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):= inf_{xεX}( φ(x) +K(x,y) ).

There is similarly a dual notion which takes functions on *Y* to functions on *X*.

K^{^}(ψ)(x):= sup_{yεY}( 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?