Is there a connection between tropical mathematics and the Lawvere enriched category theory approach to metric spaces? I guess I will give a partial answer to this below, but I mean can they be formally be put on the same level in some sense?
In the Lawverian point of view one does category theory with the extended non-negative real numbers, [0,∞] or R≥0∪∞, equipped with + as the 'tensor' product and max as the 'categorical' product or sum. In tropical mathematics you work (it seems) with the the extended reals R∪∞ equipped with the 'product' + and the 'sum' max (or min depending on your point of view I think).
In the enriched category theory approach to metric spaces, one has the notion of a kernel (or bimodule or profunctor depending on your point of view) between two metric spaces X and Y which is just a distance non-increasing function K:X×Y->[0,∞]. The correct notion of function on a metric space here is a distance non-increasing function φ:X->[0,∞]. Then the transform of a function φ by a kernel K is a function on Y defined by
K(φ)(y):= infxεX ( φ(x) + K(x,y) ).
There is similarly a dual notion which takes functions on Y to functions on X.
K^(ψ)(x):= supyεY ( ψ(y) - K(x,y) ).
This is explained in a bit more detail in a post in at the n-Category Café.
It was pointed out to me that these look similar to the Legendre transform. And looking on the internet I found that tropical mathematics is one way to interpret the Legendre transform as an 'integral transform'.
So has anyone ever considered any formal connections between these two points of view?