[contravariant]

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.

[autopoiesis]

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.

[drdeca]

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.)

[consilient]

> 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.

[contravariant]

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.

[autopoiesis]

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