt he didn’t have time to read it.

Well, it can be annoying when someone asks you to summarize something you wrote, but you should always do it when asked in a *public* forum.

Why? Because it gives *lots* of people a chance to learn what you’ve done - not just the lazy bum who asked the question. It also gives you a chance to look magnanimous. That’s why I was always willing to explain my work to people when they asked me to on sci.physics.research. That’s also the principle I was trying to exploit by getting Tom to explain his work now.

But, it turns out Toby was the one who broke down and typed in a definition of the Euler characteristic of a category, so everyone can see it without leaving this blog. Thanks, Toby! That’s a crucial step, because having the definition “sitting there for all to see” is the best way to get mathematicians to start poking at it and trying to prove stuff about it. If you make them leave the room and read a paper, they may get distracted and not come back.

(Our discussion is occuring in a kind of “room” here, albeit a virtual one, and for some reason opening a PDF file counts as leaving that room. I’m not sure why, but I know it’s true, and one has to just accept it.)

Anyway: now that Toby has shown us Tom’s definition, and Urs and Eric have raised an analogy with graph Laplacians, ideas start to bubble and brew…

For example, I just realized that Tom’s weightings and coweightings are functions from vertices of a directed graph to numbers. Our graph happens to be a category, but that’s not necessary for the definition to make sense.

And, the equations Toby wrote down are reminiscent of Poisson’s equation for a graph Laplacian.

Normally a graph Laplacian is defined for undirected graphs (I think). Say we have a function that maps vertices of our graph to numbers:
$a \mapsto k_a$
Then applying the graph Laplacian gives a new function of this sort, namely:
$a \mapsto k_a - \sum_b \zeta^a_b k_b$
where $\zeta^a_b$ is the number of edges joining the vertices $a$ and $b$.

In other words, if we call the graph Laplacian $\Delta$, we have

$(\Delta k)_a = k_a - \sum_b \zeta^a_b k_b$

You can stick some extra numbers in this definition if you want:

$(\Delta k)_a = \alpha k_a - \beta \sum_b \zeta^a_b k_b$

and you often do. For example, if your graph is an infinite square grid, you should take $\alpha = 1$ and $\beta = 1/4$, so the Laplacian takes a function of vertices and creates a new function by subtracting 1/4 times the sum of the nearest neighbor’s values. This is good because then

$\Delta k = 0$

holds when $k$ is constant.

Practical people - electrical engineers who use “finite element methods” - know all about this stuff. That’s one reason Eric got interested in this business, I guess - he was doing electrical engineering or something like that, back
before he decided to make vast sums of money in LA.

Anyway, Poisson’s equation is where you fix a function $f$ and look for functions $k$ such that

$\Delta k = f$

If we take $f = 1$, this says

$\alpha k_a - \beta \sum_b \zeta^a_b k_b = 1$

If we take $\alpha = 0$ and $\beta = -1$, we get Tom’s equation

$\sum_b \zeta^a_b k_b = 1$

But, Tom goes ahead and looks at *two* equations,

$\sum_b \zeta^a_b k^b = 1$

and

$\sum_b \zeta^a_b k_a = 1$

for two different functions $k_a$ and $k^b$, the weighting and coweighting.
And, if both these equations have solutions, Tom shows

$\sum_a k_a = \sum_b k^b$

and this number is the Euler characteristic of the category whose underlying graph I’ve been discussing.

So, some funny stuff is going on - but there is *some* relation to graph Laplacians. And we *know* that Laplacians show up when we compute the Euler characteristic of a manifold using the heat equation on that manifold.

So, it’s vaguely possible that Tom is secretly generalizing the usual way of computing the Euler characteristic of a manifold, by replacing the manifold with a directed graph, and modifying everything accordingly.

Okay - now someone figure out the details.

## Re: Euler Characteristic of a Category

David wrote:

No, not that I know of. However, Dan Christensen has been telling me some wonderful things about homotopy cardinality versus Euler characteristic. I’m too tired to retell these things now, but anyone curious about the general idea should look here:

Lots of links… and I’ll have to add one to Tom’s new paper!

Does anyone have the energy to

summarizeTom’s new paper? This is the place to do it!Tom?