Ah I was wondering about that. Their formula look suspiciously like the definition of Renyi entropy.

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.

Metric spaces are enriched categories. They are enriched over the positive reals. The 'hom' between a pair of points is then simply a number: their distance.

And, these non-negative real numbers, which are these homs, 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’ “ ? Is that right?

So, I guess, (\R_{>= 0}, >=, +, 0) is like, a monoidal category with + as the monoidal operation?

So like, for x,y,z in the metric space, the

well, from hom(x,y) and hom(y,z) I guess the idea is there is a designated composition morphism

from hom(x,y) monoidalProduct hom(y,z) to hom(x,z)

which is specifically,

hom(x,y)+hom(y,z) >= hom(x,z)

(I said designated, but there is only the one, which is just the fact above.)

I.e. d(x,y)+d(y,z) >= d(x,z)

(Note: I didn’t manage to “just guess” this. I’ve seen it before, and was thinking it through as part of remembering how the idea worked. I am commenting this to both check my understanding in case I’m wrong, and to (assuming I’m remembering the idea correctly) provide an elaboration on what you said for anyone who might want more detail.)

> 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 < b` with its "witnesses"). See https://www.emis.de/journals/TAC/reprints/articles/1/tr1.pdf for more.

Indeed they are. I'm saying it may not be the right context in this case.

At least what they seem to be doing has little to do with metrics, and a lot more to do with probability distributions.

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

Why not define ecological diversity as number of distinct biological species living in the area?

This is precisely the question answered by the OP. The answer is, "because there is a whole spectrum of things you might mean by 'diversity', of which 'number of distinct species' is only one extremum".

And also, I assume, because the concept of "species" isn't all that well defined?

It is well defined: a group of living organisms consisting of similar individuals capable of exchanging genes or interbreeding

Ring species make your definition non-transitive. The same with species that can interbreed but exhibit hybrid breakdown.

I invite you to examine the notes of the international ornithological congress... The difference between species and subspecies is quite subtle, and subject to interpretation, because no one is really going to do the experiment to find out if two individuals of geographically district populations can actually still interbreed.

So if you have a few grams of soil and want to know how many species of micro organisms are in there, you're setting them up with dates to see which ones will end up breeding?

One of a number of definitions. It is one that allows lions and tigers to be the same species.

Does he provide an example of other definition of diversity that makes sense in biological context?

Yes, the two extremes are captured by the common metrics of "species richness" which is the pure "how many unique species are there", and "species evenness", which depends on how evenly distributed the species are. A community in which 99% of individuals are species A and the remaining 1% are from species B-G is exactly as species rich as a community in which there are equal numbers of individuals of each species, but it is much less even (and therefore, under one extreme of diversity, less diverse). In different contexts and for different ecological questions, these two different versions of diversity can matter more or less, and there are metrics which take both into account, but this is a fully generalized solution which shows you relative diversity along the entire spectrum from "all I care about is richness" to "all I care about is evenness".

-edit- by the way, since it may not be obvious to everyone, the reason why an ecologist might care bout evenness is because extremely rare species are often not very important to the wider community. From an ecological function perspective, there is very little difference between my above example of the 99%/1% community and a community that is 100% species A. So an community with two, equally populous species might have more functional diversity than a community with one very abundant species and several more, very rare species.