Osman wrote:

I would like to have experts’ opinions regarding this attempt of rational
combinatorics (i.e. categorification of rational numbers).

Hey, thanks for pointing out this paper:

It’s interesting… especially since it refers heavily to stuff I did!

It begins with an intro to categorification.

Then it describes how James Dolan and I categorified the nonnegative rational numbers using groupoids in our paper From finite sets to Feynman diagrams, back around 2000. The idea is to define the cardinality of a finite groupoid, which can be any nonnegative rational number.

(This is precisely the stuff we’re developing further now with the help of Todd Trimble — the stuff I’m starting to talk about in The Tale of Groupoidification!)

Next, the paper develops a theory of **groupoid-valued species**, namely functors

$F: FinSet_0 \to Gpd$

from the groupoid of finite sets and bijections to the groupoid of groupoids. I’ll have to check, but I’m afraid they’re considering *strict* functors instead of what they should be using: weak 2-functors. By an old result of Grothendieck, a weak 2-functor

$F: FinSet_0 \to Gpd$

is essentially the same as a functor

$p: X \to FinSet_0$

where $X$ is a groupoid. James Dolan and I call these things **stuff types**. We proposed these as a generalization of Joyal’s species, for the exact same reason Blandin and Diaz use groupoid-valued species. But, stuff types are slightly more general, and their formal properties are better.

Blanding and Diaz also make the mistake of limiting attention to groupoid-valued species with $F(\emptyset) = \emptyset$. One of the joys of stuff types is that this is no longer necessary!

It’s quite possible they didn’t realize that stuff types were almost the same as groupoid-valued species, only better. This is explained in Jeff Morton’s paper, and my course notes.

But never mind. The really new and interesting thing in Blandin and Diaz’s paper — at first glance — is their attempt to categorify the *whole* ring of rational numbers.

They categorify $\mathbb{Q}$ using *pairs* of groupoids — or if you prefer, $\mathbb{Z}_2$-graded groupoids. This works in the obvious way: the cardinality of a pair $(G_0,G_1)$ is the cardinality of $G_0$ minus the cardinality of $G_1$, so it can be any rational number.

So, they define a **rational species** to be a functor

$F: FinSet_0 \to Gpd^2$

satisfying a couple of conditions… where again, they should really leave out these conditions and let $F$ be a weak 2-functor.

This idea is clearly related to the older concept of ‘virtual species’, mentioned earlier in this discussion by Todd Trimble.

A rational species has a ‘generating function’, which can be any power series

$\sum_n a_n z^n$

with rational coefficients $a_n$.
And, the really fun part is that they get rational species whose generating functions have $a_n$ being the Bernoulli numbers!!!

I’d been trying to do that for a while.

However, nowadays I’m more inclined to think of a categorified rational number as a category — not a pair of groupoids. After all, Tom Leinster’s generalization of groupoid cardinality to categories can be negative!

Getting negative cardinalities by taking a pair of things and subtracting their cardinalities is more artificial than what Tom does: namely, *discover the right concept of cardinality of a category, and notice that it can be negative!*

## Re: This Week’s Finds in Mathematical Physics (Week 252)

How many?

The distances between stars are quite large compared to the size of solar systems. On the other hand, there are many stars, and the number of collisions probably goes as the number of stars squared.

Can we estimate the number of collisions? What is the probability the solar system will be involved in such an event? Is our position within the galaxy lucky or unlucky in this respect?