Take the tour ×  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