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.
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.
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.)