The n-Category Café

Hmm, I don’t know about that. Does your category satisfy the hypotheses of the theorem in section 5 of my talk? (The slides aren’t currently accessible online, as our IT staff are doing some maintenance work, but they should reappear soon.)

On another eventual-image point, I’ve just discovered (courtesy of Vidit Nanda) a highly relevant paper:

Marian Mrozek, Normal functors and retractors in categories of endomorphisms. Universitatis Iagellonicae Acta Mathematica 29 (1992), 181–198.

It’s detailed enough that I’m not immediately able to grasp the main point, but clearly it has a great deal in common with my work on this. I should spend some time with it. Maybe it would also be useful for your purposes, Rod?

Thanks very much. I don’t know whether most “public” talks are like this, but what was predicted to happen at mine and what actually did happen was that many of the audience had some university connection. Quite a few colleagues from my department came, some maths undergraduates, some students from computer science, some people from ecology, and so on. I don’t know what the overall balance was, but my impression was that at most half of them were genuine public, if you see what I mean.

As for entropy: for readers too lazy to click, I was using the quantity

$$D(p) = 1/p_1^{p_1} p_2^{p_2} \cdots p_n^{p_n},$$

where $p = (p_1, \ldots, p_n)$ is a probability distribution. This is the exponential of Shannon entropy,

$$H(p) = -\sum_{i = 1}^n p_i \log(p_i).$$

Whereas $D(p)$ ranges between $1$ and $n$, the genuine Shannon entropy $H(p)$ ranges between $0$ and $\log(n)$. (Here “exponential” and “log” can be to any base you like; it doesn’t matter.)

A reason to preferr $D(p)$ over $H(p)$ is that it’s an effective number. This means that if you have a population of $n$ equally abundant species, $p = (1/n, \ldots, 1/n)$, then the quantity assigned to it is $n$. If you’re trying to measure diversity (and this is what $D$ stands for), that’s important.

To explain why, I’ll steal an example from Lou Jost. Suppose a continent contains a million species of plant, all equally abundant. A meteor strikes and wipes out 99% of the species entirely, leaving the other 1% untouched. The diversity $D(p)$ drops by 99%, as you’d expect. But the Shannon entropy drops by only 33%.

So percentage changes of Shannon entropy aren’t useful or meaningful, because it’s not an effective number. But when you do have an effective number, such as $D$, you can reasonably interpret a conclusion such as “for this population, $D(p) = 12.2$” to mean “our community is about as diverse as a population of 12 equally abundant species”, or more briefly “we’ve effectively got 12 species”.