An invariant from category theory solves a problem in mathematical ecology [pdf]

An invariant from category theory solves a problem in mathematical ecology [pdf]
maths.ed.ac.uk

"The starting point is the connection between diversity and entropy. We will discover:

• how Shannon entropy, originally defined for communications engineering, can also be understood through biological diversity (Chapter 2);

• how deformations of Shannon entropy express a spectrum of viewpoints on the meaning of biodiversity (Chapter 4);

• how these deformations provably provide the only reasonable abundance-based measures of diversity (Chapter 7);

• how to derive such results from characterization theorems for the power means, of which we prove several, some new (Chapters 5 and 9).

Complementing the classical techniques of these proofs is a large-scale categorical programme, which has produced both new mathematics and new measures of diversity now used in scientific applications. For example, we will find: [...]"

"The question of how to quantify diversity is far more mathematically profound than is generally appreciated. This book makes the case that the theory of diversity measurement is fertile soil for new mathematics, just as much as the neighbouring but far more thoroughly worked field of information theory"

The presentation gives us a whole range of views on biodiversity, making it clear that we need to think about both common and rare species when we're figuring out diversity. It hints that this magnitude idea might give us a more detailed picture of diversity by considering how important different species are relative to each other. Then the presentation gets into the weeds of category theory, chatting about enriched categories and size-like invariants. It talks about stuff like monoidal categories, V-enriched categories, and linear categories. These ideas are brought up as tools to help us understand and calculate magnitude.

The presentation also explores how these category theory ideas relate to metric spaces. It suggests that getting a handle on this relationship could give us even more insight into how to max out diversity. Lastly, the presentation brings up the Euler characteristic, which is a concept from algebraic topology. It hints that the Euler characteristic and magnitude are pretty tight, and understanding this connection could give us even more insight into how to max out diversity.

So, to wrap it up, the presentation is a thorough look at how category theory ideas, especially magnitude, can be used to tackle problems in mathematical ecology. It suggests that these ideas can give us a more detailed understanding of diversity and might even help us figure out how to get the most diversity.

References:

https://arxiv.org/abs/2012.02113

https://arxiv.org/abs/1606.00095

I'm not too sure where the category theoretical stuff enters though. They mention that metric spaces have a magnitude, but their end result looks more like a channel capacity (with the confusion matrix being the probability to confuse one species with another). Which, you know, makes sense, if you've got 'N' signals but they're so easily confused with one another that you can only send 'n' signals worth of data then your channels are not too diverse.

They do mention that this is equivalent to some modified version of the category theoretical heuristic, but is that really interesting? The link to Euler characteristic is intriguing, but from the way they end up at their final definition I'm not sure if metric spaces are really the natural context to talk about these things. It almost feels like they've stepped over an enriched category that would provide a more natural fit.