M Theory Lesson 13

John Baez's TWF244 is really cool. He talks about a recent paper by Leinster on the Euler characteristic of a finite category. Consider the following. Order the objects in the category $1,2,3,\cdots ,n$. The integral adjacency matrix for the category sets $a_{ij}$ to be the number of arrows from $i$ to $j$.

For example, in the category

we would have a 3x3 matrix with all entries 1, since we must not forget the identity arrows. If the inverse to this matrix $A$ existed, the Euler characteristic would be the sum of entries in $A^{-1}$. Let's fill in a few more arrows. Imagine there were $k$ arrows from $1$ to $2$, $2$ to $3$ and $1$ to $3$ which forms a basic composition triangle. Similarly, imagine the dual triangle had $m$ arrows at each edge. Then the adjacency matrix would be a circulant based on $1,k$ and $m$. Such triangles look like idempotent equations $k^2=k$, but here $k$ is an ordinal and we should wonder about composing one of the $k$ arrows from $1$ to $2$ with another one from $2$ to $3$, because the number of such compositions would naively be more than $k$. If $k^2=k$ were true, and $k$ was ordinal, then it must be zero or one, which gives a particularly simple kind of category otherwise known as a poset.

Anyway, to cut a long story short, this characteristic works nicely for all sorts of things, such as orbifolds. In M Theory we counted the number of particle generations using an orbifold Euler characteristic, which might be a rational number in general. So we can think of this as a cardinality of a category! This is wonderful, because the physical result follows from the universality of $\chi$.

Moerdijk looked at Lie groupoids as a foundation for orbifolds, which seems like a logical thing to do from the Symmetry point of view. But remember that we encountered the orbifold Euler characteristic in the work of Mulase et al on ribbon matrix models.