  This is a survey paper on the connection of enriched category theory over a
quantale and tropical mathematics. Quantales or complete idempotent semirings,
as well as matrices with coefficients in them, are fundamental objects in both
fields. We first explain standard category-theoretic constructions on matrices,
namely composition, right extension, right lifting and the Isbell hull. Along
the way, we review known reformulations (due to Elliott and Willerton) of
tropical polytopes, directed tight spans and the Legendre--Fenchel transform by
means of these constructions, illustrating their ubiquity in tropical
mathematics and related fields. We then consider complete semimodules over a
quantale $\mathcal{Q}$, a tropical analogue of vector spaces over a field, and
mention Stubbe's result identifying them with skeletal and complete
$\mathcal{Q}$-categories. With the aim to bridge a gap between enriched
category theory and tropical mathematics, we assume no knowledge in either
field.
