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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?

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The Legendre transform is the tropical version of the Laplace transform (golem.ph.utexas.edu/category/2006/11/…). –  David Corfield Dec 1 '09 at 14:31
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Perhaps you know that over the years we've had many discussions about matrix mechanics at the Cafe. Depending on the rig (ring without negatives) used, you end up with a different form of mechanics - classical, quantum, statistical - where you multiply along paths and the sum over paths. We even took a look at homotopy theory as a truth-valued mechanics, e.g., for path connectedness between two points sum (in this case OR, so does there exist) paths for which the product (in this case AND) of truth values for points on the path being in the space. I'd hoped we'd do something interesting transforming the different mechanics using rig morphisms. So we'd know, say, that if a classical particle could move from x to y, that x and y were path connected.

In your question, you're looking beyond matrices corresponding to a single space to rig-valued kernels on two spaces X and Y. In the case of the Legendre transform the choice is for related spaces, e.g., a vector space and its dual. This transform is often a good way to form a corresponding (possibly easier to solve) dual optimisation problem. Interesting, though, to think of more general pairs of spaces.

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