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

I found the paper easier to follow than the slides: https://www.maths.ed.ac.uk/~tl/mdiss.pdf

(Less emphasis of the category theory, and more attention to the basic math behind entropy-like diversity measures)

(TLDR) There are two important aspects:

1. Generalize diversity measures to when the categories are not fully distinct (as assumed for the Shannon entropy calculation) but have similarities parametrized in a "Z"-matrix here.

2. A parameter "q" to represent whether you value a category highly (for diversity purposes) even when it has only a single distinct example highly (q --> 0) or whether you value it highly only when it has many many examples (q --> infinity)

* With Z = identity matrix (categories completely distinct) they show how different values of q reproduce different measures that have been considered before. (nicely summarized in a table)

* The generalization (parameterized by Z) when the categories can/do have some overlap is very elegant, and feels like an important step forward. (especially how it makes the measure robust to how we partition a bunch of examples into distinct categories, so long as we keep track of the similarity between the categories)

* The paper also summarizes a bunch of sensible properties that we would want any diversity measure to satisfy (like the one I just mentioned above).

Fun stuff!

> are “hom objects”, so regarded as objects in “the category with as objects the non-negative real numbers, and as morphisms, the ‘being greater than or equal to’ “ ?

This works, but it's not quite what you want in most cases. There's a lot of stuff that requires you to enrich over a closed category, so instead we define `Hom(a,b)` to be `max(b - a, 0)` (which you can very roughly think of as replacing the mere proposition `a https://www.emis.de/journals/TAC/reprints/articles/1/tr1.pdf for more.

This is the most confusing possible title. Why is it written this way?

Wow. The fact that there is an objective answer that is independent of any perspective on the importance of rare species is a rare gift, at least for this part of the problem.

Some questions and thoughts.

It seems that the result could vary based on how you construct the similarity matrix Z, e.g. is it purely taxonomic? or does it try to account for the ecological roles that a species is playing in the community, etc.

A seeming limitation is that the optimization works only for a fixed set of n species. While it is useful for managing existing communities, it means that there is still a question of whether larger n is strictly better, and leaves open questions of how to deal with transient or migratory members (if the community is spatially bound).

The answer I think, is that it depends on how the similarity matrix is constructed. If every species is fully dissimilar then increasing n is always a good thing. If you use niche space to construct it and new species do not some add or enter new niches so they overlap with others, then they will be close to another species in the matrix and increasing n will not have much impact. On the other hand if you use a purely taxonomic approach then you wind up balancing the number of birds and mammals regardless of niche.

It is not clear to me whether it is possible to construct a similarity matrix that can account for the interaction between n, the carrying capacity of the ecosystem, and the number of available niches (or the ability of species to create new niches). By analogy if you have a stream (sunlight) powering water wheels, how many wheels and how many levels of gears (layers in the ecosystem) can be added, created, and/or sustained? At what point does adding an additional species mean that either two species are forced to be close together in the similarity matrix or both their populations must shrink in size because they must compete for the same energy sources?

Does the model sometimes produce impractical results, e.g. that it is good to have a single member of a sexually reproducing species (this is probably an orthogonal concern and you would want to scale to real population sizes such that the minimum corresponded to the smallest viable a self sustaining population)?

Is there evidence that maximizing diversity using this measure actually produces more robust and stable ecologies?

It's not clear what you're seeking. Probabilities appear because the magnitude of a space is a way of 'measuring' it -- and thus magnitude is closely related to entropy. Of course, you can follow your nose and find your way beyond mere spaces, and this may lead you to the notion of 'magnitude homology' [1]. But it's not clear that this generalization is the best way to introduce the idea of magnitude to ecology.

[1] https://arxiv.org/abs/1711.00802